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City size distributions: foundations of analysis Mulligan, Gordon Fredrick 1972

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CITY SIZE DISTRIBUTIONS: FOUNDATIONS OF ANALYSIS by GORDON FREDRICK MULLIGAN B . S c , Univers i ty of B r i t i s h Columbia, I969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of GEOGRAPHY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1972 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the 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 study. I f u r t h e r agree 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 copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of 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 not 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 . Department of GKO£^/1 /> s4 y The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date V/7 2-ABSTRACT While many observers recognize the s ignif icance of the c i t y size d i s t r i b u t i o n t o p i c , the r e s o l u t i o n of several apparent inconsistencies i n the body of l i t e r a t u r e has not yet been achieved. This may explain why geographers, s o c i o l o g i s t s , demographers, h i s t o r i a n s , economists, and planners e s s e n t i a l l y tend to describe i n t e r c i t y patterns, are biased toward ad hoc interpreta t ions , and are prone to making i n t u i t i v e statements i n t h e i r research. The primary purpose of t h i s thesis i s to evolve a more consistent methodological v i e w p o i n t wi thin the community size t o p i c . E f f o r t s are made to unite a n a l y t i c a l statements res t ing upon a common premise, to q u a l i f y , i n t h i s l i g h t , the approaches prevalent i n empirical research, and to relate theory and empiricism by adopting a f l e x i b l e explanatory framework. The discussion necessari ly involves a c r i t i q u e of exis t ing arguments and c e r t a i n extensions that, we can devise from those arguments. While there i s considerable attention directed to presenting empirical methodologies, no o r i g i n a l data analysis i s included. Contending that the notions should be bound "together within a systems framework, we natura l ly devote i n i t i a l emphasis to the features of central place systems i I i i a s o u t l i n e d i n t h e p a r t i a l e q u i l i b r i u m t h e o r y o f C h r i s t a l l e r (I966) and L o s c h (195 )^. We p l a c e p a r t i c u l a r s t r e s s u p o n t h e C h r i s t a l l e r m o d e l , t h e s i m p l e r and a p p a r e n t l y more r e a l i s t i c o f t h e two a p p r o a c h e s . A m a j o r t h r u s t o f t h e p a p e r i s a n i n t e g r a t i o n o f s e v e r a l c i t y s i z e m o d e l s , a l l o f w h i c h d i s p l a y a C h r i s t a l -l a r i a n h i e r a r c h y . The s i m p l e s t m o d e l s a r e shown t o be s p e c i a l c a s e s o f a more g e n e r a l f o r m u l a t i o n g i v e n b y Dacey (1966). B e s i d e s , we i l l u s t r a t e t o what d e g r e e t h e c h a r a c t e r i s t i c p r o p e r t y ( t h a t i s , t h e c o n s t a n t p r o p o r t i o n -a l i t y f a c t o r ) o f t h e most e l e m e n t a r y m o d e l (Beckmann, 1958) may be c o n s i d e r e d a l i m i t o f e m p i r i c a l g e n e r a l i z a t i o n . U s i n g t h e h i e r a r c h i a l c o n c e p t , we a l s o p r o v i d e some r a t h e r n o v e l v i e w s on t h e r e l a t i o n b e t w e e n community economic b a s e and t h e d i s t r i b u t i o n theme. I t i s f e l t t h a t t h i s s u b t o p i c may be u s e f u l i n b r i d g i n g t h e i n t r a -and i n t e r u r b a n s c a l e s . The w i d e l y expounded r a n k - s i z e r u l e , e s s e n t i a l l y a c o n s e q u e n c e o f e m p i r i c a l r e s e a r c h , i s t h e n f o r m a l l y a t t a c h e d t o t h e h i e r a r c h i a l m o d e l s . A t t h i s s t a g e o u r a r guments become i n c r e a s i n g l y r i g o r o u s i n o r d e r t o q u a l i f y c e r t a i n i n t u i t i v e n o t i o n s t h a t seem a c c e p t e d i n t h e l i t e r a t u r e . The i d e a o f h i e r a r c h i a l s e t s i s c r u d e l y d e v e l o p e d t o complement t h e u n i - h i e r a r c h y a r g u m e n t s . The b a s i c c o n c l u s i o n h e r e i s t h a t e x i s t i n g c i t y s i z e m o d e l s h a r d l y e x p l a i n t h e r a n k - s i z e phenomenon b u t t h a t t h e two n o t i o n s c a n n o t be c o n s i d e r e d t o t a l l y i n c o m p a t i b l e . i * • 11 Empirical research methodologies are stressed as another fundamental subtopic. We suggest c e r t a i n avenues along which empirical e f f o r t s must be strengthened before ei ther (i) rigorous inductive generalizations or ( i i ) f i rm theory substantiation become more r e a l i z a b l e . P a r t i c u l a r at tention i s given to del imita t ion of the study area (and, therefore to the scale problem), the comparison of frequency curves, and the value of inferences we can make using rather crude s t a t i s t i c a l t o o l s . At t h i s stage we introduce other skew d i s t r i b u t i o n s that are g e n e t i c a l l y s i m i l a r to the rank-size curve. Furthermore, the stochastic models that seemingly account for these d i s t r i b u t i o n s are taken to complement the deterministic theory mentioned above. Here we support the central place argument as the only exis t ing source of models that explicate those factors inducing s p a t i a l d i f f e r e n t i a t i o n of economic a c t i v i t i e s and, as a consequence, urban populations. F i n a l l y , we pursue the idea of growth within the interurban structure. At t h i s time, however, discussion i s ce r ta in ly exploratory and so i s l i m i t e d to developing notions concerning the inter re la t ions of growth var iables (population, income, etc . ) and h i e r a r c h i a l structure i n the broadest sense. Within t h i s analytic framework we can suggest only the most general factors that may be associated with low degrees of primacy (a q u a l i t y of interurban s t ruc -ture that we view as a deviation from a charac ter is t i c skew d i s t r i b u t i o n ) . This par t i cular subtopic promises to be i v an exci t ing research theme i n i t s own r i g h t as investigators move from equilibrium to dynamic modelling. TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS v LIST OF TABLES v i i i LIST OF FIGURES i x Chapter 1. INTRODUCTION 1 2 . THE CENTRAL PLACE SYSTEM 7 A Review of Central Place Theory 7 Introductory Remarks 7 Case of the Single Good 8 Case of Many Goods 15 Differences Between C h r i s t a l l e r and Losch 22 Scope and Nature of the C l a s s i c a l Argument 26 Extensions of the C l a s s i c a l Li terature 29 Aggregate Relations and Elemental Components 30 H i e r a r c h i a l Structure 35 The Central Place System Reconsidered 37 3. CITY SIZE MODELS AND DISTRIBUTIONS 38 A Review of the Hierarchia l Models 38 Terms and Notation . 38 Model It The General Case 41 v v i Model I I J The Aggregate Approach 4 5 Model III« The Geometric M u l t i p l i e r 49 Model I V i The Constant M u l t i p l i e r 52 H i e r a r c h i a l Models and the Economic Base 55 H i e r a r c h i a l Models and the Rank-Size Rule 60 H i e r a r c h i a l S e t s and the Rank-Size Rule 73 4 . EMPIRICAL ANALYSIS AND INTERPRETATION 78 Background 78 The Study Area 80 C i t y S i z e Patterns» Skew D i s t r i b u t i o n s and R e l a t e d Concepts 89 The Rank-Size and Pa r e t o D i s t r i b u t i o n s 90 Steady S t a t e D i s t r i b u t i o n s 96 Comparison o f D i s t r i b u t i o n s 100 I n t e r p r e t a t i o n and E x p l a n a t i o n 103 S t o c h a s t i c Approaches 103 General Systems Theory 107 The Aggregate Model Recon s i d e r e d 113 5 . CHANGING PATTERNS OF INTERURBAN STRUCTURE 116 Growth i n a T h e o r e t i c a l Context 117 P o p u l a t i o n 117 Income 121 I n n o v a t i o n s 125 A B r i e f S y n t h e s i s 131 Graphing the Aggregate Model 132 6. CONCLUDING REMARKS BIBLIOGRAPHY LIST OF TABLES Table Page 1 . Service M u l t i p l i e r s and Basic/Non-Basic Ratios of Four Central Place Systems 57 2. Fundamental Properties of Midway C i t i e s i n Related Central Place Systems v i a Diverse Modelling Approaches 71 3 . Constant Rank-Size Products Given by Independent H i e r a r c h i a l Sets v i a Model II 76 v i i i LIST OF FIGURES Figure Page 1. Price and Output Conditions for the Individual Producer with no Competition and with Free Entry 11 ix Chapter 1 INTRODUCTION Concern over the quest i o n of community s i z e d i s t r i -b u t i o n i s widespread i n the geographical l i t e r a t u r e . In f a c t , i t i s a t o p i c that i n t r i g u e s s o c i a l s c i e n t i s t s i n many f i e l d s . The theme i s given impetus on the e m p i r i c a l side through J e f f e r s o n ' s ( 1 9 3 9 ) study of the primate c i t y and Z i p f 's (19^-9) account of rank-size r e g u l a r i t i e s . However, contemporary e f f o r t s on the t o p i c o f t e n feature a mixture of i n t u i t i o n , weak l o g i c , and r a t h e r loose s t a t i s t i c a l a n a l y s i s . Among the few accepted g e n e r a l i -z a t i o n s are those that primacy i s a s s o c i a t e d with over-u r b a n i z a t i o n , c o l o n i a l i s m , and underdevelopment while rank-size tendencies are as s o c i a t e d with the in t e r u r b a n ! i n t e g r a t i o n of economically advanced r e g i o n s . Perhaps the most serious recent e f f o r t s made to e x p l a i n c i t y s i z e d i s t r i b u t i o n s come from those f o l l o w i n g Beekmarm ( 1 9 5 8 ) who adhere to c e n t r a l place models and from others pursuing Simon (1955) who p r e f e r the s t o c h a s t i c argument. But with l i t t l e agreement on both t h e o r e t i c a l and e m p i r i c a l f r o n t s , and disparate approaches flavoured with i n c o n s i s t e n c i e s and redundancies, i t i s not s u r p r i s i n g that the t o p i c i s enveloped by an a i r of d i s s a t i s f a c t i o n . 2 From the standpoint of strengthening harmony among the diverse e f f o r t s , there i s alone s u f f i c i e n t reason to attempt a rigorous review of existing contributions. Besides, the tenor of present argument i n the f i e l d of regional development and planning i s that a much sounder knowledge of the relationships among urbanization, economic growth, and c i t y size arrangements i s decidedly needed. Friedmann ( I 9 6 6 ) , for example, emphasizes that l i t t l e i s understood about the substructures of the space-economy and the influence of spatial a c t i v i t y patterns upon regional growth. Hopefully, then, t h i s study w i l l show prac t i c a l benefits as well as satisfying personal c u r i o s i t i e s . In this thesis, we analyse the logic of existing theoretical and empirical statements about community size distributions and, when i n disagreement, present our counter-arguments. With th i s i n mind we attempt to resolve some of the apparent differences between the deterministic and proba b i l i s t i c interpretations that support (to some extent) the rank-size principle. Also, attention i s devoted to r e l a t i n g seemingly independent geographical concepts (for example, economic base and diffusion) to the discussion of c i t y sizes. The purposes of the thesis are c l e a r l y twofold: (i) To examine and attempt to refine ( i n e x p l i c i t fashion) the existing methodology of the general problem area 1 and 3 ( i i ) To o f f e r new ideas w i t h i n the s p e c i f i c subtopics and to extend notions that bond the general problem area to the growing body of geographical l i t e r a t u r e and theory. As the chapters are devised to be somewhat independent, a concise sketch of the e n t i r e study seems appropriate a t t h i s time. We are f i r s t concerned with p r e s e n t i n g a comprehensive review of c e n t r a l place theory as developed by C h r i s t a l l e r ( I 9 6 6 ) , Losch ( 1 9 5 ^ ) i and l a t e r students. The review i s e s s e n t i a l i n that i t e l u c i d a t e s the drawbacks of the theory and the s i g n i f i c a n t d i f f e r e n c e s between C h r i s t a l l e r i a n and Loschian fundamentals, both of which are needed to r e a l i z e the domain of e x i s t i n g h i e r a r c h i a l models. S p e c i a l emphasis i s placed on i d e n t i f y i n g the q u a l i t i e s of h i e r a r c h i a l s t r u c t u r e w i t h i n a set of i n t e r r e l a t e d communities. The f o l l o w i n g chapter i s the most r i g o r o u s of the t h e s i s . Here, we d i r e c t a t t e n t i o n to r e l a t i n g the v a r i o u s h i e r a r c h i a l models i n e x p l i c i t f a s h i o n . Furthermore, we employ the notions of the c e n t r a l place models to l i n k i n t e r - and i n t r a u r b a n s c a l e s v i a the concept of economic base. The remainder of the chapter i s given to i n t r o d u c i n g the rank-size r e g u l a r i t y (the most c h a r a c t e r i s t i c concern i n the c i t y s i z e d i s c u s s i o n ) w i t h i n the c e n t r a l place frame-work. Some mathematical arguments d i s p l a y the nature of the a s s o c i a t i o n between the h i e r a r c h i a l and u n i - s i z e c l a s s ideas, while q u a l i f y i n g any p r e s e n t l y accepted statements 4 t h a t are seen to be i n v a l i d . The t h r u s t of t h i s d i s c u s s i o n i s a demonstration that the r a n k - s i z e r u l e and the e x i s t i n g C h r i s t a l l e r models are probably, but not n e c e s s a r i l y , incompatible and t h a t the chances of coincidence may r i s e when other independent systems are i n c l u d e d as w e l l . The methodology of e m p i r i c a l r e s e a r c h i s n o t i c e a b l y weak with regard to the study of i n t e r u r b a n s t r u c t u r e . Observers c o n s i s t e n t l y f a i l to give care and thought to the e f f e c t s of ( i ) a r b i t r a r i l y d e f i n i n g study areas, ( i i ) blending d i f f e r e n t means of comparing frequency d i s t r i b u t i o n s , and ( i i i ) i m p r e c i s e l y e v o l v i n g s t a t i s t i c a l analyses. Much of the next chapter i s devoted to questions l i k e these i n hope that we may become i n c r e a s i n g l y aware of the value of i n f e r e n c e s made from e m p i r i c a l study with improved methodology. On the other hand, the l a t t e r p o r t i o n of t h i s s e c t i o n shows how d e t e r m i n i s t i c and s t o c h a s t i c i n t e r p r e t a t i o n s of the skewed frequency d i s t r i b u t i o n s are not n e c e s s a r i l y i n o p p o s i t i o n . The f i f t h chapter completes a c i r c u i t with the second, i n that i t b u i l d s upon the ideas of the i n t e r v e n i n g d i s c u s s i o n but a l s o concerns simple micro-economic reasoning. I t s primary purpose i s to examine w i t h i n an assumptive framework how growth f a c t o r s a f f e c t i n t e r c i t y s t r u c t u r e . Besides, the e f f e c t s of s t r u c t u r e upon growth are suggested w i t h i n the fundamentals of item d i f f u s i o n . No attempt i s made toward developing a flow chart or feedback model, 5 even of the simplest k i n d ; rather , e f f o r t s at t h i s stage are t o t a l l y directed to displaying impact tendencies alone. A charac ter is t i c feature of the thesis i s the adherence to a systems framework for studying the i n t e r -re la t ions of population c lusters i n a s p a t i a l s e t t i n g . I t seems absolutely necessary to evoke t h i s framework when t r y i n g to integrate the various facets of the l i t e r a t u r e into a more meaningful whole. Being aware that no r e a l world system i l l u s t r a t e s the precise q u a l i t i e s of the central place system, hierarchial notions may, of course, be somewhat relaxed (see Marshal l ; I969). The geographical l i t e r a t u r e i s replete with systems thinking but only recently do we f i n d the concepts formally appl ied . To be b r i e f , the his tory of systems thinking i s t i e d up with funct ional and ecological approaches, the organismic analogy, and the idea of regional synthesis (Harvey, 1969). E x p l i c i t to the d e f i n i t i o n of a system i s that we are concerned not only with a sum of elements whose at tr ibutes are directed by causal laws, but by a sum of re la t ions among those units and some environment. The c r i t i c a l point , then, i s that a system possesses propert ies , functions, or purposes that are d i s t i n c t from i t s constituent objects, re la t ionships , and attr ibutes (Hall and Fagen, 1956). In our immediate study there i s some intent of complementing ex is t ing l i n e s of argument with simple fundamentals of general systems theory. I n any case, the systems framework i s e s p e c i a l l y f l e x i b l e with regard to our l e v e l of a b s t r a c t i o n and serves as a reminder of the ever-present s c a l e problem. When we t a l k of r e g i o n a l c i t y systems as opposed to n a t i o n a l c i t y systems, the value of a c o n s i s t e n t approach should c r y s t a l l i z e . Before c l o s i n g t h i s i n t r o d u c t i o n , we must comment b r i e f l y on the most troublesome aspect of the c i t y s i z e t o p i c ; t h a t i s , the q u e s t i o n of "explanation" per se. On the one hand, we have an, ever-improving e q u i l i b r i u m theory d e a l i n g with f u n c t i o n a l a l l o c a t i o n s i n space, but whose domain i s t y p i c a l l y r e s t r i c t e d to a c t i v i t i e s where input p r i c e s vary l i t t l e over d i s t a n c e . Most e m p i r i c a l s t u d i e s , however, concern a domain much gr e a t e r than t h i s and include centers of s p e c i a l s i t e and s i t u a t i o n f e a t u r e s . In a d d i t i o n , we have an a p r i o r i s t o c h a s t i c model that e s s e n t i a l l y avoids the s p a t i a l dimension. I t i s argued here that despite the f a c t the s o - c a l l e d entropy approach may describe a l a r g e r domain, i t f a i l s to s a t i s f y our c u r i o u s i t y to the same degree as the c e n t r a l place approach does. The increased a t t e n t i o n we give to the c e n t r a l place scheme, combined with our growing awareness of what the theory l a c k s , promises to be the best route f o r s u i t a b l e explanation i n the f u t u r e . Chapter 2 THE CENTRAL PLACE SYSTEM A Review of Central Place Theory Our discussion of centra l place theory pursues a synthesis of the fundamental contributions of C h r i s t a l l e r , Losch, and more recent advocates of the subject . We plan to e f f e c t i v e l y defend the notion of a central place system, while developing a strong framework for t rea t ing the topic of c i t y size models. Only i n t h i s l i g h t may the methodology of theory extension be properly understood. Introductory Remarks The route to comprehension of the s p a t i a l economic systems of C h r i s t a l l e r (I966) and Losch (1938, 1952*) i s through the independent study of single goods or services . While t h e i r i n i t i a l assumptions are not e n t i r e l y i d e n t i c a l , we can nevertheless isolate four general postulates that appear ei ther i m p l i c i t l y or e x p l i c i t l y commons (i) A homogeneous p l a i n with uniform r u r a l densit ies ; ( i i ) A system of f . o . b . p r i c i n g ; ( i i i ) Equal demand by a l l consumers (consuming units) at any r e a l p r i c e ; (iv) Free entry of producers into the market. 7 8 A c lear interpreta t ion of these p r i c i n g r e s t r i c t i o n s i s v i t a l to analysis i n terms of cost and demand fac tors . F . o . b . p r i c i n g i s simply the case where the consumer pays the price for a good at the production s i t e (the f . o . b . price) plus the cost of transportation to his l o c a t i o n (the t o t a l being the r e a l p r i c e ) . Such a p o l i c y seems suitable for firms dealing with ( i ) large numbers of customers and ( i i ) goods and services whose distance decay ( spat ia l e l a s t i c i t y of demand) i s high. Losch puts forward his argument i n a more rigorous manner, while inc luding settlement geography as only a portion of the general l o c a t i o n problem. By presenting his case within the confines of formal economic theory he attaches a strong theore t i ca l tone to the settlement p r i n c i p l e s of C h r i s t a l l e r . Case of the Single Good L e t ' s imagine the world as defined by the assump-tions of C h r i s t a l l e r and Losch. F i r s t we consider an i n d i v i d u a l good or service that i s offered at site "0" on the p l a i n . The desires of a consumer res iding at the production s i te are indicated by the usual convex downward-sloping demand curve that intersects both the price and the quantity axes. Since demand " q " i s a continuous function of the f . o . b . p r i c e , we may consider how demand changes for d i s t i n c t f . o . b . l eve ls "p^" i n the in terval p m i n £ P A f P m a x (where " P m i n " represents 9 the price at which a consumer at " 0 " w i l l purchase a maximum quantity of the good and " P m a x " i s that price where the same consumer w i l l purchase a zero quant i ty ) . L e t ' s consider now an i d e n t i c a l consumer who resides "x" units distant from the production point . This customer must pay an addi t ional "xt" (where "t" i s the transport cost per uni t distance) to cover the movement of the commodity to his l o c a t i o n . In other words, demand " q " i s a continuous function of the r e a l price "p^ + xt" i n the general case. With t h i s knowledge we can determine the distance "r^" to the l a s t customer exerting ef fec t ive demand for the good or service supplied at " 0 " . This defines a market area of radius "r^" for any f . o . b . price "p^"? It should be apparent that l i n e a r demand i s , then, a function of marketing ( f . o . b . price "p^") and transport (cost "t") technologies. Using our f i r s t assumption, we may compute the areal demand facing the firm at " 0 " . By rota t ing the distance-demand response curve (for given "p^" and "t") about a v e r t i c a l axis through " 0 " we can trace out a demand curve for the t y p i c a l consumer. Now, when we mult iply the volume beneath the demand cone by a constant "D" representing population (consumer) density, the t o t a l demanded quantity " Q . " i n the area about " 0 " i s given i n 10 i n t e g r a l f o r m : 2fT/ r . Q. = D Y r i f ) J f ( p i + x t ) x dx dO . . . ( 2 . 2 ) I f t h i s c a l c u l a t i o n i s r e p e a t e d f o r a v a r i e t y o f f . o . b . p r i c e s ( i n the i n t e r v a l P m j _ n — P^ _ — pmax) we c a n d e r i v e d i f f e r e n t l e v e l s o f t o t a l demand "Q^" as t h e cones v a r y i n h e i g h t and r a d i i . When we p l o t the v a l u e s o f " p ^ " v e r s u s t h o s e o f " Q ^ " , an a g g r e g a t e demand c u r v e i s c o n -s t r u c t e d f o r the m a r k e t a r e a d e l i m i t e d by some r a d i u s " r "(where p.= p . ) . S i n c e we are i n f a c t d e a l i n g max v * i * m i n ' & w i t h an i n i t i a l p r o d u c e r and c o m p e t i t i o n i s a b s e n t , t h i s p a r t i c u l a r demand c u r v e i s termed the f r e e s p a t i a l demand c u r v e . A l t h o u g h i n the o r i g i n a l l i t e r a t u r e L o s c h r e p r e -s e n t s t h i s c u r v e as b e i n g concave t o the o r i g i n i t may be shown t h a t , w i t h our i n i t i a l p o s t u l a t e s , the demand c u r v e must be convex (Denike and P a r r , 1 9 7 0 ) . W i t h t h e aggregate demand c u r v e "D^" f a c i n g our i n i t i a l p r o d u c e r the n e x t s t e p i s t o d e t e r m i n e the p r o f i t m a x i m i z i n g p r i c e and o u t p u t r e l a t i v e t o the c u r v e . P r o d u c t i o n c o s t s are r e p r e s e n t e d by an average c o s t c u r v e "AC" t h a t d e s i g n a t e s the c o s t o f p r o d u c t i o n per u n i t o f o u t p u t , w h i l e a m a r g i n a l c o s t c u r v e "MC" shows the i n c r e m e n t s i n t o t a l c o s t as o u t p u t i s e x t e n d e d . L o s c h i l l u s t r a t e s the c o s t c u r v e s as f a l l i n g a t each o u t p u t 11 AC Q 2 Q, Demand/Output F i g . 1 . P r i c e a n d O u t p u t C o n d i t i o n s f o r t h e I n d i v i d u a l P r o d u c e r w i t h n o C o m p e t i t i o n a n d w i t h F r e e E n t r y ( f r o m P a r r a n d D e n i k e , 1 9 7 0 ) . l e v e l u n d e r m o n o p o l i s t i c c o n d i t i o n s , h u t t h i s d o e s n o t s e e m t o h i n d e r t h e g e n e r a l i t y o f h i s a r g u m e n t . M a r g i n a l r e v e n u e " M R " , o n t h e o t h e r h a n d , r e f e r s t o t h e i n c r e m e n t s o f r e v e n u e b r o u g h t i n t o t h e f i r m t h r o u g h s m a l l p r o d u c t i o n e x p a n s i o n s . I f we a s s u m e t h a t n o r m a l p r o f i t s ( i n c l u d i n g t h e r a t e o f r e t u r n t h a t c o u l d b e e a r n e d i n o t h e r i n v e s t m e n t s ) a r e p r e s e n t i n p r o d u c t i o n c o s t s , 12 then the p r o f i t maximizing price " p 1 " and output "Q 1" are determined by the intersec t ion of the marginal revenue and marginal cost curves. Losch and Berry (1967) argue that the price charged w i l l he determined by the in tersec t ion of the average cost curve and demand curve but t h i s i s c l e a r l y not a p r o f i t maximizing interpre ta t ion . This p a r t i c u l a r price l e v e l " P m £ n n (where P m ^ n V-^) allows the maximum number of customers to be provided from " 0 " and may wel l improve t o t a l revenuer unfortunately, these gains are more than offse t by climbing operation costs . The s i t u a t i o n changes somewhat when we consider free entry into production a c t i v i t y . The p o s s i b i l i t y of a t ta ining excess p r o f i t s encourages new entries into the market while disrupt ing the i n i t i a l equil ibrium s i t u a t i o n . New producers continue to enter u n t i l each can only earn normal p r o f i t s . Now the competitive demand curve " D 2 " facing the single i n i t i a l producer (and a l l new producers) i s shi f ted to the l e f t of " D ^ " . This i s the case because with unrestr ic ted entry the single o r i g i n a l f i rm loses customers along the edges of i t s i n i t i a l market area. The new output equilibrium i s indicated by the tangency of the new demand curve with the average cost curve. We not ice , too, that the demand curve s h i f t s even farther to the l e f t i f an excessive number of new producers enter the market. Now the demand curve l i e s below the average production cost at every output l e v e l and various 13 s e l l e r s axe forced out of business. Hence, the point of tangency indicates the minimum or threshold size of the f i r m . By observing F i g . 1, we see that with unrest r ic ted entry the equil ibrium output"Q 2 n I s lower and the equil ibrium price " p 2 " i s higher r e l a t i v e to the p r i o r monopolistic condit ions . This change i s explained by the f a l l i n g cost curve: as entry into the market continues, production at any one s i te i s l i m i t e d and prices r i s e as the opportunities for scale economies are l o s t . Two general condit ions , then, ar ise as a consequence of unlimited entryJ ( i ) the loss of excess p r o f i t s shows that the number of firms i s maximized and ( i i ) each producer seeks a loca t ion as distant as possible from his neighbours' i In the ideal case where a l l suppliers are equally spaced over the homogeneous p l a i n , a uniform t r iangular arrangement pers is t s , C h r i s t a l l e r and Losch argue that t h i s i s the most favourable s p a t i a l equil ibrium pattern and that , as a r e s u l t , a net of hexagonal market areas i s provided. The monopolistic state defines , i n C h r i s t a l l e r ' s terms, the ideal range of the good or service being offered. The new ideal range i d e n t i f i e d by the higher competitive f . o . b . price cannot be attained, however, since the extent of each s e l l e r ' s market i s r e s t r i c t e d by his adjacent competitors* market areas. The new boundary that del imits the competitive market area i s c a l l e d the r e a l range of the good. 14 Obviously t h i s r e a l range i s not equal i n a l l d i r e c t i o n s , (since i t defines the extent of i d e n t i c a l hexagonal c e l l s ) and for t h i s reason we define i t as being one-half the distance between adjacent producers of the same good. As Parr and Denike (1970) point out, what C h r i s t a l l e r terms the upper l i m i t on the range refers to e i ther the r e a l or the ideal form, depending on whether or not s p a t i a l competition e x i s t s . There also exis ts a minimum l i m i t on the range of a good which C h r i s t a l l e r c a l l s the lower l i m i t . Getis and Getis (1966i222) state that t h i s encloses " . . . the number of consumers necessary to provide the mimimum sales volume for the good to be produced and d i s t r i b u t e d p r o f i t a b l y . . . . " For the single o r i g i n a l producer t h i s threshold range i s equal i n a l l d i r e c t i o n s . With competition, however, the firm earns only normal p r o f i t s and only a minimum l e v e l of aggregate demand determines the threshold range. Now the lower l i m i t of the range i s coincident with the r e a l range and i s not equal i n a l l d i rec t ions . The fundamental contributions of C h r i s t a l l e r and Losch toward a general understanding of the single good pattern are roughly i d e n t i c a l . B a s i c a l l y , the former r e l i e s upon the concept of threshold range while the l a t t e r s t ipulates that the attainment of normal p r o f i t s i s paramount. Since the demand and cost factors underlying the two concepts are e s s e n t i a l l y the same, we usual ly 15 consider Losch's treatment as only a more e x p l i c i t or sophisticated approach to the same problem that confronted C h r i s t a l l e r . Case of Many Goods Both C h r i s t a l l e r and Losch develop schemes for integrat ing the features of the various s ingle good nets . We consider Losch's analysis f i r s t , since i t i s the more general of the two, and then go on to summarize C h r i s t a l l e r ' s ideas. Losch*s derivations rest upon a modif icat ion of our f i r s t postulate . He further assumes that the r u r a l pop-u l a t i o n i s discontinuously d is t r ibuted over the i s o t r o p i c p l a i n and that inhabitants reside i n basic settlement uni ts (farmsteads or hamlets) that are arranged on a uniform t r iangular l a t t i c e . Reasoning that the d u a l i t y of agriculture and industry i d e a l l y leads to t h i s punctiform d i s t r i b u t i o n (compromises between proximity to food and i n d u s t r i a l production, s u i t a b i l i t y to most aspects of a g r i c u l t u r a l production), he s t ipulates that these basic settlement units l i e at the centre of hexagonal farms. The s ignif icance of t h i s approach unfolds when he demon-atrates that , with t h i s discontinuous r u r a l stratum of population, ( i ) the possible sizes of the complementary areas for d i f f e r e n t goods and services and ( i i ) the number of basic settlement units these areas enclose, likewise grow discontinuously. 16 To i l l u s t r a t e t h i s condi t ion , a concept that i s fundamental to a l l central place discussions i s introduced. Losch formulates a method for determining the number of equivalent "basic settlements i n any market area. This number equals the sum of the following three: ( i ) the number of units (or preferably l a t t i c e points) i n t e r i o r to the c e l l , ( i i ) one-half the number of units on edges of the c e l l , and ( i i i ) one-third the number of uni ts at ver t ices of the c e l l (on a t r iangular l a t t i c e , that i s ) . Using t h i s concept, Losch derives the possible market area sizes i n terms of how many basic settlement uni ts are provided. In a s i m i l a r v e i n , s p a t i a l extent of the "nA market areas i s given by /3"» where " A " i s the area of the smallest hexagonal c e l l and " n " represents the number of equivalent settlements enclosed. The resul ts of th is r e s t r i c t i v e approach should be obvious: Losch i s arguing that minimum demand f o r commodities offered at various farmstead locations i s usual ly met by market sizes that o f f e r an unnecessarily large number of basic consuming u n i t s . The i n f l e x i b i l i t y of Losch's der ivat ion means that moderate surplus p r o f i t s cannot be eliminated by further entry and some producers are cer ta in to benef i t . He (1954:120) emphasizes further that " . . . not a l l possible market areas need occur i n r e a l i t y . , . but conversely, every actual market area must be on the l i s t of possible ones," 1? From his formulation of market s i z e s , Losch proceeds to discuss integrat ion of the d i f f e r e n t market nets . He combines them by ( i ) ensuring that each good has one common supply center (the metropolis) and ( i i ) r o t a t i n g the nets so as to y i e l d a cogwheel pattern of s ix sectors with few and s ix sectors with many production s i t e s . He (1954:124) states that : . . . with t h i s arrangement the greatest number of locat ions coincide , the maximum number of purchases can be made l o c a l l y , the sum of the minimum distances between i n d u s t r i a l locat ions i s l e a s t , and i n conse-quence not only shipments but also transport l i n e s are reduced to a minimum. E s s e n t i a l l y he i s applying r a t i o n a l agglomeration assumptions i n order to derive a related set of market nets i n hope of def ining some reasonable notion of an economic region. The underlying theme of his entire analyt ic argument i s , i n fac t , that t h i s derived arrangement i d e n t i f i e s the most orderly and s p a t i a l l y confined closed system of market areas. As Losch (1938 !75) points outs "How many of these s e l f - s u f f i c i e n t systems w i l l come into existence on our p l a i n depends merely upon the commodity which has the largest shipping radius , as long as there are no economic l i m i t s to the size of the central c i t y . " While the thrust of Losch*s approach involves the concept of regional integrat ion, a very important port ion of the discussion concerns the numbers of coincident settlement units at various points on the t r iangular l a t t i c e . Losch does not disc lose , however, his interpre-t a t i o n of the size of these aggregate settlement units 18 except through a l i s t i n g of the functions they provide. I t seems, though, that several in teres t ing properties arise when we superimpose the various market nets i n t h i s way: ( i ) Some l a t t i c e points possess more economic functions than others; hence there i s d i f f e r e n t i a t i o n among farmsteads, towns, e t c . ; ( i i ) Some l a t t i c e points possess the same number of functions but these functions may be d i f f e r e n t ; hence there i s s p e c i a l i z a t i o n among centers; ( i i i ) A l l l a t t i c e points possess at least one function but there are few with many funct ions ; hence, a numerical pyramid i n the number of multiple-good supply centers i s suggested. In c los ing o f f the Loschian case, we should emphasize that he consistently requires that only one producer of a given good i s located i n the center where that good i s offered. This charac ter is t i c l i m i t a t i o n i s based sole ly on the r a t i o n a l scheme used to combine the independent market area nets. C h r i s t a l l e r ' s approach to the multi-good system i s less general and we present a summary of his i n t e r -pretation i n a considerably more rigorous manner so as to avoid r e p e t i t i o n at a l a t e r time. Generally we might consider a large region i n which "y" d i f ferent goods and services are provided. Designating the f i r s t of these as " t^ B , we may rank these 19 c e n t r a l goods from " t 1 " to " t " i n ascending order of t h r e s h o l d need? a center o f f e r i n g " t then, r e q u i r e s the g r e a t e s t amount of consumer purchasing power f o r supply-to p e r s i s t i n the long run. We term such a place an "M" l e v e l center and, according to our i n t r o d u c t o r y p o s t u l a t e s , i t i s a s s o c i a t e d with the l a r g e s t complementary area on the homogeneous p l a i n . Of course, only as many "M" l e v e l centers emerge i n the r e g i o n as there are t h r e s h o l d markets a v a i l a b l e to support those firms o f f e r i n g " t Since these f i r m s compete s p a t i a l l y , production s i t e s become arranged so that supply i s most e f f i c i e n t l y sustained. In other words, by e n f o r c i n g an i m p l i c i t assumption that firms o f f e r i n g the good of minimum distance decay organize the s p a t i a l p a t t e r n of "M" l e v e l centers, a t r i a n g u l a r l a t t i c e develops on the p l a i n . The boundaries between the v a r i o u s "M" l e v e l places are determined by the r e a l range of " t " and form hexagon shaped market areas about each c e n t r a l place. I f t o t a l s a l e s l e v e l s are an exact m u l t i p l e of thresholds f o r good " t these firms earn only normal p r o f i t s (since they l o c a t e so as to minimize consumer movement). Excess p r o f i t s may be earned i f s a l e s i n the r e g i o n are s l i g h t l y greater than t h i s exact m u l t i p l e . As we noted e a r l i e r , the ranges f o r d i f f e r e n t goods and s e r v i c e s d e c l i n e with lower t h r e s h o l d r e q u i r e -ments; t h e r e f o r e , greater and greater numbers of surplus 2 0 consumers l i e between the t h r e s h o l d market areas of "M" l e v e l c e n t e r s f o r these same commodities. There may be some good " t i " f o r which the i n t e r s t i t i a l p u r c h a s i n g power reaches t h r e s h o l d volume i t s e l f . I n t h i s c a s e , a l t e r -nate c e n t e r s e v o l v e t o s u p p l y " t y _ ^ " (and a l l o t h e r goods and s e r v i c e s o f lower t h r e s h o l d need) a t p r i c e s below those a t the "M" l e v e l p l a c e s . These "M-l" l e v e l c e n t e r s s e r v i c e the areas between the t h r e s h o l d ranges o f those goods s u p p l i e d e x c l u s i v e l y from " M " l e v e l c e n t e r s . B e r r y and G a r r i s o n (1958d) c a l l " t ." a h i e r a r c h i a l m a r g i n a l good. A s i m i l a r argument c a l l s f o r the emergence o f 1 - 2 " l e v e l c e n t e r s where some commodity " t ( j i ) i s the new h i e r a r c h i a l m a r g i n a l good. These c e n t e r s s e r v i c e the areas between the t h r e s h o l d ranges of those goods s u p p l i e d o n l y from h i g h e r o r d e r c e n t e r s ( i . e . "M" and " M -l" l e v e l p l a c e s ) . L i k e w i s e , goods " t ." through t o " t - ^ " , are p r o v i d e d a t these lower o r d e r c e n t e r s . A c o n s i s t e n t p r o p e r t y o f the C h r i s t a l l e r scheme i s t h a t a c e n t e r o f a g i v e n order develops e q u i d i s t a n t from i t s n e i g h b o u r i n g c e n t e r s of the next h i g h e s t o r d e r . G e t i s and G e t i s ( 1 9 6 6:224) add t h a t " I n t h i s way, consumer movements are kept t o a minimum, and a maximum number o f demands are s a t i s f i e d from a mimimum number of c e n t e r s . " T h e r e f o r e , j u s t as i n the L o s c h i a n case, a l l c e n t r a l p l a c e s are l o c a t e d on a t r i a n g u l a r l a t t i c e . ^ ^Losch and C h r i s t a l l e r are aware o f d i f f e r e n t geomet-r i e s as w e l l as d i f f e r e n t market nets on the same l a t t i c e . 21 C e r t a i n fundamental c h a r a c t e r i s t i c s o f the s p a t i a l p a t t e r n we have o u t l i n e d seem t o e x i s t : ( i ) A l l c e n t e r s hut the s m a l l e s t have o t h e r c e n t e r s dependent upon them f o r the p r o v i s i o n o f goods and s e r v i c e s ? hence, the s e t of c e n t r a l p l a c e s d i s p l a y s i n t e r d e p e n d e n c y i ( i i ) Each c e n t r a l p l a c e o f f e r s a l l the goods and s e r v i c e s t h a t dependent c e n t e r s s u p p l y p l u s a d d i t i o n a l ones? hence, the c r i t e r i o n o f i n c r e m e n t a l b a s k e t s o f goods suggest t h a t these communities show d i s c r e t e s t r a t i f i c a t i o n o f c e n t r a l i t y i ( i i i ) While the s c a l e o f the p a t t e r n i s changeable, the i n t e r s t i t i a l placement o f o r d e r s i s a d i s t i n c t i v e f ormi ( i v ) There e x i s t s a d e f i n i t e n u m e r i c a l pyramid a c c o r d i n g t o the o r d e r s o f the c e n t e r s . B a s i c a l l y C h r i s t a l l e r i s f o r w a r d i n g a simple geometric argument i n which the market a r e a s i z e s i n c r e a s e i n e x t e n t by a f a c t o r "q". T h i s scheme c o n t r a s t s w i t h Losch's where t h e r e i s a c o n s i d e r a b l y smoother p r o g r e s s i o n o f p o s s i b l e market a r e a s i z e s . I f we f u r t h e r assume a d i s c o n t i n u o u s r u r a l p o p u l a t i o n , then i n a C h r i s t a l l e r q = 3 system (where "q" r e p r e s e n t s the n e s t i n g f a c t o r f o r market a r e a s ) , the p o s s i b l e market a r e a s i z e s i n terms of e q u i v a l e n t b a s i c s e t t l e m e n t s are 3 t 9 , 2 7 , 81, e t c . S i m i l a r l y , i f we denote the a r e a l e x t e n t o f the s m a l l e s t market as "A", then the m u l t i p l i e r "q zA" (z = 0, 1, 2 , , , . ) 22 represents the progression of a l l possible market s i z e s . Differences Between C h r i s t a l l e r and Losch Some of the basic differences between the two approaches have already been mentioned. The c r i t i c a l divergence between the schemes arises out of the d i f f e r e n t methods employed i n combining the market networks of i n d i v i d u a l goods. Losch considers f i r s t the commodity with the smallest market area and then introduces commodities with progressively larger threshold requirements. In other words, Losch's approach i s a n a l y t i c ! i t develops i n stages from the most general ideas of Chamberlinian economic theory (where the d i f f e r e n t i a t i o n of the producer's l o c a t i o n i s but one type of product d i f f e r e n t i a t i o n ) . C h r i s t a l l e r ' s case i s r e l a t i v e l y inductive as he argues from the most p a r t i c u l a r to the 2 most general . Since C h r i s t a l l e r begins with the most "national commodity" while Losch begins with the most " l o c a l commodity", von Boventer (1963:171) suggests: ""Central place theory has a l u c i d deductive structure for the general arguments proceed from a p r i o r i premises to statements concerning par t i cular instances (for example, the number of functions coincident at a cer ta in l a t t i c e point ) . Furthermore, C h r i s t a l l e r ' s interpretat ion r e a l l y assumes that a community system exists and that a p a r t i c u l a r community (the "M" l e v e l place) i s dominant therein, Losch, however, does not r e l y upon the f i r s t of these assumptions i n the same sense. As he envisages agglo-meration from the most general case while C h r i s t a l l e r approaches i t from the most p a r t i c u l a r , we f e e l the l a t t e r has an added grain of inductive reasoning. 23 In economic-historical terms, C h r i s t a l l e r ' s method of der iving his system may he thought of as descr ibing the population growth i n an area which at the beginning i s very t h i n l y populated. Losch's system would appear to be a more adequate descr ipt ion of a l a n d -scape i n which a cer ta in dense ground structure ex is ts , with, i n the beginning, e n t i r e l y s e l f - s u f f i c i e n t small s p a t i a l units ( i f new commodities with ever-increasing i n t e r n a l economies of production are introduced). I t i s s o l e l y th is difference i n the derivat ion of the systems which has the e f f e c t that Losch's system becomes much more complicated than C h r i s t a l l e r ' s . As a consequence of these opposite approaches we may i d e n t i f y numerous s i g n i f i c a n t differences between the two schemes: ( i ) The deviations from the optimal layout for the i n d i v i d u a l goods and services are smaller i n the Loschian system since a greater number of possible market area sizes exist} the idea that r e l a t i v e l y few market area sizes are permissible i n the C h r i s t a l l e r system provides the opportunity for i n i t i a l excess prof i ts? ( i i ) While the general geometric appearance (triangular l a t t i c e ) i s i d e n t i c a l for both, the s p a t i a l arrangement of centers i s at variance. The Loschian system has one extra degree of freedom l e f t a f ter the metropolis i s s p a t i a l l y f ixed (hence the c i t y - r i c h and the c i ty-poor sectors) while C h r i s t a l l e r * s system i s e n t i r e l y symmetrical? ( i i i ) Losch does not consider the addi t ional demands of a supplying population i n a central place nor does he include the possible effects of multi-purpose t r i p s . Concomitantly, ei ther we must expect s i g n i f i c a n t 24 v a r i a t i o n i n the sizes of hexagonal c e l l s for the same commodity (Isard, 1956s2?0-273) or, i n order to preserve geometric r e g u l a r i t i e s , we must compose new assumptions to eliminate such change i n the s p a t i a l demand function (von Boventer, I963 1 I 7 I - I 7 2 ) , C h r i s t a l l e r (1966:50-55) appears to i m p l i c i t l y include these features i n his scheme. Therefore, the physical extent of the market areas for most commodities i n a multi-good system w i l l be smaller i n the C h r i s t a l l e r case than i n the Loschian} (iv) The l e v e l of urban concentration also varies considerably between the two approaches. I f we suppose that the smallest market area size i s the same i n both, then we f i n d fewer central places (and therefore a greater concentration of urban population) i n the C h r i s t a l l e r formulation. Besides, th is l e v e l of urban concentration i s d i r e c t l y related to the value of the " q " factor i n the simpler system? (v) Losch takes better account of the p a r t i a l s p e c i a l i z a t i o n of production i n smaller centers? C h r i s t a l l e r 1 assumption that each higher order central place supplies a l l the commodities (plus some addit ional ones) i s a rather r e s t r i c t i v e one. In the r e a l world, smaller communities frequently supply larger places with special ized goods and services? (vi) Losch's approach i s more r e s t r i c t i v e i n considering entry of competitors at the same place. While the loca t ion of more than one producer of the same commodity 25 at the same c e n t r a l place runs c o n t r a r y to Loschian a n a l y s i s , Parr and Denike (1971:572) suggest t h a t f u r t h e r entry (at l e a s t i n the short run) may be taken as another q u a l i f i c a t i o n of the i n c o n s i s t e n c i e s r e l a t e d to i n ( i i i ) above. On the other hand, C h r i s t a l l e r ' s more i n d u c t i v e d e r i v a t i o n does allow f o r entry of competing producers where excess p r o f i t s may be gained; ( v i i ) C h r i s t a l l e r ' s incremental baskets o f goods suggest c e r t a i n n a t u r a l agglomerative tendencies ( i n t e r -i n d ustry) among f i r m s , but t h i s c o n d i t i o n i s not c h a r a c t e r -i s t i c of the Loschian landscape; ( v i i i ) A c l e a r l y i d e n t i f i a b l e h i e r a r c h y i s found i n the C h r i s t a l l e r scheme but not i n the Loschian ( i d e n t i -f i a b l e , t h a t i s , i n terms of orders and not i n d i v i d u a l f u n c t i o n s ) ; ( i x ) I t i s much more d i f f i c u l t to make i n f e r e n c e s about the popula t i o n s i z e s of centers i n the Loschian model; C h r i s t a l l e r ' s r i g i d h i e r a r c h y suggests that d i s c r e t e p o p u l a t i o n l e v e l s (that i s , d i f f e r e n t s i z e c l a s s e s ) may be assigned to centers o f d i f f e r e n t order. To make t h i s d i f f e r e n c e even more apparent, we can r e l a x the assumption of uniform purchasing power i n the C h r i s t a l l e r model and s t i l l develop d i s c r e t e s t r a t i f i c a t i o n (Berry and Garrison, 1958d). Nevertheless i t i s s i g n i f i c a n t that both observers reach q u i t e s i m i l a r conclusions while u s i n g somewhat d i f f e r e n t l i n e s o f reasoning. In both cases complete 26 systems of networks are derived from an indefinite number of goods and services through p a r t i a l equilibrium suggestions. In both systems the triangular arrangement of production sites and the hexagonal shaping of market areas for each commodity are found to be optimal. Perhaps von Boventer (1963*173) best sums up the consequences of the two formulations: . . . (i) Losch*s system i s r e a l i s t i c and capable of an extension i n that on a homogeneous pl a i n a specialization of production i n different centers, an interregional or inter-urban exchange of i n d u s t r i a l goods and a complicated network of markets i s derived, ( i i ) In i t s f i n a l r e s u l t , as far as the overall system of a hierarchy of c i t i e s or central places i s concerned, where the individual economic a c t i v i t i e s are neglected, C h r i s t a l l e r ' s system gives both a better description of r e a l i t y - at least with regard to Southern Germany i n the ' t h i r t i e s - and has the advantage of being simpler, or more elegant, whereas the Losch system i s very d i f f i c u l t to test at a l l . Scope and Nature of the C l a s s i c a l Argument The c l a s s i c a l l i t e r a t u r e attracts considerable comment from geographers for diverse reasons. It i s not our purpose here to rigorously qualify the central place approach but to make certain that the reader i s f u l l y aware of the more important drawbacks of the theory. A f i r s t group of criticisms deals with the actual assumptions employed by C h r i s t a l l e r and Losch. Isard (1956:274), for one, points out that Loschian analysis i s limited only " . . . i n situations where raw materials are not required (as i n service industries) or are ubiquitous and everywhere available at the same costs." In other 2? words, the central place approach i s omitting too many production s i t e s (whether market or material oriented) that are strongly influenced by the nature of input p r i c e s . A l s o , the theory i s r e s t r i c t e d to those a c t i v i t i e s not affected by more select ive p r i c i n g p o l i c i e s , the introduction of which substant ia l ly a l t e r s the s p a t i a l extent of markets f o r many goods. However, t h i s only means that central place theory has a somewhat smaller domain of economic operations than some optimist ic observers would give i t . The theory does possess an analyt ic framework that allows i t to explain cer ta in hypotheses, with at tention being devoted to the a p p l i c a b i l i t y of i t s assumptions, \ On the other hand, some c r i t i c i s m i s devoted to the s ta t ic or deterministic nature of the theory. As Pred (1967*99) s tates : In order for the fundamental precepts of central place theory to be f a i t h f u l l y reproduced i n a r e a l world s i t u a t i o n i t would be necessary for every t e r t i a r y - a c t i v i t y supplier (entrepreneur, firm) to make an optimal loca t ion decision (si te and s i t u a t i o n selection) and every t e r t i a r y - a c t i v i t y consumer (service c l i e n t ) to make a t o t a l l y r a t i o n a l journey-to-consume decis ion . This difference seemingly arises from disagreement as to what enta i l s a sa t isfactory framework for explanation. C l a s s i c a l theory rests strongly upon economic assumptions and Eucl id ian geometric notions. E f f o r t s continue to be channelled along functional (roles of phenomena within organizations) and morphometric ( spat ia l structure and 28 form) l i n e s (see Harvey, 1 9 6 9 : 7 8 - 8 3 ) . Inherent i n such c e n t r a l place t h i n k i n g i s that the system of market networks i s c o n s t a n t l y a d j u s t i n g i t s e l f toward some long run optimal s p a t i a l l a y o u t . The b e h a v i o r a l approaches r e s t upon p s y c h o l o g i c a l and s o c i o l o g i c a l p o s t u l a t e s so as to avoid a mechanistic view o f decision-making. Decisions concerning where to l o c a t e and where to buy are v a r i a b l e due to d i f f e r e n c e s i n a c t o r s ' ( f i r m s , buyers) information and c a p a b i l i t y to employ that information. In many r e s p e c t s , production decision-making does involve non-optimal cause and e f f e c t behavior, but improvement of the l o c a t i o n s i t u a t i o n seems to occur with time. In t h i s way, b e h a v i o r i a l matrix approaches may a f f o r d good p r e d i c t i o n s i n the short run. On the other hand, consumer decision-making i s s t i l l c onfined to d e s c r i p t i v e t h i n k i n g (see Curry, 1962). Besides, the more temporal or genetic approaches a l s o tend to be d e s c r i p t i v e i n the sense that t r a n s p o r t routes, economic a c t i v i t i e s , migration, etc, are randomly assigned. However, with stage-by-stage q u a l i f i c a t i o n s , s i m u l a t i o n studies do a f f o r d e x c e l l e n t p i c t u r e s of r e a l i t y ( M o r r i l l , 1963). In any case, no other i n t e r p r e t a t i o n challenges the c l a s s i c a l view i n the way i t l i n k s the governing processes and the r e s u l t i n g s p a t i a l s t r u c t u r e and form, a synthesis that Harvey ( I969sl27) suggests i s c e n t r a l to geographic theory. Or to take another view, " . . . the 29 synoptic f e a s i b i l i t y of a model i s enough to j u s t i f y i t s use as a b a s i s f o r e m p i r i c a l research. I t i s v a l i d to enquire whether a s i t u a t i o n which co u l d e x i s t , does e x i s t , even i f one has no a i r - t i g h t l o g i c to account f o r i t s emergence from non-existence or chaos." ( M a r s h a l l , 1 9 6 9 s 4 o ) . A l s o , i t i s c e r t a i n l y to t h e i r c r e d i t t h a t C h r i s t a l l e r ( 1 9 6 6 : 1 1 1 - 1 1 2 ) and Losch ( 195^»*iii) seem e n t i r e l y aware of the shortcomings i n t h e i r c e n t r a l place d e r i v a t i o n s . We should view t h e i r models as the f i r s t attempts to complement the ideas of s p a t i a l d i f f e r e n t i a t i o n (due to economic f a c t o r s s o l e l y ) with those of i n t r a r e g i o n a l e q u i l i b r i u m (that i s , the simultaneous determination o f market l o c a t i o n s , production c e n t e r s , t r a n s p o r t a t i o n routes, etc.).-^ B a s i c a l l y , they are d e r i v i n g a system o f economic order through a minimum of assumptions i n hope of e x p l a i n i n g the e s s e n t i a l s of s p a t i a l d i f f e r e n -t i a t i o n . Extensions of the C l a s s i c a l L i t e r a t u r e We have o u t l i n e d c e n t r a l place theory as the p a r t i a l theory of the l o c a t i o n , s i z e , nature, and d i s t r i b u -t i o n of a c t i v i t y c l u s t e r s . In many instan c e s , f u n c t i o n a l -'Depending upon the scale at which we examine the c e n t r a l place system, e q u i l i b r i u m may be considered i n t r a r e g i o n a l ( a l l a c t i v i t i e s ) or i n t e r r e g i o n a l (subsets of a c t i v i t i e s ) . 30 interdependence amongst centers and t h e i r market areas has "been s t r e s s e d : t h e r e f o r e , i t i s not unnatural t h a t the term "system" has "become l o o s l y a s s o c i a t e d with the combination of market nets. In the f o l l o w i n g d i s c u s s i o n we e x p l i c i t l y develop some of the more b a s i c f e a t u r e s of a c e n t r a l place system i n order to have added r a t i o n a l e f o r the formulation of c i t y s i z e models. I t may be p r o f i t a b l e , then, to f i r s t expand on the term "system" by seeing how c e n t r a l place theory t i e s i n t o general systems theory: A system i s a s e t of objects ( f o r example, c e n t r a l p l a c e s ) , a t t r i b u t e s of the objects (population, estab-lishments, business types, t r a f f i c generated), i n t e r r e l -a t i o n s among the objects (midpoint l o c a t i o n s f o r lower l e v e l c e nters, uniform spacing at any g i v e n l e v e l ) and among the a t t r i b u t e s (the graphs o f l o g -l o g r e l a t i o n s h i p s ) and interdependencies of objects and a t t r i b u t e s (the c e n t r a l place h i e r a r c h y ) . (Berry, 1967:76-77). Aggregate R e l a t i o n s and Elemental Components Considerable e f f o r t i s d i r e c t e d toward summarizing the fundamental interdependencies (that i s , e m p i r i c a l s t r u c t u r a l r e l a t i o n s h i p s ) of c e n t r a l place systems i n a c l o s e l y k n i t set of equations, (Berry and Barnum, 1962; Berry, Barnum, and Tennant, 1962; M a r s h a l l , I 9 6 9 ) . The study areas f o r these e m p i r i c a l i n v e s t i g a t i o n s are, t y p i c a l l y , r u r a l regions so as to f u l f i l l the p o s t u l a t e s of c e n t r a l place theory to a reasonable degree. E m p i r i c a l research i s a l s o c a r r i e d on at the intraurban l e v e l i n order to f a c i l i t a t e i n t e g r a t i n g r e g u l a r i t i e s at d i f f e r e n t s c a l e s , (Berry, I 9 6 7 ) , To describe these basic r e l a t i o n s h i p s 31 we must provide d e f i n i t i o n s f o r s e v e r a l v a r i a b l e s ! p: the p o p u l a t i o n of a c e n t r a l p l a c e ; r : the p o p u l a t i o n of the complementary area served by a c e n t r a l place ; P: the t o t a l p o p u l a t i o n served by a c e n t r a l p l a c e ; A: the s p a t i a l extent of the complementary areas being s e r v i c e d ; Qpj the p o p u l a t i o n d e n s i t y of the e n t i r e area s e r v i c e d by a center, i n c l u d i n g the o u t l y i n g area and the center i t s e l f ; Q n the p o p u l a t i o n d e n s i t y of the o u t l y i n g trade area; y: the number o f c e n t r a l f u n c t i o n s (separate business types) o f f e r e d by a c e n t r a l p l a c e ; hence, the highest l e v e l c e n t r a l f u n c t i o n performed by the center; Dyt the maximum distance that customers t r a v e l to a c e n t r a l place; t h e r e f o r e the range o f good "y"; f : reads "some f u n c t i o n of"; logt i n d i c a t e s base 10 logarithms. To begin with, we may define s e v e r a l e q u a l i t i e s as w e l l i P = p+r , . . ( E l ) A = f(Dy) . . . (E2) r = AQr = f(Dy)Qr . . . (E3) p = AQp = f(Dy)Qp . . . (E4) 32 One g e n e r a l l y expects t h a t l a r g e r c e n t r a l places have more c e n t r a l f u n c t i o n s , more establishments, and l a r g e r market areas than smaller centers. L o g l i n e a r r e l a t i o n s h i p s seem to p e r s i s t between the number of fun c t i o n s performed i n c e n t r a l places and ( i ) the populations i n those places or ( i i ) the t o t a l populations served by those p l a c e s . (Berry, Barnum, and Tennant, 1962*69; Berry, 1 9 6 8 : 3 7 - 3 8 « M a r s h a l l , 1969:163-164). Besides, a l i n e a r p a t t e r n seems t o i l l u s t r a t e the a s s o c i a t i o n between the maximum distance consumers are w i l l i n g to t r a v e l to a c e n t r a l place and the number of f u n c t i o n s o f f e r e d t h e r e . ^ (Berry, Barnum, and Tennant 1962:100-101; Berry, 1968:28). These and s i m i l a r arguments may be fo r m a l i z e d i n s t r u c t u r a l equations: l o g p = a-j^  + b^y . . . (2.3) l o g P = a 2 + b 2 y . . . (2.4) Dy = a^ + b^y . . . (2.5) where a 2 > a^> a^ Various i m p l i c a t i o n s may be drawn from these equations: from (2.3) & (2.4) l o g p = a ^ - a ^ + b x l o g P . . ,(2.6a) b 2 b 2 (2.3) & (2.5) l o g P = a 1 b 3 - a ^ j , + \ Dy . . .(2 .6b) b 3 b 3 " (2.3) & (2.4) l o g P = ai2b1 - a^bg + b 2 l o g p . . . ( 2 . 7 a ) \ b l This c o n t r a d i c t s a statement i n Berry and Barnum, 1962 but seems j u s t i f i e d by the e m p i r i c a l evidence r e f e r r e d to above. 33 (2.4) & (2,5) l o g P = a 2 b 3 - a 3 b 2 b £ Dy . . . (2.7b) t 3 b 3 " (2.6a) & (E4) l o g A = a ^ - a-jbg b2 l o g p - l o g Qp + b, b, 1 1 . . . (2.8a) " (2.6b) & (E4) l o g A = a 2 b 3 - a 3 b £ b £ Dy - l o g Qp b~ b_ 3 3 . . . (2.8b) " (2.6a) & (E3) l o g r = a 2 b r a l b 2 b2 l o s p + l o g bT™ + bT Q p 1 1 . . . (2.9a) " (2.6b) & (E3) l o g r = a,b^ - a-b9 b9 Dy + l o g Or 2_£ + _£ Qp *3 *3 . . . (2.9b) These statements supplement those that may be formulated f o r establishments, f u n c t i o n a l u n i t s , e t c , i n a s i m i l a r way. The r e s u l t s of these equations, however, suggest e m p i r i c a l f e a t u r e s of c e n t r a l place systems that are simpler than those c i t e d elsewhere (Berry, Barnum and Tennant, 1962} Berry, 1964): ( i ) C e n t r a l place populations are constrained only by the t o t a l populations they s e r v i c e . T h i s i n t e r p r e -t a t i o n strengthens an elementary economic base r a t i o n a l e f o r c e n t r a l place systems since i t avoids gross d e n s i t y as an e x p l i c i t v a r i a b l e . Besides, the c o e f f i c i e n t s "b^" and "b 2" determine how the r a t i o P/P " changes as centers 34 take on more and more f u n c t i o n s . E m p i r i c a l evidence (Berry, Barnum, and Tennant, 1962: F i g s . 5 & 6) suggests that "b 2" i s s l i g h t l y g reater than "b^" and t h a t , as a consequence, community populations assume a decreasing p r o p o r t i o n of t o t a l market populations as they grow l a r g e r . ( i i ) The s p a t i a l extent of the complementary area about a c e n t r a l place i s constrained by the t o t a l p o pulation and gross density. T h i s suggests that the area i s a f u n c t i o n of the number of business types o f f e r e d by a c e n t r a l place but that t h i s area diminishes as o v e r a l l d e n s i t i e s i n c r e a s e . ( i i i ) The non-central place p o p u l a t i o n of the complementary area (that i s , r e s i d e n t s i n smaller c e n t r a l places of the trade area or r u r a l i n h a b i t a n t s ) depends on the number of business types i n the market center and "Or / the nature of the d e n s i t y r a t i o /Qp". These e x t e r n a l populations account f o r an i n c r e a s i n g p r o p o r t i o n of t o t a l p o pulation as market areas expand (other things equal). In short summary, the important aggregate r e l a t i o n s of c e n t r a l place systems appear to be exponential: t r i b u t a r y area populations, t o t a l market area populations, and the p h y s i c a l extent of these areas are a l l exponential f u n c t i o n s of the population s i z e s of c e n t r a l places. A l s o , (2,6b) suggests that the range of the highest l e v e l good provided by a c e n t r a l place i s e x p o n e n t i a l l y r e l a t e d to that community's population. Moreover, the r e l a t i o n s h i p between the growth r a t e s of the s p a t i a l components and that 35 of the a s s o c i a t e d market center depends upon p a r t i c u l a r c o n s t r a i n t s i n each case. E m p i r i c a l i n v e s t i g a t i o n i n d i c a t e s t h a t there must be fewer l a r g e r centers with l a r g e r trade areas and that these l a r g e r centers are more widely spaced than s m a l l e r c e n t e r s . Such p r o p e r t i e s e n t i r e l y r e - i n f o r c e the o r i g i n a l a n a l y t i c statements of c e n t r a l place theory concerning the s i z e , spacing, and f u n c t i o n s of urban centers. Berry and Barnum (1962) add to t h e i r d e r i v a t i o n s a s e t of e m p i r i c a l l y based i n e q u a l i t i e s t h a t i d e n t i f y d i s c o n t i n u i t i e s of area and p o p u l a t i o n served at any gross p o p u l a t i o n d e n s i t y . I f we r e c a l l t h a t major t h r u s t of the C h r i s t a l l e r model concerning the existence o f d i s c r e t e orders of c e n t r a l p l a c e s , then these l i m i t s express the maximum s i z e of communities at p a r t i c u l a r l e v e l s of c e n t r a l i t y with regard to d e n s i t y c o n s t r a i n t s . H i e r a r c h i a l S t r u c t u r e Those readers f a m i l i a r with the c e n t r a l place l i t e r a t u r e may w e l l be questioning the avoidance of the term "hierarchy" to t h i s p o i n t . I t i s c l e a r that c o n s i d -erable confusion a r i s e s over the common use of that term and i t remains the author's contention that proper i n t e r -p r e t a t i o n can only come a f t e r a review of the t h e o r e t i c a l l i t e r a t u r e . The most important n o t i o n to remember about "hierarchy" i s that i t i s a s p a t i a l term when employed to describe features of a c e n t r a l place system. Therefore, I 36 i t confines to a c i t y system only s p a t i a l l y r e l a t e d centers among any set of centers. Lukermann ( 1966 ) s t a t e s t h a t we must be e x p l i c i t about the d i r e c t i o n s of p h y s i c a l c i r c u l a t i o n and movement when d i s c u s s i n g h i e r a r c h i a l c o n t r o l ? i n other words, i t i s not s u f f i c i e n t to only enumerate f u n c t i o n s , populations, e t c , i n a set of c i t i e s and extend our knowledge of h i e r a r c h i a l s t r u c t u r e . Therefore, "hierarchy" implies both s p a t i a l and f u n c t i o n a l (order) r e s t r i c t i o n s . T h i s should be immediately apparent, since the h i e r a r c h y bridges the interdependencies of a t t r i b u t e s and objects f o r the e n t i r e c e n t r a l place system, F u n c t i o n a l r e s t r i c t i o n i s measured by ( i ) the number of c i t i e s having the f u n c t i o n , ( i i ) the s i z e o f the p o p u l a t i o n served by the f u n c t i o n , and ( i i i ) the area of the popula t i o n served by the f u n c t i o n (Lukermann, 1966). S p a t i a l r e s t r i c t i o n , on the other hand, i s determined by ( i ) the interdepedence of centers and ( i i ) the i n t e r s t i t i a l placement of orders. The h i e r a r c h y determines the o r g a n i z a t i o n of a c i t y system i n space. Seen as a consequence of t e r r i t o r i a l s p e c i a l i z a t i o n , f u n c t i o n a l d i f f e r e n t i a t i o n , and degree of i n t e r a c t i o n among a c t i v i t y nodes, i t emerges only with some maturity i n the r e g i o n a l urban s t r u c t u r e . Once there, though, i t tends to define the l i m i t s o f i n d i v i d u a l growth among the urban places. 37 The C e n t r a l Place System Reconsidered The s p e c i f i c a t i o n of a c i t y system r e s t s upon the d e l i m i t a t i o n of an i n i t i a l center f o r i n q u i r y . Using the h i e r a r c h y concept, we can i d e n t i f y those centers of lower order that are commercially l i n k e d to the c e n t r a l p l a c e . R e c a l l i n g our d i s c u s s i o n of the C h r i s t a l l e r model, we began with the emergence of an "M" l e v e l center o f f e r i n g the s e t ^ t ^ , t g , . . t ^ of f u n c t i o n s . On the other hand, the i n t e r s t i a l l y s i t u a t e d "M-1" centers o f f e r the set { t 1 # t 2 , , . ., t ^} where f u n c t i o n type "M" i s the d i f f e r e n c e between the two s e t s . In t h i s manner, a c i t y system i s developed with "M" h i e r a r c h i a l l e v e l s and c o n t r o l i s maintained by the property that f u n c t i o n s provided by a center a t one l e v e l are proper subsets of those fun c t i o n s given a t higher h i e r a r c h i a l l e v e l s . Besides, a concomitant feature of any c e n t r a l place system i s i t s cl o s u r e or f u n c t i o n a l wholeness. This economic i n t e g r a t i o n i s determined by the l i n e s o f interdependence and the orders i n the hi e r a r c h y . I t i s a c r e d i t to C h r i s t a l l e r and Losch that they o f f e r c e n t r a l place models that combine f u n c t i o n a l and s p a t i a l c o n t r o l , i f only i n a p a r t i a l sense. Chapter 3 CITY SIZE MODELS AND DISTRIBUTIONS Review of the H i e r a r c h i a l Models The h i e r a r c h i a l approach i s e x p l i c i t to the d e r i v a t i o n of e x i s t i n g c i t y s i z e models. These models are p r e s e n t l y confined to the simpler but more p l a u s i b l e C h r i s t a l l e r i n t e r p r e t a t i o n ? indeed i t would be i n t e r e s t i n g i f a model based upon the Loschian landscape were s i m i l a r l y developed. As a r e s u l t , the c i t y s i z e models evade the postulate of even purchasing power d i s t r i b u t i o n (although we r e t a i n i t f o r i l l u s t r a t i v e ease), but are r e s t r i c t e d to cases of d i s c r e t e f u n c t i o n a l ordering. Terms and Notation Beckmann (1958) provides the i n i t i a l model o f c i t y s i z e s but the r a t h e r debatable p r o p e r t i e s of t h i s approach coupled with the more recent e f f o r t s i n the subject by Beckmann and others, r e q u i r e s that we f i r s t study a g e n e r a l -iz e d model. However, before departing on a r a t h e r r i g o r o u s d i s c u s s i o n , the reader should be acquainted with the terminology and n o t a t i o n of the subject. A c e n t r a l place that provides the "m"th bundle (basket) of goods and s e r v i c e s i s s a i d to possess f u n c t i o n type "m" (where "m" 38 39 represents one of the d i s t i n c t f u n c t i o n subsets between "1" and "M"); a l s o , i f that place provides f u n c t i o n type "m" but not "m + 1", i t i s s a i d to have order "m". Since the c e n t e r provides the "m"th basket f o r a complementary area, i t i s s a i d to "m"-dominate the e n t i r e p o p u l a t i o n i n that surrounding area ( i n c l u d i n g the r u r a l p o p u l a t i o n and the urban p o p u l a t i o n i n t h a t center and a l l smaller c e n t e r s ) . Dacey (1966) r e f e r s to the c e n t r a l place system where "q" i n d i c a t e s the n e s t i n g f a c t o r (see Chapter 2; that i s , the number of places with f u n c t i o n type "m-1" that are "m"-dominated by an order "m" place) and "M" denotes the t o t a l number of f u n c t i o n types o f f e r e d through-out the system,* The f o l l o w i n g n o t a t i o n i s common i n the l i t e r a t u r e s mt The f u n c t i o n provided by a places hence, the l e v e l i n the h i e r a r c h y as w e l l (m = 1, 2, . , ,, M); smallest centers o f f e r only f u n c t i o n one; nt The s i z e c l a s s (n = 1, 2, , . ., M) t f o r the s i n g l e l a r g e s t center and i t s a s s o c i a t e d market area, n = 1; M: the t o t a l number of functions provided i n the system or the number of l e v e l s on the h i e r a r c h y ; n o t i c e m = M - n + 1 and ^"Implicit to the c e n t r a l place scheme i s that these f u n c t i o n subsets remain r e l a t i v e l y constant i n nature 1 therefore we u s u a l l y r e f e r to them as simply " f u n c t i o n s " . n = M - m + 1 ? r m : the p o p u l a t i o n of the complementary area on the "m Mth l e v e l of the hierarchy? when m = 1 the p o p u l a t i o n i s e n t i r e l y r u r a l ? Pjjjt the po p u l a t i o n of a center on the "m"th l e v e l of the hierarchy? Pj^ x the popula t i o n of the l a r g e s t center i n the system? P m« the t o t a l p opulation served by a center on the "m"th l e v e l of the hierarchy? k : a s e r v i c e or technology m u l t i p l i e r t h a t m denotes the p r o p o r t i o n of the po p u l a t i o n i n an "m" l e v e l complementary area plus an "m*" (m 6. m* - M) l e v e l c e n t r a l place ( s e r v i c i n g the complementary area i n the c a p a c i t y o f an "m" l e v e l place) that i s r e q u i r e d to r e s i d e i n the "m*n l e v e l place i n order to provide f u n c t i o n "m* " to both; a necessary H c o n d i t i o n e x i s t s that > " k < 1 ; £.—i m m = 1 k i a simple p r o p o r t i o n a l i t y f a c t o r that r e l a t e s the population of a c i t y to the t o t a l p opulation served by that c i t y ; a necessary c o n d i t i o n e x i s t s that 0 < k < 1; q« the ne s t i n g f a c t o r f o r market areas; s i the equivalent number of centers of the "m - l " s t l e v e l that are dominated by an 41 order "m" p l a c e j the geometry of c e n t r a l place systems r e q u i r e s that s = q - 1 where "s" and "q" are "both constants. Model I: The General Case Dacey (1966) f i r s t o u t l i n e s the general c i t y s i z e model that i n t e r e s t s us i n t h i s d i s c u s s i o n . The development of h i s model i s r a t h e r sketchy, though, and the reader i s greeted hy s e v e r a l complicated formulations that are not e x p l i c i t l y d e r i v e d . Beckmann and McPherson (1970) evolve an i d e n t i c a l model i n a more elegant f a s h i o n . The d e r i v a t i o n of urban populations r e s t s upon three p o s t u l a t e s , besides those e s s e n t i a l to C h r i s t a l l e r ' s model. The f i r s t assumption s t a t e s that the amount of employment a s s o c i a t e d with a f u n c t i o n depends on the e n t i r e p o p u l a t i o n supporting that f u n c t i o n . The second assumption s t a t e s that population i n a c e n t r a l place i s a l i n e a r f u n c t i o n of employment (see Dacey, I 9 6 6 ) . The combination of these postu l a t e s leads to a s e r v i c e m u l t i p l i e r " k ^ c h a r a c t e r i s t i c of each f u n c t i o n . A t h i r d assumption i s that "k " i s i d e n t i c a l f o r a l l centers o f f e r i n g f u n c t i o n "m". m We begin d e s c r i p t i o n of the model with those centers p r o v i d i n g only the f i r s t f u n c t i o n to a u n i f o r m l y dispersed r u r a l population r^» P X = k x ( p x + r x ) = h?l . . . (3.D l - k x 42 In t h i s case "Tz^" denotes the p r o p o r t i o n of the t o t a l p o p u l a t i o n demanding f u n c t i o n one to the population of the c e n t e r p r o v i d i n g i t . Now, consider the case of a l a r g e r center that provides both the f i r s t and second f u n c t i o n s : P 2 = k i ( P 2 + r i ) + k 2 ( p 2 + r 2 ^ ' • * ( 3 , 2 ^ T h i s simply means that the population of a second order center i s determined hys ( i ) A population group that i s r e l a t e d to the supply of f u n c t i o n one to the second l e v e l center and a f i r s t l e v e l complementary area (note "P 2" serves " r ^ " i n the c a p a c i t y of a "p^" c e n t e r ) : ( i i ) A p o p u l a t i o n group that i s r e l a t e d to the supply of f u n c t i o n two to the second l e v e l center and a second l e v e l complementary area. Reasoning i n t h i s f a s h i o n , we may determine the p o p u l a t i o n r e s i d e n t i n a "m"th l e v e l center: m P m = l ? i k i ( P m + r i ) • • * ( 3 , 3 ) T h i s premise rooted i n C h r i s t a l l e r t h i n k i n g i s s u f f i c i e n t f o r generating a model i n which center and complementary area populations are p r o p o r t i o n a l to the b a s i c r u r a l p o pulation served by a f i r s t order center. The next step i s to determine the nature of the complementary area populations " r ^ " , . Centers of order "m" have "s" s a t e l l i t e c i t i e s of order "m-1", each of which i s surrounded by a complementary area of population 43 B r m ^". In other words, the population of the complementary area about a "m" l e v e l center c o n s i s t s of " r " i n the market area of order "m-l" surrounding the center plus a population of s ( p m _ 2 + rm_i^  i n s a t e l l i t e c i t i e s and t h e i r t r i b u t a r y areas. That is? rm = s p m - l + ( 1 + s ) rm-l • • • ( 3 ' 4 ) To s i m p l i f y the s u b s t i t u t i o n method, we employ Beckmann and McPherson's d e f i n i t i o n s ! m K = 22 k m i = 1 m D = p - p i m m^ m^-1 so that (3.3) becomesi m P m d - K j = S k.r. . . . (3.5) *m m _^2_ x i where: W = " Pm-1 ( 1 " Km-1> • • • ( 3- 6> and since P„ = p m + r » m m m D = km Pm m-l . . . (3.7) But from (3.4)! Pm - ( 1 + s ) Pm-1 + Dm • • • ( 3' 8> 44 or, P , (1 + s) (1 - K ^ ) m (1 - m-1 . . . (3.9) Through repeated s u b s t i t u t i o n s * m-1 P M =TT (1 + s) (1 - K.) PX i = l (1 - K i + 1 ) r , J J ^ (1 + s) ( 1 - K . ) 1 - k l i - 1 (1 - K i + 1 ) . . . (3.10) and from the d e f i n i t i o n of D t m m P m = P i + S k i i=2 1-K i_ 1 P i Through s u b s t i t u t i o n s i n (3.7)« m ?m = * l r l + S . . . (3.11) ^ ^ 1-2 ^ i - l P i k-.r, r v J L k. i l l (1+s) ( 1 - K j = "l'l , * l - ^ - v ~ i -pr -1 l-kl l-kx 2^ 1 - K 1 . 1 _U d - K . + 1 ) . . . (3.12) I t should he obvious from (3.12) that a c i t y of order "m" i s depicted " . . . as being constructed of l a y e r s or segments supplying a nested set of markets, each def i n e d by the bundle of goods and s e r v i c e s s u p p l i e d " . (Beckmann and McPherson, 1970i27-28). In the c e n t r a l place framework the po p u l a t i o n of any community i s d e t e r -mined by the nature of d e c l i n e of the s e r v i c e m u l t i p l i e r s ( k ^ ) , the geometry ( s ) , and the r u r a l d e n s i t y (r-^). Model l i t The Aggregate Approach Given the p r o p e r t i e s of the general c i t y s i z e model we how t u r n to the d i s c u s s i o n of the simple models. The f i r s t of these i s Beckmann's o r i g i n a l hierarchial scheme which employs an assumption that the s i z e of any c e n t e r i s a constant p r o p o r t i o n of the p o p u l a t i o n i t serves} t h a t i s i p m = k P a . . . (3.13) The model i s a p r i o r i since i t r e s t s more upon i n t u i t i o n than development from a theory. Therefore we must be wary of making p r e d i c t i o n s with t h i s model, at l e a s t u n t i l we understand b e t t e r how i t r e l a t e s to c e n t r a l place theory. Beckmann's i n i t i a l model, however, displays a g l a r i n g inconsistency with central place r e l a t i o n s h i p s . On interpret ing the geometry of the system, he overstates the t o t a l population served by a c i t y on the "m"th l e v e l . (Since he seems to equate "s" with " q " ) . I t appears that t h i s error ar ises from the difference between the t o t a l number of settlements i n an economic region and the apportioning of those settlements among various h i e r -a r c h i a l l e v e l s . In any case, Beckmann (1968) and Parr ( I 9 6 9 ) r e c t i f y the misinterpretat ion i n independent contr ibutions . By adding " p m " to both sides of (3 .4) i t should be obvious that 1 P m = pm + s P m - l + r m - l . . . (3 .14) Using ( 3 . 1 3 ) and (3 .14) together i t i s a simple matter to demonstrate that both c i t y size and t o t a l population served increase exponentially with the h i e r a r c h i a l l e v e l i pm = (3.15) 47 . . . (3.16) Parr (I969) i l l u s t r a t e s the nature of the e r r o r i n the e a r l y Beckmann model by c o n s i d e r i n g the change i n the b a s i c progression component from " q " to "s+1". 1-k 1-k At t h i s p o i n t i n the d i s c u s s i o n i t may be p r o f i t a b l e to compare the a t t r i b u t e s of t h i s simple model and the more complex a p o s t e r i o r i model o u t l i n e d e a r l i e r . To begin with, the r a t i o n a l e f o r the f a c t o r s "k m" and "k" r e s t on q u i t e d i f f e r e n t c e n t r a l place r e l a t i o n s h i p s . The proposal of a d i s t i n c t "k m" value f o r each of the "m" f u n c t i o n s seems to be a reasonable d e r i v a t i v e of C h r i s t a l l e r i a n theory i n that i t focuses upon the changing r o l e s of ( i ) employment-function and ( i i ) c e n t e r - t r i b u t a r y area a s s o c i a t i o n s as we move through the h i e r a r c h y . In other words, while we suppose that the technology used i n p r o v i d i n g i d e n t i c a l f u n c t i o n s at d i f f e r e n t l e v e l s remains unchanged, we are i n t r o d u c i n g systematic changes i n c i t y s i z e s through the unique s e r v i c e mix at each l e v e l . On the other hand, the postulate of a constant "k" value i s t o t a l l y a r b i t r a r y , though i t may indeed have some e m p i r i c a l merit. For instance when we r e c a l l (2.6a) we n o t i c e that as "b 2" approaches "b^" i n value, a constant r e l a t i o n s h i p between center and t o t a l market p o p u l a t i o n i s neared (that i s , as k—*-log~" L (a^ - a 2 ) ) . 48 In a d d i t i o n to t h i s variance i n terms of r a t i o n a l e we note that the two f a c t o r s cannot be compared by assuming that "k " i t s e l f i s a constant, since the repeated a p p l i c a -nt t i o n of the f a c t o r s a f f e c t s the models i n d i f f e r e n t ways. For instance, a comparison of ( 3 . 1 ) and ( 3 . 1 3 ) i n d i c a t e s that "k" would equal "k^", but t h i s would introduce c o n t r a d i c t i o n s i n the case when ( 3 . 2 ) and ( 3 . 1 3 ) are compared with k = k^ = k 2. I t should be c l e a r that the two f a c t o r s have no obvious i n t e r r e l a t i o n s h i p and t h a t , t h e r e f o r e , Dacey ( 1 9 6 6 : 3 1 ) i s u n j u s t i f i e d i n c r i t i c i z i n g Beckmann*s r e s u l t . However, Model II can be shown to be only a p a r t i c u l a r case of the general model by the use of decreasing "k^" f a c t o r s . For t h i s to be t r u e , i t i s only necessary t h a t : m . . . ( 3 . 1 7 ) which i m p l i e s , as we noted above, that k = k-^ The determination of remaining "k^" values i s performed one step at a time; f o r i n s t a n c e t K2 ~ k l r 2 " k l r l p 2 + r 2 or, i n general: m-l m-l k = k,r_ - 52 k,r. - 2 D k,p„ . . . (3.18) Apparently, then, the a p r i o r i model and the a p o s t e r i o r i model have a fundamental premise (3.1) i n common. The f l e x i b i l i t y of the general model, however, comes from a higher l e v e l of a n a l y s i s with the a d d i t i o n of f u r t h e r premises. Rather than using urban centers as study u n i t s i n the c e n t r a l place system, the a p o s t e r i o r i model i s e f f e c t i v e l y employing f u n c t i o n a l l y determined population subsets (that i s : k,p , k„p , . , ., k p ) i m d^m mm of those centers as elements i n a more complex s p a t i a l system. Therefore the f a c t o r "k" emerges as the aggregate counterpart of the set ( k 1 # k 2, . . ., k ) i n the simpler system f o r one p a r t i c u l a r case. The e s s e n t i a l n o t i o n i s that Models I and II r e a l l y apply to d i s t i n c t systems t h a t have the same hi e r a r c h y and that are s p a t i a l l y c o i n -c i d e n t ( i n t h a t we d e p i c t centers as nodes i n a geometric network), Model I I I - The Geometric M u l t i p l i e r Dacey (I966) suggests i n t e r p r e t i n g "k m" as an exponential "k r a" to reasonably account f o r s p e c i a l i z a t i o n through the s e r v i c e m u l t i p l i e r s . Unfortunately he f a i l s to o f f e r any a n a l y t i c i n t e r p r e t a t i o n f o r h i s choice. However, a v a r i a t i o n of t h i s scheme i s an immediate d e r i v a t i v e of the Beckmann-McPherson formulation. I t i s based upon the proposal that market area populations "Pm" i n c r e a s e f r o m l e v e l to l e v e l by a constant f a c t o r . From '(3,10) i t should be obvious that a s u f f i c i e n t c o n d i t i o n f o r t h i s i s t h a t : 50 1-K , 1_^~ x = c o n s t a n t = 1 + h . . . (3.20) m D e f i n i n g k Q = 0, (3.20) h o l d s f o r a l l m 0: however t h i s i n d i c a t e s t h a t : = 1 + h . . . (3.21) l - k x o r , k l - k x h- = h . . . (3.22) B e s i d e s , the meaning o f (3.20) i s : •*» ^xkj1"*1 k i = < 1 - l t i > m " 1 k i • • • < 3 - 2 3 ) where the s e r v i c e m u l t i p l i e r decreases i n a geometric f a s h i o n f o r the second and h i g h e r h i e r a r c h i a l l o a d s . T h i s f o r m u l a t i o n and the Dacey s u g g e s t i o n are i d e n t i c a l f o r o n l y one v a l u e , v i z . = £. Now, u s i n g (3.10)s m-1 Pm = P l T T ( 1 + s ) ( 1 + h ) i = l = P x ( l + s ) 1 0 - 1 ( l + h ) * - 1 . . . (3.24) and s i n c e : Dm = . . . (3.25) 51 i t f o l l o w s t h a t i m - r, 1-k, 1 [ ( l - k x ) (1+hr ( 1 + s ) J ( l + h ) m + 1 (l+s)m+1 - (1+s) 2 (1+h) 2 (1+h) (1+s) - 1 which s i m p l i f i e s (see 3.21 or 3.22) t o : k, | | { r i k ~ ) m 1 (3.26) Pm = r l i k l 1-k, T T ^ 72 m (1+s) -.(nq) (1+s) " 1. . . (3.27) The r a t i o n a l e f o r t h i s geometric m u l t i p l i e r model i s not c l e a r though. Beckmann and McPherson suggest, however, that the growth f a c t o r i n (3.24) i s the same as that i n (3.16). Unfortunately, i t i s e a s i l y demonstrated that t h i s i n t e r p r e t a t i o n i s i n e r r o r . For inst a n c e , assuming t h a t : means: (1+s) (1+h) = _§_ + 1 1-k 1-k = , _k_ 1-k, " 1+s . . . (3.28) . . . (3.29) But since k = ^ i f " P ^ i s i d e n t i c a l i n (3.16)& (3.24) and since k , k^ > 0 we have a c o n t r a d i c t i o n (L.S. > R.S.) i n (3.29): i n other words, Model II and Model I I I cannot generate i d e n t i c a l market area populations and therefore must he considered d i s t i n c t . \ \ 52 B e s i d e s , ( 3 . 2 5 ) i n d i c a t e s t h a t t h e p o p u l a t i o n d i f f e r e n c e s b e t w e e n c e n t e r s on a d j a c e n t l e v e l s a r e a c o n s t a n t p r o p o r t i o n o f t h e t o t a l p o p u l a t i o n on t h e h i g h e r l e v e l j i n f a c t t h i s n e c e s s i t a t e s t h a t t h e p o p u l a t i o n s o f u r b a n c o m m u n i t i e s become a n i n c r e a s i n g p r o p o r t i o n o f t h e t o t a l m a r k e t a r e a p o p u l a t i o n s a s we a s c e n d t h e h i e r a r c h y . W h i l e t h i s i s a d e r i v a t i v e o f t h e a p o s t e r i o r i g e n e r a l m o d e l , we have no r e a s o n t o e x p e c t ( 3 . 2 0 ) i s n o t a c o m p l e t e l y a r b r i t r a r y p r o p o s a l . T h e r e f o r e , s i n c e t h i s i n t e r p r e t a t i o n seems i n c o n s i s t e n t w i t h a v a i l a b l e e m p i r i c a l e v i d e n c e we c o n s i d e r t h e s i m p l e a g g r e g a t e m o del a more v a l i d a p p r o a c h . M o d e l I V - The C o n s t a n t M u l t i p l i e r A t h i r d e l e m e n t a r y m o d el i s s u g g e s t e d i n t h e l i t e r a t u r e b u t i s nowhere d i s c u s s e d e x p l i c i t l y . D acey (1966) i n t r o d u c e s t h e i d e a o f c o n s t a n t s e r v i c e m u l t i p l i e r s b u t we have a l r e a d y d e m o n s t r a t e d t h a t t h i s c o n t r a d i c t s t h e a s s u m p t i o n s o f t h e g e n e r a l model. However, we may i n q u i r e what e f f e c t t h e r e w o u l d be on t h e s i z e d i s t r i b u t i o n o f c e n t e r s i f a c o n s t a n t m u l t i p l i e r were t o emerge a t t h e s e c o n d l e v e l . Beckmann and M c P h e r s o n ( 1 9 7 0 0 3 ) s u g g e s t : " . . . t h a t t h e l a r g e gap b e t w e e n k^ and k 2 i s common, b u t no c l e a r p a t t e r n i n t h e h i g h e r s e r v i c e m u l t i p l i e r s h a s a p p e a r e d . . ."? n e v e r t h e l e s s , t h e y do p r o v i d e d a t a t h a t i n d i c a t e a c o n s t a n t m u l t i p l i e r f o r a l l l e v e l s above t h e f i r s t i s n o t u n r e a s o n a b l e . The a s s u m p t i o n f o r t h i s model i s t h a t (3.3) may be e x p r e s s e d a s : 53 m i=2 . . . (3.30) ^ > k', or, m p m (1-^ - {m-l} k') = V l + *' £ 2 r i . . . ( 3 . 3 D which leads to (see 3A)t Pm 8 8 k i r l . x 1 - ( k x 4\m-j} k9 „m-i m-l k fm m-l . -A E £ p , s (l+s)3 + r (1+B) ( l + s ) 1 0 - 1 - ! j=0 !•! 1 8 I - J J l<*l+{»-W . ' . . (3.32) While the proposal f o r t h i s model i s s i m i l a r t o Model I I I i n t h a t a r e l a t e d p a t t e r n of s e r v i c e m u l t i p l i e r s begins a t the second h i e r a r c h i a l l e v e l , i t s r e s u l t s are more l i k e those of Model I I . I t seems that f o r c e r t a i n values i n the i n t e r v a l k m < k 'Ckg, where k 2, k^, . . . »k m are determined by (3.19)• t h i s new c i t y s i z e model approx-imates the use o f a b a s i c progression component. Fo r instance, i f we equate "k'" to the mean of k,, k~, . , .,k as determined i n Model I I , then Model IV underestimates t h e i r populations at higher l e v e l s . Moreover, c e n t r a l places with small to medium populations form a lower p r o p o r t i o n of t h e i r t o t a l market populations than the f i r s t l e v e l centers do, hut the l a r g e r centers tend to become a gre a t e r p a r t of the t o t a l populations they s e r v i c e . Unfortunately, i t i s d i f f i c u l t to defend t h i s model with the l i m i t e d e m p i r i c a l i n d i c a t i o n s we have a t t h i s time. A l s o , i f such a " k / H e x i s t s , we have l i t t l e evidence to s t i p u l a t e t h a t i t emerges at the second l e v e l . The best we can do i s hypothesize that a l a r g e gap between "k^" and "k 2" brings some s o r t of steady s t a t e i n t o being. On the other hand, the e m p i r i c a l evidence we have c i t e d f o r the support of the aggregate model covers only a number of the l a r g e r s i z e c l a s s e s , and i t remains to be e m p i r i c a l l y s u b s t a n t i a t e d (though i t seems i n t u i t i v e l y reasonable) that the very l a r g e s t centers assume a smaller p r o p o r t i o n of t h e i r t o t a l market populations. A l l i n a l l , though, i t seems improbable that we can d i s c a r d the constant p r o p o r t i o n a l i t y model i n f a v o r o f e i t h e r o f the two remaining elementary models. The f a c t t h a t i t i s not completely unsubstantiated by e m p i r i c a l study plus i t s extreme s i m p l i c i t y suggests that the e a r l y Beckmann model ( i n r e v i s e d form) i s the most p r a c t i c a b l e of the three. We say p r a c t i c a b l e because the a p o s t e r i o r i model i s not so f i r m l y attached to theory that we can suggest the n o t i o n of dec l i n e i n the "k m" values and, t h e r e f o r e , we do r e q u i r e some i n t u i t i v e s p e c u l a t i o n as to a systematic d e c l i n e (see 3.13, 3 . 2 0 , 3 . 3 0 ) . In other words, the general model has, a t t h i s time, e x t r a unknowns t h a t cannot be deduced from c e n t r a l place theory and t h e r e f o r e i t i s not workable i n generating c e n t e r populations. H i e r a r c h i a l Models and the Economic Base The economic base concept may be attached to the c i t y s i z e models with l i t t l e d i f f i c u l t y (see Dacey, 1966). Basic a c t i v i t i e s provide goods and s e r v i c e s f o r consumers outside the urban community while non-basic production i s d i r e c t e d t o the r e s i d e n t s of the center. With the assump-t i o n t h a t a l l employment i s b a s i c or non-basic, we may devise r a t i o s between the two types of employment f o r each of the h i e r a r c h i a l models. I t i s intended that the economic base concept should c l a r i f y our i n t e r p r e t a t i o n of the c i t y s i z e models t besides, we may g a i n s i g n i f i c a n t evidence toward understanding the changing character o f the basic/non-basic r a t i o (at l e a s t w i t h i n the confines of a c t i v i t i e s explained by c e n t r a l place theory) as urban centers r i s e or d e c l i n e i n s i z e . To begin with, we r e c a l l the b a s i c premise (3.3) o f the general h i e r a r c h i a l model. The p o p u l a t i o n i n center "P m" t h a t s e r v i c e s i t s complementary area i s zr? k.r., while the population f u l f i l l i n g l o c a l need i s i ^ i 1 1 56 m p ^ k.. T h i s i n d i c a t e s s e v e r a l p r o p e r t i e s of the ... in m ^ I T: basic/non-basic r a t i o : r?^ k i r i / p m i = l k i * ( i ) The r a t i o i s maximized at i = l and minimized a t i=m > ( i i ) The value of the r a t i o i n the i n t e r v a l 1 =j i - m depends upon the nature of d e c l i n e i n s e r v i c e m u l t i p l i e r s ; ( i i i ) The r a t i o i s a f u n c t i o n of the geometry or t r a n s p o r t topology of the c e n t r a l place system. Table 1 i n d i c a t e s the nature of these p r o p e r t i e s i n four c e n t r a l place systems, each with d i f f e r e n t char-a c t e r i s t i c s but a l l having seven h i e r a r c h i a l l e v e l s . The f i r s t and second systems use d i f f e r e n t m u l t i p l i e r s ( g e o m e t r i c a l l y d e c l i n i n g and constant p r o p o r t i o n a l i t y f a c t o r ) but generate data t h a t i s t o p o l o g i c a l l y comparable to the q=3 C h r i s t a l l e r data (see Beckmann and McPherson, 1970) with a r e l a t i v e l y constant m u l t i p l i e r "k /n beginning at the second l e v e l . The f o u r t h system i s a geometrical v a r i a n t of the second i n that i t i s depicted by the simple aggregate model. 57 Table 1 Service Multip l i e r s and Basic/Non-Basic Ratios of Four Central Place Systems (1) q = 3 (2) q = 3 (3) q = 3 (4) q = 4 r l " 2000 r x = 2000 r l = 2700 r l = 2000 km = empirical k' k = • 1 6" m km ra t i o 1^ r a t i o k r a t i o m k : m ratio 7 . 0 0 0 1 .00 . 0 5 3 . 6 5 . 0 3 4 1 .30 .054 . 5 6 6 . 0 0 1 1.01 . 0 6 0 .81 . 0 3 0 1 .49 .062 .71 5 .004 1 .01 .067 1 .02 .028 1.64 .072 . 9 1 4 .012 1 .02 .076 1 .34 . 0 3 1 1 .93 .082 1.18 3 .037 1.08 . 0 8 5 1.86 . 0 3 7 2 .22 . 0 9 5 1 .69 2 .111 1 .25 .098 2 . 7 9 . 0 4 5 2 . 6 7 .109 2 . 6 3 1 .333 2 , 0 0 .16? 5 . 0 0 .228 3 . 3 8 .167 5 . 0 0 ! 58 The nature of the s e r v i c e m u l t i p l i e r s i s c e r t a i n l y the most s i g n i f i c a n t determinant of the basic/non-basic r a t i o . The i n i t i a l c o n s t r a i n t i s induced by "k^" i n each case but the v a r i a b i l i t y of d e c l i n e b r i n g s out some very i n t e r e s t i n g p atterns. For instance, when c e n t r a l place f u n c t i o n s become extremely s p e c i a l i z e d (advanced technology, c a p i t a l i n t e n s i v e perhaps) and r e l y very l i t t l e on employment, we might expect a system s i m i l a r to the f i r s t . In t h i s case, the e x p o n e n t i a l l y d e c l i n i n g f a c t o r l e v e l s o f f the basic/non-basic r a t i o very q u i c k l y . Urban populations i n l a r g e communities are r e s t r i c t e d i n the sense that the capture of markets f o r higher order goods and s e r v i c e s brings i n l i t t l e employment} i n f a c t , the community assumes a smaller and smaller p r o p o r t i o n of the t o t a l market population as both grow l a r g e r . Employment becomes i n c r e a s -i n g l y balanced between the b a s i c and non-basic s e c t o r s m since 2 ^ k. —•*:§• as "M" becomes grea t e r . i = l 1 The second and f o u r t h systems are c h a r a c t e r i z e d by gradual f u n c t i o n a l s p e c i a l i z a t i o n . Since employment does not taper o f f r a p i d l y f o r higher order goods and s e r v i c e s , a v a r i e t y of basic/non-basic r a t i o s i s permitted. In both systems, c e n t r a l places form a constant p r o p o r t i o n of t h e i r t o t a l market populations but, as we ascend the hierarchy, both s e r v i c e m u l t i p l i e r s and the r a t i o s d e c l i n e . I t seems that the percentage increases i n b a s i c a c t i v i t y b r i n g f o r t h even greater percentage increases i n non-basic endeavours u n t i l absolute increases favor the l a t t e r i n the smaller s i z e c l a s s e s . The gr e a t e s t increments i n populat i o n increase are apparently determined byi ( i ) The capture of the a d d i t i o n a l markets f o r the highest order f u n c t i o n ; and ( i i ) The a d d i t i o n a l demands placed on the f i r s t order goods and s e r v i c e s by the new members of the b a s i c s e c t o r . The C h r i s t a l l e r system introduces another type of m u l t i p l i e r v a r i a t i o n , where k 2» k^, . . H k m are r e l a t i v e l y constant. Nevertheless, the basic/non-basic r a t i o s t e a d i l y d e c l i n e s as we move up through the hierarchy. This seems f u r t h e r proof that the p r o v i s i o n of f i r s t - o r d e r commodities i n response to demands made by a d d i t i o n a l b a s i c employees i s an extremely important determinant of the s i z e of the urban community. We n o t i c e , too, that i n t h i s e m p i r i c a l m example the r a t i o always exceeds u n i t y ( t h a t i s , k. < i ) . i = l 1 The geometry of the c i t y system a l s o i n f l u e n c e s the nature of the r a t i o but i n a l e s s s pectacular f a s h i o n . I t appears that with an increase i n the number of s a t e l l i t e c i t i e s , b a s i c a c t i v i t y gives way to l o c a l s e r v i c e s i n a more r a p i d f a s h i o n as we ascend the hie r a r c h y . Higher m u l t i p l i e r s are needed to meet the demands of more smaller centers i n the q=4 system; l i k e w i s e , t h i s introduces the need f o r f u r t h e r expansion of the non-basic sector (where lower order goods have higher m u l t i p l i e r s ) . I t seems that as the i n d i v i d u a l m u l t i p l i e r s converge at high l e v e l s of the h i e r a r c h y , the r e s u l t a n t basic/non-basic r a t i o s remain s i g n i f i c a n t l y d i f f e r e n t . The c i t y s i z e models i n d i c a t e some r e l e v a n t patterns i n the v a r i a b i l i t y of basic/non-basic r a t i o s , a t l e a s t w i t h i n the domain of a c t i v i t i e s t h a t c e n t r a l place theory seems to cover. Besides, we see more c l e a r l y how the s i z e d i s t r i b u t i o n of urban communities both c o n s t r a i n s and i s i n f l u e n c e d by the i n d i v i d u a l urban economies through the c e n t r a l place h i e r a r c h y . I t seems r e l e v a n t , then, t h a t we should be more aware of the c h a r a c t e r i s t i c s of c e n t r a l place s i z e d i s t r i b u t i o n s , H i e r a r c h i a l Models and the Rank-Size Rule Geographers d i r e c t considerable e f f o r t toward d e s c r i b i n g the frequency d i s t r i b u t i o n s of urban centers as based on c i t y s i z e models or e m p i r i c a l evidence. We leave d i s c u s s i o n of the l a t t e r issue u n t i l the next chapter and here we examine the course of arguments concerning the s i z e and frequency d i s t r i b u t i o n of centers i n a c e n t r a l place h i e r a r c h y . The rank-size r u l e continues to be the dominant t o p i c of i n t e r e s t i n r e l a t i o n to c i t y s i z e models. I t may be represented i n the f o l l o w i n g f o r m t % = R b p R . . . (3.33) where "R" i s the rank of the c i t y , 'p^" i s the population of the c i t y of rank " R i , " p ^ i s the population of the l a r g e s t c i t y , and ' V i s a derived constant. I f we graph t h i s f u n c t i o n on double l o g a r i t h m i c paper, we have a s t r a i g h t l i n e : l o g PR = l o g P M - b l o g R . . . (3.3^) As o r i g i n a l l y (and u s u a l l y ) i n t e r p r e t e d , "b" has a value of u n i t y and the p o p u l a t i o n of the "R"th l a r g e s t center mul-t i p l i e d by i t s rank "R" equals "Pj^". Hoover ( 1 9 5 5 ) appears to be one of the f i r s t to s e r i o u s l y q u e s t i o n the r e l a t i o n s of C h r i s t a l l e r ' s c e n t r a l place h i e r a r c h y and the rank-size p r i n c i p l e . He notes . . that the C h r i s t a l l e r system a u t o m a t i c a l l y y i e l d s a s e r i e s of c i t y t r i b u t a r y areas arranged according to the rank-size r u l e . . . ", but f a i l s to suggest a scheme that l i n k s c e n t r a l place populations and the p r i n c i p l e (Hoover, 1 9 5 5 ' 1 9 6 ) . Beckmann ( 1 9 5 8 ) i s again the f i r s t to e x p l i c i t l y comment on t h i s r e l a t i o n s h i p . While h i s o r i g i n a l model has been shown to be f a u l t y , h i s c l e v e r approach to t h i s new issue merits p r a i s e . We r e c a l l that Beckmann's e a r l i e s t model ( i n c o r -r e c t e d form) employs a constant b a s i c progression component s " 1-k + 1 " . However, i f we consider t h i s m u l t i p l i e r as a random v a r i a b l e about that stated constant, then a l l c i t i e s on the same h i e r a r c h i a l l e v e l do not n e c e s s a r i l y have i d e n t i c a l populations. Besides, the component has greater v a r i a t i o n s as "m" increases? i n other words, the c i t y s i z e s approach a continuous r a t h e r than s t e p l i k e d i s t r i b u t i o n . Beckmann i s e s s e n t i a l l y a l t e r i n g the r i g i d 62 C h r i s t a l l e r system to random disturbances so that only the midway center of any given h i e r a r c h i a l l e v e l i s r e p r e s e n t a t i v e of a l l c i t i e s on that l e v e l . Parr ( 1 9 6 9 ) demonstrates that the o v e r a l l rank "R " of a c i t y midway i n the "n"th s i z e c l a s s can be n expressed as: R n = q° + ( q 1 - q°) + ( q 2 - q 1) + . . . + ( q ^ 1 - q n ~ 2 +41 . . . ( 3 . 3 5 ) f o r n > l , U = 0 or 1 , where i s u n i t y i f the number of centers i n the s i z e c l a s s i s even and "t>C" i s zero i f that number i s odd. Since only the second (that i s f o r n > 1 ) s i z e c l a s s can possess an odd number of c e n t r a l p l a c e s , "*<" i s u s u a l l y one and "Rn" i s w r i t t e n more conveniently as: R = ( o " " 1 + Q n~ 2 + 1 ) - ( l + s ) " - 1 + ( l + s ) n - 2 + 1 n 2 " 2 . . . ( 3 . 3 6 ) Let's f i r s t of a l l consider the aggregate model with the p a r t i c u l a r b = 1 case of ( 3 . 3 3 ) . Now i f Model I I can accommodate the rank-size d i s t r i b u t i o n , then the product of the o v e r a l l rank of a midway c i t y on a p a r t i c -u l a r h i e r a r c h i a l l e v e l and the population of that c i t y must equal the population of the l a r g e s t c i t y i n the system. Hence, 63 or, ( l + s ) " " 1 + ( l + s ) n " 2 + 1 / s 2 l l - k But since 1 + s ^ j ^ s / l - k j + l j , i t i s simple to demonstrate that the l e f t side (rank) of (3.38) i s always exceeded by the r i g h t side (power of the p r o g r e s s i o n component) and no c o m p a t i b i l i t y e x i s t s between a c e n t r a l place system based on a constant center/market popul a t i o n r a t i o and a rank-size d i s t r i b u t i o n with an exponent of one. Parr a l s o considers the p o s s i b i l i t y t h a t coincidence of the aggregate model and the rank-size p r i n c i p l e may e x i s t f o r b ^ l j i n t h i s case: or. b = ( l o g R n + 1 - l o g R n ) . . . (3.40) By demonstrating that the denominator on the r i g h t side of (3.40) i s v a r i a b l e , he i s able to s t i p u l a t e that the value of "b" i n (3.39) and (3.40) v a r i e s with "n" and t h a t , t h e r e f o r e , the i n i t i a l assumption of a constant "b" i s v i o l a t e d . In Parr's (1969«249) words: ". . . i t may therefore be concluded that a c e n t r a l place system based on the constant p r o p o r t i o n a l i t y f a c t o r i s not I 64 compatible with a rank-size d i s t r i b u t i o n even where the value of the constant "b" assumes a value other than 2 u n i t y . " Beckmann and McPherson f e e l that a s u f f i c i e n t c o n d i t i o n f o r rank-size c e n t r a l place coincidence i s that market area populations increase by a constant m u l t i p l i e r from l e v e l to l e v e l (see 3.2k), the assumption they maintain to devise Model I I I . At t h i s point i n the d i s -c u s s i o n we present an argument that seems to r e f u t e t h i s t h i s a s s e r t i o n . To begin with, we consider the b = 1 case of ( 3 . 3 3 ) . Using ( 3 . 1 ) and (3.2?) we see t h a t : But employing ( 3 . 3 6 ) , i f c o m p a t i b i l i t y occurs then: Note our s u b s t i t u t i o n of "b" f o r "q" i n Parr's a r t i c l e ; t h i s discrepancy i s due only to a d i f f e r e n c e i n n o t a t i o n . o r , 65 + ( l + s ) * 1 " 2 - 1 . . < (3.^3) w h e r e t h e l e f t s i d e e q u a l s : ^ £ ( l + s ) M + k x ( l + s ) ^ 1 - ( l - k l ) ( l + s ) M ~ 2 -( i + s ) + d - ^ y j T h e r e f o r e , t o s t a t e t h a t t h e l e f t a n d r i g h t s i d e s o f (3.43) a r e i d e n t i c a l m e a n s : ( l + s ) M + k x ( l + s ) 1 ' 1 ' 1 - ( 1 - k ^ ( l + s ) M _ 2 - ( 1 + s ) + (1-*!) l\ ((ri^f'1 ( l + s ) " - ( l + 8 ) • • « 3.44) o r , k x ( I + s ) 1 " 1 - 1 - ( 1 - ^ ) ( l + s ) M " 2 + ( 1 + s ) + ( l - k x ) 2 2 . . . (3.45) B u t f o r a l l s > 0, 0 < k 1 < 1, we know: 2 ( l + s ) M - 1 + 2 ( l + s ) M " 2 + . . . + 2(l+s)° ^ 2 ( l + s ) M 66 H e n c e , b y n o t i n g t h a t t k x ( l + s ) M _ 1 - ( l - k x ) ( l + s ) M " 2 + ( 1+s) + ( l - k ^ 4 2 ( l + s ) 1 * ' 1 + . . . + 2(l+s)° a n d t h a t t h e f a c t o r / fl \ M _ 1 1 I - _ Q , „„<+,, < | I - 2 ) e x c e e d s u n i t y f o r M 0 , t h e n t h e l e f t s i d e o f (3 .45 ) i s e x c e e d e d b y t h e r i g h t s i d e ; t h e r a p i d l y e x p a n d i n g g r o w t h f a c t o r d i s a l l o w s t h e m o d e l f r o m b e i n g c o i n c i d e n t w i t h t h e r a n k - s i z e a r r a n g e -m e n t f o r b = 1. N e x t we c o n s i d e r c o m p a t i b i l i t y o f t h e s e c o n d e l e m e n t a r y m o d e l w i t h t h e r a n k - s i z e p r i n c i p l e w h e n t h e e x p o n e n t i s n o t r e s t r i c t e d t o u n i t y . We s h o u l d n o t e t h a t B e c k m a n n a n d M c P h e r s o n do n o t a l l u d e t o t h i s g e n e r a l c a s e b u t d i r e c t t h e i r a r g u m e n t t o t h e p a r t i c u l a r c a s e j u s t r e f u t e d . T h e p r o o f i n t h i s c a s e i s n o t a s e l e g a n t a s t h e one j u s t o u t l i n e d s i n c e s e v e r a l o f i t s s t a t e m e n t s i n v o l v e t a k i n g l i m i t s w h e n M >^ 0 . We r e c a l l ( 3 . 4 1 ) , w h i c h g i v e s i n s i m p l e r f o r r a i i • f A j-'-w .d -v s + k l . . . (3 .46) T h e r e f o r e t Now i f the rank-size r u l e holds f o r the model and b ^ 1, thent P l RM = P2 RM-1 = • • • = PM Here we have two cases (among "M-l") of immediate i n t e r e s t : © RM* 2 % p l . . . (3.48) p l . . . (3.^9) meaning: b © = l o g f y p j L O G RM . . . (3.50) b (2) = l o g (P2/PI) l o g ( V Rm-l) . . . (3.5D Considering b Q , we can define b / @ > b (£) where: L 0 G RM . . . (3.52) (M-l) l o g (iZiT-) M l o g (l±^ L O G RM L 0 G RM . . . (3.53) Now the f i r s t of these terms i s l e s s than M-1 L O S ( AT) M-2 l o g (1+s) 68 and the second term i s l e s s than M l o g |" (1+s)^ M-2 l o g (1+s) (by s u b s t i t u t i n g (3.36) i n t o the denominator of (3.53)). As M >^ 0, these terms converge toward -,. _ / 1 \ l o g ( i - v l o g (1+s) and l o g C^^-) r e s p e c t i v e l y , l o g (1+s) In other words, b ^ ( p i t s e l f converges at the sum 1 1 - fe)+ ^ M l o g (1+s) or, * ' © - . i o g r r ^ ) ( i + s ) < l f o r M " ° l o g l 0 S < 1 + 8> . . . (3.54) On the other hand, we may rewrite (3.51) as: b © = log^ l o g (1+s) {(rar)1*'*2 - (1-kl) \ s + k l - f o r M 0 . . . (3.55) However, when we compare (3.54) and (3.55) we f i n d that the numerator i n the former i s always exceeded by that i n the l a t t e r . Therefore, f o r s ? 0 , 0 4 ^ 4 1 , M » 0, we have b Q * b ' © * b ® and the rank-size r u l e i s not v a l i d f o r an exponent "b" unequal to u n i t y . 69 The d i s a s s o c i a t i o n of the geometric m u l t i p l i e r model from the r a n k - s i z e approach should come as no s u r p r i s e i n l i g h t of Parr's e a r l i e r a n a l y s i s . In that case, market populations grow at a constant r a t e but c i t y s i z e s grow too q u i c k l y f o r c i t y rank d e c l i n e s when these centers expand at the same r a t e . In t h i s l a t e r case, market populations grow at a constant r a t e but c i t y growth exceeds that r a t e j hence, we can expect again t h a t c i t y s i z e w i l l outweigh the rank value and that a constant product of rank and s i z e cannot be r e a l i z e d . The t h i r d simple model, as t y p i f i e d by a constant s e r v i c e m u l t i p l i e r that emerges at the second l e v e l , i s more d i f f i c u l t to r e l a t e to rank-size t h i n k i n g . As we noted before, no e x p l i c i t r a t i o n a l e determines the nature of "k " and we cannot develop an argument s i m i l a r i n form to that f o r Models I I and I I I . Nevertheless, we i n t u i t i v e l y expect that s i z e s expand too r a p i d l y f o r rank d e c l i n e s , since t h i s model overestimates the aggregate model at high l e v e l s (that i s when " k / n depends on the sum of k 2, k^, . . . ,k m i n Model I I ) . Therefore we do not consider Model IV and the rank-size p r i n c i p l e as being compatible concepts. Dacey i s unsuccessful i n d e f i n i n g a sequence of s e r v i c e m u l t i p l i e r s that permits the general model to conform to a rank-size d i s t r i b u t i o n , but from the tone of h i s a r t i c l e he may w e l l be r e s t r i c t i n g h i s search to a set of f u n c t i o n a l l y r e l a t e d "k^'s". Nevertheless, i f 70 we can i d e n t i f y any set of m u l t i p l i e r s t h at gives compati-b i l i t y , then we cannot accept that populations i n a c e n t r a l place system are at variance with the rank-size r u l e . Beginning with the lowest l e v e l s of the h i e r a r c h y , a s u f f i c i e n t c o n d i t i o n f o r "p^" and "P 2" to be rank-size r e l a t e d i s : RM ^ _ ^ %Tl P l " P 2 . . . (3.56) Now by employing ( 3 . 5 6 ) to define c e n t r a l place populations, one can s t i p u l a t e a s e r v i c e m u l t i p l i e r "kg" by i n t r o d u c i n g ( 3 . 1 7 ) . In other words we can c o n s t r u c t a more general statement than ( 3 . 1 9 ) i n which s e r v i c e m u l t i p l i e r s (one at a time) are designated so as to generate a r a n k - s i z e d i s t r i b u t i o n among urban communities: m-1 km - pm - ^ k i ( pm + r i } pm + rm . . . ( 3 . 5 7 ) While t h i s approach i s i n d u c t i v e and t o t a l l y l a c k i n g i n t h e o r e t i c a l r a t i o n a l e , i t does e s t a b l i s h some a s s o c i a t i o n between the deductive features of c e n t r a l place systems and the more e m p i r i c a l l y founded (see Chapter 4) r a n k - s i z e p r i n c i p l e . On the other hand, there i s no suggestion as yet that the rank-size r u l e may be i n t e r p r e t e d as a law ststement w i t h i n the framework of c e n t r a l place theory. In table 2 we present i n a rather comprehensive f a s h i o n the various p r o p e r t i e s of four c e n t r a l place 71 h i e r a r c h i e s , each generated from M = 7 , k^ = 0 , 3 3 3 , s = 2 , r j = 2000t ( i ) Model I formulated to conform to constant rank-size products; ( i i ) Model I I ; ( i i i ) Model I I I ; ( i v ) Model IV with "k' M estimated from the s e r v i c e m u l t i p l i e r s ( k 2 , k^, . . .t k^,) defined i n ( i ) . The t a b l e i s u s e f u l f o r q u a l i f y i n g any of the statements we have made to t h i s p o i n t i n the d i s c u s s i o n . Table 2 Fundamental P r o p e r t i e s of Midway C i t i e s i n Related C e n t r a l Place Systems v i a Diverse Modelling Approaches Model I: The General Case - Rank-Size Pa t t e r n Rank m rank Pm rm Pm k m m X S i z e 7 1 486 , 5 0 0 4 , 2 2 3,940 4 , 7 1 0,440 . 0 2 1 . 1 0 3 486 , 5 0 0 6 2 . 5 1 9 4 , 7 0 0 1 ,278,180 1 ,472,880 . 0 2 9 . 1 3 2 486 , 5 0 0 5 6 . 5 7 5 , 0 0 0 3 7 6 , 0 6 0 4 5 1 , 0 6 0 .044 . 1 6 6 486 , 5 0 0 4 18 . 5 2 6 , 3 0 0 107,820 1 3 4 , 1 2 0 . 0 6 0 . 1 9 6 486 , 5 0 0 3 5 4 . 5 8,940 2 9 , 9 8 0 3 8 , 9 2 0 .084 . 2 2 9 486 , 5 0 0 2 1 6 2 . 5 2 , 9 9 0 8 , 0 0 0 1 0 , 9 9 0 . 1 2 0 . 2 7 3 486 , 5 0 0 1 486 . 5 1 , 0 0 0 2 , 0 0 0 3 , 0 0 0 . 3 3 3 . 3 3 3 486 , 5 0 0 72 Table 2 (Continued) Model l i t The Aggregate Approach 7 1 4,096,000 8,192,000 12,288,000 .040 .333 4,096,000 6 2.5 1,024,000 2,048,000 3,072,000 .053 .333 2,560,000 5 6.5 256,000 512,000 768,000 .070 .333 1,664,000 4 18.5 64,000 128,000 192,000 .094 .333 1,184,000 3 54.5 16,000 32,000 48,000 .125 .333 872,000 2 162.5 4,000 8,000 12,000 .167 .333 650,000 1 486.5 1,000 2,000 3,000 .333 .333 486,500 Model I I I : The Geometric M u l t i p l i e r 7 1 10,450,000 14,143,000 24,593,000 .029 .428 10,450,000 6 2.5 2,330,000 3,161,000 5,491,000 .043 .428 5,800,000 5 6.5 532,000 699,000 1,231,000 .067 .428 3,460,000 4 18.5 115,500 156,000 271,500 .099 .427 2,140,000 3 54.5 25,500 35.000 60,500 .148 .422 1,390,000 2 162.5 5,500 8,000 13,500 .222 .406 892,000 1 486.5 1,000 2,000 3,000 .333 .333 486,500 Model IV: The Constant M u l t i p l i e r - "k'" est. from Model I 7 1 1,025,000 3,727,300 4,752,300 .060 .222 1,025,000 6 2.5 251,000 1,075,100 1 ,326,100 .060 . 190 628,000 5 6.5 64,000 315,700 379,700 .060 .169 416,000 4 18.5 17,300 93,700 111,000 .060 .156 320,000 3 54.5 5,150 27,800 32,950 .060 .155 281,000 2 162.5 1,900 8,000 9,890 .060 .191 309,000 1 486.5 1,000 2,000 3,000 .333 .333 486,500 73 H i e r a r c h i a l Sets and the Rank-Size Rule We have alr e a d y i n d i c a t e d t h a t the aggregate model appears to he the most s u i t a b l e approach i n l i g h t of ( i ) e x i s t i n g theory, ( i i ) e x i s t i n g e m p i r i c a l evidence, and ( i i i ) elegance. However, t a b l e 2 i l l u s t r a t e s t h a t apparently s i m i l a r d e c l i n e s of the s e r v i c e m u l t i p l i e r s i s not a s u f f i c i e n t reason to expect s i m i l a r i t y i n the nature of r a n k - s i z e products. Parr suggests that the s i z e d i s t r i b u t i o n of centers on the endpoints of each h i e r a r c h i a l l e v e l gives d e s c r i p t i v e support to coincidence of the b a s i c p r o g r e s s i o n component model and the rank-size r u l e . On the other hand, he r i g h t f u l l y notes that c o m p a t i b i l i t y of the two notions cannot be i n f e r r e d as such. This r a i s e s the q u e s t i o n of whether or not the aggregate model, through any reasonable m o d i f i c a t i o n , can be a l i g n e d to rank-size t h i n k i n g ? Surely, though, i n t e r n a l m o d i f i c a t i o n defeats the purpose of a model whose strength l i e s i n i t s s i m p l i c i t y . For instance, v a r i a t i o n of the "k" f a c t o r from l e v e l to l e v e l adds more unknowns to the argument, while changes i n the number of s a t e l l i t e c i t i e s erases the concept o f a general progression m u l t i p l i e r . Since we may i n t e r p r e t the d i s t r i b u t i o n of urban s i z e s through one r a t h e r i n t r i c a t e a p o s t e r i o r i model, i t seems unreasonable to manipulate an elementary model having i t s own d i s t i n c t advantages. 74 However, an extension of our s i n g l e system framework allows a new a s s o c i a t i o n between Model II and the r a n k - s i z e p r i n c i p l e to a r i s e . I f we consider a set of independent c e n t r a l place systems, then we may consider the o v e r a l l p a t t e r n of c i t y s i z e formed by the v a r i o u s independent h i e r a r c h i e s . We may pursue the approach that we used to compute the "k^'s" f o r the general model so as to r e l a t e to the rank-size p r i n c i p l e (see ( 3 . 5 6 ) and ( 3 . 5 7 ) ) . In t h i s case we generate a h y p o t h e t i c a l "M" l e v e l system of c e n t r a l places and demonstrate that, with the a d d i t i o n of v a r i o u s smaller systems (that i s , ones with M-1, M -2 , , . , , 2 , or 1 l e v e l s ) , an o v e r a l l r a nk-size arrangement may evolve. C l e a r l y , the idea i s to determine ranks when a constant growth f a c t o r " s + 1 " i s supposed f o r an 1-k e n t i r e set of c i t i e s . We use the same conceptual method of the i n i t i a l Beckmann c o n t r i b u t i o n where a c t u a l populations vary about t h i s modal value and add, too, that an increased number of centers i n each aggregate s i z e c l a s s (except the f i r s t ) means that a smoother d e c l i n e i n c i t y s i z e s i s now more l i k e l y . The s i n g l e l a r g e s t center of the "M" l e v e l system gives us the c e n t r a l place of rank one f o r the e n t i r e s e t . Our next step i s construct a h y p o t h e t i c a l s i z e c l a s s of population P ^ _ i so that the midpoint of the group has rank = 1 R e c a l l i n g ( 3 . 3 6 ) , we know that there are "s" centers of that second s i z e c l a s s i n the complete 75 "M" l e v e l system; hence, we add "x^" more centers so t h a t : 1 + s + (x1 +1) g ~~ 2 = I^k + 1 . . . ( 3 . 5 8 ) Obviously we are b u i l d i n g up "x^" independent c e n t r a l place systems of "M-1" l e v e l s each to supplement the f i r s t system. Likewise, we continue to the t h i r d c l a s s where we already have " s ( s + l ) " centers i n the complete system plus "sx^" centers i n the smaller systems. Now we must add "Xg" new centers of "M-2" l e v e l systems by s o l v i n g : 1 + s + x, + s(s+l) + s(x, ) + (Xp+1) / s ^ 2 r — =(i-k + y . . . ( 3 . 5 9 ) In t h i s manner we can develop a d d i t i o n a l smaller c e n t r a l place systems f o r v a r y i n g s i z e c l a s s e s "n" so t h a t : . . . ( 3 . 6 0 ) f o r ' i n i t i a l c e n t r a l place system i n the f o l l o w i n g t a b l e : To c l a r i f y t h i s argument, "x^* s" are determined o r the Table 3 Constant Rank-Size Products Given by Independent H i e r a r c h i a l Sets v i a Model II Size Popula- Rank h Q a h 1 h 2 h~ h^ h- h^ T o t a l Class t i o n 1 4 ,096,000 1 1 - - - - - - 1 2 1,024,000 4 2 3 ( x x ) - - - - - 5 3 2 5 6 , 0 0 0 16 6 6 7(x 2) - _ 1 9 4 64,000 64 18 18 14 2 7 ( x 3 ) - - . 7 7 5 16,000 256 54 54 42 54 103 (x^) - - 307 6 4,000 1,024 162 162 126 162 206 4 l l ( x 5 ) - 1 , 2 2 9 7 1,000 4,096 486 486 378 486 618 822 l,639(x 6) 4,915 ^ h e "hi " columns i n d i c a t e the t o t a l number of places i n each s i z e c l a s s of independent h i e r a r c h i a l sets» " i " r e f e r s to the number of l e v e l s missing from "M". 77 We should emphasize, however, t h a t the v a r i o u s systems must be assumed to be i n t e g r a t e d i n a s i z e d i s t r i -b u t i o n sense alone, since we are assuming that s p a t i a l i n t e g r a t i o n of the systems i s non-existent. I t i s d i f f i c u l t to j u s t i f y t h i s s u p p o s i t i o n i n a r e a l world case, but the idea of a very l a r g e t e r r i t o r y that d i p l a y s r e g i o n a l economic closu r e approximates the idea. We w i l l summarize the e m p i r i c a l s i z e d i s t r i b u t i o n l i t e r a t u r e i n the next chapter and t h i s should shed some l i g h t on the nature of l a r g e t e r r i t o r i e s and c l o s u r e i n the r e a l world. On the other hand, we must consider t h i s argument as an extremely h y p o t h e t i c a l one that only demonstrates how the aggregate model can conform to the rank-size r u l e and not why i t does. Chapter k EMPIRICAL ANALYSIS AND INTERPRETATION In t h i s chapter we have three general and i n t e r -r e l a t e d o b j e c t i v e s i ( i ) To c r i t i c a l l y examine the p a r t i c u l a r t e c h -niques employed i n e m p i r i c a l c i t y s i z e s t u d i e s ; ( i i ) To review e x i s t i n g s t o c h a s t i c i n t e r p r e t a t i o n s so that they supplement one another; ( i i i ) To s t i p u l a t e whether or not these i n t e r -p r e t a t i o n s are s a t i s f a c t o r y explanations of c i t y s i z e p a tterns. D i s c u s s i o n i s arranged to h i g h l i g h t the improvement of methodological concern i n the subject, while emphasizing the e x p l i c i t r o l e of theory i n e x p l a i n i n g the popul a t i o n d i s t r i b u t i o n amongst urban places. Background Pick any l a r g e area. I t w i l l l i k e l y c o n t a i n many small c i t i e s , a l e s s e r number of medium-size c i t i e s , and but few larg e c i t i e s . T h i s p a t t e r n of c i t y s i z e s has been observed to be quite r e g u l a r from one area to another. That i s , when the frequency of occurrence of c i t y s i z e s i n any area i s compared with the frequency of occurrence of s i z e s i n another area, the two f r e -quencies are very much a l i k e , . . Such e m p i r i c a l r e g u l a r i t i e s of c i t y s i z e have been noted many times and have long posed a challenge to those who would e x p l a i n or i n t e r p r e t them. (Berry and Garrison, 1958a t83) 78 79 In a few words we are b r i s k l y introduced to the more r e l e v a n t features of e m p i r i c a l c i t y s i z e d i s c u s s i o n s . The l i t e r a t u r e i s c h a r a c t e r i z e d by numerous di s p a r a t e c o n t r i b u t i o n s and, u n f o r t u n a t e l y , few reviews attempt to i n t e g r a t e the v a r i o u s concepts and schemes i n t o a meaningful whole. In the i n t e r e s t s of avoiding r e p e t i t i o n i n terms (as w e l l as i n hypotheses f o r that matter) we adopt a c o n s i s t e n t model framework f o r explanations that being, of course, the concept of c i t y system. Berry (1964), f o r one, p a r t i c u l a r l y advocates the grounding or urban theory i n a general systems approach. The f l e x i b i l i t y o f systems i n q u i r y and i t s s t r e s s i n g of i n t e r a c t i o n s and interdependencies suggest t h a t the s p a t i a l system i s a most adequate conceptual device to bridge t h e o r e t i c a l and e m p i r i c a l c o n t r i b u t i o n s . However, where the n o t i o n i s a p p l i e d with regard to c i t y systems, i t i s o f t e n done i m p l i c i t l y and the student wonders why a t t e n t i o n i s devoted to the idea at a l l . With e x p l i c i t use of the c i t y system idea however, e m p i r i c a l g e n e r a l i z a t i o n s may become i n c r e a s i n g l y substantive since i t a f f o r d s a c o n s i s t e n t base to r a t i o n a l l y order sense-perception data (Harvey, 1969:33). Besides, c e n t r a l place theory, as we have seen, i s n a t u r a l l y couched i n t h i s framework and the system no t i o n seems advantageous ( i f not e s s e n t i a l ) f o r s t a t i n g deduced p r o p o s i t i o n s so that they may be e m p i r i c a l l y t e s t e d . In other words, p e r c e i v i n g c i t y sets as systems may be defended as a most s a t i s f a c t o r y methodological 80 d e v i c e i n t h a t i t s e r v e s t o ( i ) i n i t i a t e and ( i i ) s u b s t a n t i a t e g e o g r a p h i c t h e o r y . Upon c o n s i d e r i n g such a d i f f u s e t o p i c as c i t y s y s t e m s , where debate c o v e r s a number o f i s s u e s , i t may be more p r o d u c t i v e t o i s o l a t e s e v e r a l p o i n t s f o r d i s c u s s i o n r a t h e r t h a n a t t e m p t t o u n i t e t h e f r e q u e n t l y i n d e p e n d e n t i d e a s i n a c h r o n o l o g i c a l s e n s e . The s t a t e m e n t u s e d t o i n t r o d u c e t h i s s e c t i o n a p p e a r s t o h i g h l i g h t t h e s e i s s u e s : ( i ) What c o n s t i t u t e s "any l a r g e a r e a " ? ( i i ) How do we d e s c r i v e a " p a t t e r n o f c i t y s i z e s " ? ( i i i ) How do we demonstrate t h a t "two f r e q u e n c i e s a r e v e r y much a l i k e " ? ( i v ) I n what manner do we " e x p l a i n o r i n t e r p r e t " t h e s e e m p i r i c a l r e g u l a r i t i e s ? I t i s hoped t h a t through an e x a m i n a t i o n o f t h e s e b a s i c q u e s t i o n s we c a n d e f i n e w h i c h m e t h o d o l o g i c a l q u a l i t i e s a r e w a n t i n g i n e x i s t i n g i n v e s t i g a t i o n s . The S t u d y A r e a The i n i t i a l m a t t e r i s by f a r the most n e g l e c t e d a l t h o u g h i t s h o u l d be c o n s i d e r e d c r i t i c a l t o any i n t e r -p r e t a t i o n o f c i t y s i z e d i s t r i b u t i o n s . W i t h o u t d e v o t i n g some a t t e n t i o n t o the n a t u r e o f the s t u d y a r e a a c o n s i s t e n t p o i n t o f v i e w i s forgone and c o m p a r a b i l i t y o f d i f f e r e n t i n v e s t i g a t i o n s becomes i m p o s s i b l e . I n i t s l o o s e s t i n t e r -p r e t a t i o n , a " l a r g e a r e a " i s an i n t u i t i v e g e n e r a l c l a s s -i f i c a t i o n i n t h a t we a r e a b s t r a c t i n g a s u b s e t o f u r b a n c e n t e r s from the u n i v e r s a l s e t o f a l l c e n t e r s (whatever 81 our d e f i n i t i o n o f "urban" may be). But such an a r b i t r a r y c l a s s i f i c a t i o n process seems hardly acceptable toward o f f e r i n g c o n s i s t e n t s e l e c t i o n measures throughout space and time. The e a r l i e s t s t u d i e s consider e n t i r e nations as appropriate study regions. Auerbach (1913). Lotka (1924, 1941), and Z i p f (1949) give o r i g i n a l impetus to the ran k - s i z e t h e s i s as a d e s c r i p t i o n of the s i z e of a l l c i t i e s i n a country above some designated population t h r e s h o l d . In f a c t , there i s a d e f i n i t e theme of n a t i o n a l i n t e g r a t i o n i n many of the ran k - s i z e arguments ( Z i p f , 19^9; J. Q. Stewart, 1947; Berry, 1961), J e f f e r s o n (1939t231), on the other hand, evokes the p r i n c i p l e of the primate c i t y where "A country's l e a d i n g c i t y i s always d i s p r o p o r t i o n a t e l y l a r g e and e x c e p t i o n a l l y expressive of n a t i o n a l c a p a c i t y and f e e l i n g . " He considers only the t r i o of l a r g e s t centers throughout a sample of n a t i o n a l u n i t s i n order to index s i z e r e l a t i o n s and, th e r e f o r e , the domain of h i s statement i s s e v e r e l y r e s t r i c t e d . While the l a r g e s t center appears to be much greater than the second and t h i r d centers, i t i s never c l e a r how i t i s considered d i s p r o p o r t i o n a t e l y g r e a t e r . This d i f f e r e n c e of opinion i s only resolved when we s t i p u l a t e what i s considered an appropriate sample space i n each n a t i o n a l u n i t . For example, when we generated populations i n the f i r s t two s i z e c l a s s e s o f the h i e r a r c h i a l models (see Chapter 3), the l a r g e s t center was always much 82 grea t e r than the f o l l o w i n g two, but only to a degree that was determined by the parameters of the e n t i r e system. A review of J e f f e r s o n ' s (1939:228) data i n d i c a t e s there i s s u f f i c i e n t reason to doubt h i s law statement on the small samples alone; to a l s o i n f e r a property that i s supposedly c h a r a c t e r i s t i c of the e n t i r e n a t i o n a l system from such a small sample i s yet another matter and must be t r e a t e d with a d d i t i o n a l skepticism. However, a more e s s e n t i a l problem must be s e t t l e d before we can even consider the comparison of the o v e r a l l c i t y s i z e d i s t r i b u t i o n s i n these n a t i o n a l t e r r i t o r i e s : we must state u n e q u i v o c a l l y whether the argument concerns c i t y sets or c i t y systems. The l a t t e r term, of course, c a l l s f o r a d d i t i o n a l f u n c t i o n a l r e l a t i o n s h i p s among the urban centers and l i k e w i s e suggests t h a t there e x i s t some c r i t e r i a of s e l f - s u f f i c i e n c y or clos u r e w i t h i n a c i t y s e t . The importance of t h i s dichotomy i s that i n f e r e n c e s derived from systems may be c a r r i e d i n t o sets but not v i c e v e r s a . Or, to take an example, Z i p f (19^9) cannot r e a l l y s t a t e t h a t the exponent "b" i n a ra n k - s i z e r e l a t i o n -ship i n d i c a t e s whether or not a national system of urban communities i s i n t e g r a t e d when he s e l e c t s the elements of that system i n a p r i o r i fashion. I t seems that our choice must r e s t s o l e l y upon the immediate purpose of our argument. I f we wish to suggest a r a t h e r general e m p i r i c a l r e l a t i o n s h i p that appears to p e r s i s t ( i ) i n t e r n a t i o n a l l y at one point i n time or ( i i O n a t i o n a l l y f o r s e v e r a l time periods, then the simple grouping of c i t i e s (above some minimum t h r e s h o l d l e v e l ) seems a reasonable approach. On the other hand, i f we wish to formulate ( i ) comparisons between subnational and n a t i o n a l u n i t s or ( i i ) amongst the subnational u n i t s them-s e l v e s , or i f we plan to ( i i i ) o f f e r some economic r a t i o n a l e f o r t h i s r e g u l a r i t y , then the systems concept seems a b s o l u t e l y necessary. Adherence to t h i s systems framework i s simply a measure f o r ensuring consistency or accuracy i n our i n f e r e n c e s since we have no reason to expect that the s i z e d i s t r i b u t i o n s are independent of how study areas are d e l i m i t e d . Since i t i s a most common t h e s i s that urban populations are determined by f u n c t i o n a l d i f f e r e n t i a t i o n and the degree of i n t e r a c t i o n among economic a c t i v i t i e s d i s t r i b u t e d i n space, i t i s s e n s i b l e that these urban elements be d e l i m i t e d by the very f a c t o r s that determine t h e i r magnitude. S p a t i a l i n t e r a c t i o n (socio-economic flows) between c i t y p a i r s seems adequately described by s e v e r a l interactance hypotheses ( g r a v i t y models and graph theory a p p l i c a t i o n s , i n p a r t i c u l a r ) , but d e v i a t i o n s r e s u l t from p h y s i c a l , c u l t u r a l , and p o l i t i c a l b a r r i e r s . Hence, when we postulate that a set of c i t i e s i n a n a t i o n a l t e r r i t o r y i s d i s t i n c t from sets i n adjacent t e r r i t o r i e s (and therefore corresponds to the " . . . general i n t u i t i v e n o t i o n as regards c l a s s i -f i c a t i o n , namely, that the c l a s s e s be as d i s t i n c t from one another as p o s s i b l e and i n t e r n a l l y as homogeneous as 8k p o s s i b l e " (Harvey, 1 9 6 9 : 3 3 9 ) ) , we are r e a l l y a s s e r t i n g t h a t i n t e r n a t i o n a l boundaries are b a r r i e r s of paramount i n f l u e n c e . F o r t u n a t e l y , e m p i r i c a l evidence tends to f i r m l y support t h i s s u p p o s i t i o n . Hackay (1958) and Nystuen and Dacey ( 1961 ) , f o r example, f i n d that the i n t e r n a t i o n a l boundary between Canada and the United States c o n s i d e r a b l y reduces t r a f f i c between c i t y p a i r s . However, i t i s h a r d l y c l e a r at t h i s time how i n t e r a c t i o n tapers o f f with ( i ) v a r y i n g distance decay q u a l i t i e s of the flow commodities or ( i i ) the p o p u l a t i o n p o t e n t i a l of the general area. Nevertheless there are reasonable signs that c i t y sets and c i t y systems are h i g h l y c o i n c i d e n t a t the n a t i o n a l l e v e l . Studying the changing features of n a t i o n a l c i t y s i z e d i s t r i b u t i o n s through time seems to f o l l o w i n the same v e i n . While annexation or c e s s i o n of areas (and urban centers) or development f o r c e d upon resource f r o n t i e r s (Friedmann, 1966) d i f f e r e n t i a l l y shock s t a b l e patterns of n a t i o n a l growth, the f a i l u r e of these new elements to enter the n a t i o n a l space-economy can only p e r s i s t i n the short run. However, r e t a i n i n g the systems expression i n but an i m p l i c i t r o l e would s t i l l prevent an apparent miscon-c e p t i o n concerning i n t e r n a t i o n a l comparison, Consider the nature of u r b a n i z a t i o n i n today's t y p i c a l underdeveloped (low per c a p i t a income, t e c h n o l o g i c a l and r e g i o n a l dualism) nations j u s t p r i o r to the i n t r o d u c t i o n of investment and t e c h n i c a l innovation, T h e i r urban s t r u c t u r e s at t h i s time of i n i t i a l awareness (or adoption) of the merits of economic 85 competition may be g e n e r a l l y represented by a number of comparatively small and independent a g r i c u l t u r a l communities. On the other hand, i n some of these nations there p e r s i s t s (where production surpluses allow) the lineaments of a r a t h e r elaborate s o c i a l and/or p o l i t i c a l h i e r a r c h y t h a t accounts f o r s i z e d i f f e r e n c e s among the l a r g e centers. Now the e v o l u t i o n of a progressive space-economy from t h i s subsistence base i s accompanied by the emergence of a c i t y system defined by p r i n c i p l e s of economic comparative advantage. However, i n the e a r l i e s t stages of development, a nation's economic space i s somewhat d i r e c t e d by the e x i s t i n g l i n e s of a d m i n i s t r a t i v e and s o c i a l o r g a n i z a t i o n ( f o r example, l a r g e r urban centers o f f e r more concentrated markets, t r a n s p o r t a t i o n routes tend to focus on the l a r g e r places, e t c . ) . In f a c t , the o r i g i n a l s o c i o - p o l i t i c a l h i e r a r c h y may be considered as the most prevalent of s e v e r a l i n i t i a l c o n d i t i o n s - demographic, p h y s i c a l ( t e r r a i n and amount of arable l a n d ) , and c u l t u r a l v a r i a b l e s - which c o n s t r a i n the i n i t i a l c o n f i g u r a t i o n of economic development. The suggestion i s o f f e r e d that the economic h i e r a r c h y i s more a c o n d i t i o n of convergence that dominates s p a t i a l o r g a n i z a t i o n only i n the long run (when f a c t o r s of production tend to be mobile and a c t i v i t i e s whose input p r i c e s vary r e l a t i v e l y l i t t l e i n space become i n c r e a s i n g l y demanded) and then makes po s s i b l e the thorough i n t e g r a t i o n of s o c i a l , p o l i t i c a l , and economic spaces as a u n i t (Friedmann, 1 9 6 1 ) . 86 The important poin t here, though, i s that nations with comparable low i n d i c e s of economic development may-be represented by considerable v a r i e t y i n t h e i r c i t y s i z e d i s t r i b u t i o n s . A l s o , i t becomes dangerous to assume that i n t e r n a t i o n a l cross-time data and n a t i o n a l t i m e - s e r i e s data are simply interchangeable (Lasuen, Lorca, and O r i a , 1967). T h i s implies that Berry (1961:585) i s not r e a l l y j u s t i f i e d i n making the ti m e - s e r i e s statement " . . . that d i f f e r e n t c i t y s i z e d i s t r i b u t i o n s are i n no way r e l a t e d to the r e l a t i v e economic development of c o u n t r i e s . Rank s i z e i s not the culmination of a process i n which n a t i o n a l u n i t y i s expressed i n a system of c i t i e s . " when h i s observations are of a cross-time i n t e r n a t i o n a l nature. In the more recent c o n t r i b u t i o n s , there i s i n c r e a s i n g s t r e s s on i n v e s t i g a t i n g c i t y s i z e d i s t r i b u t i o n s i n sub-n a t i o n a l or r e g i o n a l t e r r i t o r i e s , e s p e c i a l l y i n the r o l e of a crude planning device (Dziewonski, 1964? Lasuen, Lorca, and O r i a , 1967? Vapnarsky, 1969), However the same stu d i e s suggesting that n a t i o n a l a d m i n i s t r a t i v e u n i t s n e c e s s a r i l y d e l i m i t c i t y systems tend to r e f u t e the value of subnational p o l i t i c a l u n i t s as p a r a l l e l cases. Mackay's (1958) study of i n t e r a c t i o n i n d i c a t e s that i n t e r n a l a d m i n i s t r a t i v e boundaries are considerably more permeable than i n t e r n a t i o n a l boundaries, Nystuen and Dacey (I96I) i l l u s t r a t e that centers i n the p o l i t i c a l region may dominate p e r i p h e r a l communities i n an adjacent region. C l e a r l y , then, the existence of a r e g i o n a l c i t y system that d i s p l a y s 87 f u n c t i o n a l s t r u c t u r i n g i s r a t h e r independent of s t a t e or p r o v i n c i a l "boundaries, except where n a t i o n a l p o l i c y advocates high c l o s u r e i n these u n i t s . We should s t r e s s that there e x i s t s great v a r i e t y i n the nature of these r e g i o n a l economies (as d e l i m i t e d by p o l i t i c a l boundaries)j so much, i n f a c t , that one wonders i f we are drawing in f e r e n c e s from or p r o v i d i n g i n t e r p r e t a t i o n s f o r comparable sets of c i t i e s . C. T. Stewart (1958) i s perhaps the e a r l i e s t observer to propose a more adaptable d e f i n i t i o n o f the study area. He suggests that s e l f - s u f f i c i e n c y i s the c r i t e r -i o n that may lead to g e n e r a l i z a t i o n , Vapnarsky (I969) extends t h i s view by framing the complementary terms of rank-size and primacy w i t h i n the c o n d i t i o n of c l o s u r e . The tenor of h i s argument i s that unless subnational a d m i n i s t r a t i v e u n i t s d i s p l a y p r o p e r t i e s of nodal r e g i o n s , then they are " . . . t o t a l l y u n r e l a t e d to e i t h e r c l o s u r e or interdependence i n an e c o l o g i c a l sense. , ," and i t i s d i f f i c u l t to defend hypotheses and explanation based on such a r b i t r a r y areas (Vapnarsky, 1969*589). S i m i l a r l y , the urban system seems to be the only means of c a r r y i n g g e n e r a l i z a t i o n s from the r e g i o n a l l e v e l to the n a t i o n a l l e v e l without being t o t a l l y confused by the s c a l e problem (Harvey, 1969:352-353, 452-454). I t seems, though, that we should be more confident i n making i n t e r r e g i o n a l comparisons than i n c a r r y i n g g e n e r a l i z a t i o n s through d i f f e r e n t r e s o l u t i o n l e v e l s . 8 8 We see w i t h i n the mainstream of the c i t y s i z e t o p i c a c e r t a i n change i n the purpose of i n v e s t i g a t i o n and the concomitant increase i n a t t e n t i o n devoted to r e f i n i n g methodology. While opinions may s t i l l d i f f e r on how to e x p l a i n c i t y s i z e patterns i t i s c l e a r t h at an i n t e r p r e -t a t i o n r e s t i n g upon economic r a t i o n a l e r e q u i r e s some comp a r a b i l i t y i n d e f i n i n g systems. Unfortunately, t h i s c o m p a r a b i l i t y i s only given i n terms l i k e c l o s u r e and interdependence that are hardly o b j e c t i v e l y s t a t e d as yet. We note, too, that there i s a n o t i c e a b l e tendency to avoid formulating these p r o p e r t i e s along the l i n e s o f c e n t r a l place theory. In other words, c i t y systems are commonly d e l i m i t e d a p r i o r i through an a r e a l ( h i n t e r l a n d of the l a r g e s t center) point of view, when the c i r c u l a t i o n t h a t bonds the system i n a whole i s c l e a r l y l i n e a r . With an approach more i n l i n e with theory, l o g i c a l d i v i s i o n of the centers i s c a l l e d f o r , so that the l i n e s of dominance may be studied (Marshall, 1966). Unfortunately, the Loschian model (that seems more s a t i s f a c t o r y f o r the secondary sector) i s not so f l e x i b l e as the C h r i s t a l l e r model i n t h i s regard, and the idea i s s t i l l confined to more a g r i c u l t u r a l regions where the t e r t i a r y s e c t o r i s paramount. On the other hand, we expect that the coincidence of a c i t y system and the r e l a t e d nodal region of the dominant center breaks down i n only the periphery where smaller centers may be evenly a t t r a c t e d to adjacent systems. 89 In summary, "any l a r g e area" i s adequate as a comparable base i n only a q u a l i f i e d sense. Comparing the c i t y s i z e d i s t r i b u t i o n s of Korea and Washington State (Berry and G a r r i s o n , 1958a:83), f o r example, i s an i n t e r e s t i n g d e s c r i p t i v e undertaking but i t clouds the case f o r r a t i o n a l economic explanation. Only through the use of notions l i k e c l o s u r e and interdependency, which are f i r m l y based i n our only theory of the s i z e , spacing, and f u n c t i o n s of urban centers, can we hope to give a c o n s i s t e n t tone to t h i s explanation. C i t y Size Patterns: Skew D i s t r i b u t i o n s and Related Concepts A second general issue concerns the d e s c r i p t i o n of the c i t y s i z e arrangement. While a l l observers agree that the frequency d i s t r i b u t i o n of c i t i e s by s i z e appears to be h i g h l y skewed i n the shape of a r e v e r s e - J , o p i n i o n i s d i v i d e d over the nature of the s i z e c l a s s e s of urban centers and t h e i r r o l e i n determining these frequency d i s t r i b u t i o n s . To add to the confusion, c r i t i c s point to a number of p r o b a b i l i t y d i s t r i b u t i o n s (each showing a f a m i l y resemblance through p o s i t i v e skewness) that adequately describe the p a t t e r n of c i t y s i z e s i n many areas. The f i r s t of these problems i s taken up at a l a t e r time i n the d i s c u s s i o n while the l a t t e r i s of immediate concern here. 90 The Rank-Size and Pareto Distributions A large portion of the l i t e r a t u r e i s devoted to the a p p l i c a b i l i t y of the rank-size principle. For conven-ience, we r e c a l l the relevant equations stated e a r l i e r : % = R b p R . . . ( 3 . 3 3 ) log p R = log ^ - b log R . . . ( 3 . 3 4 ) In empirical approaches, however, we have no j u s t i f i c a t i o n for expecting a precise least squares f i t to ( 3 , 3 4 ) and i t i s frequently proposed that a constant "B" (where B Pj^) affords an improved s t a t i s t i c a l description of the r e l a t i o n -ship. This approximation i s defensible i n l i g h t of (i) the notion that populations may be accepted as being the same when they d i f f e r only by chance (Thomas, 1 9 6 1 ) and ( i i ) the data l i a b i l i t i e s regarding the d e f i n i t i o n of the individual urban centers (that i s , are corporate or metropolitan en t i t i e s most appropriate?), but must be determined i n a t o t a l l y objective manner. By plotting ranks versus sizes on double logarithmic paper, straight li n e tendencies may be observed where the sample follows ( 3 . 3 4 ) . Goodness of f i t may be determined through the linear regression model. The use of "B", then, i s feasible i f (i) we are confident i n the overall covariation of the two variables with "p^" as the largest center and ( i i ) "B" i t s e l f l i e s within some li m i t i n g error band determined by the sample (Thomas, 1 9 6 7 ) . 91 H e n c e , i n g e n e r a l , t h e r a n k - s i z e e q u a t i o n s may be r e i n t e r -p r e t e d a s : B = R \ o r , l o g P R = l o g B - b l o g R w h e r e B £ % S i n g e r (1936) a n d A l l e n (1954) a r g u e t h e c a s e f o r a s i m i l a r P a r e t o c u r v e r e p r e s e n t a t i o n , w h e r e r e g u l a r i t y i n t h e c i t y s i z e p a t t e r n i s c h a r a c t e r i z e d b y : A = R p R a . . . (4.3) o r , l o g R = l o g A - a l o g p R . . . (4,4) T h e P a r e t o f o r m i s l e s s u s e f u l i n t h e i n t e r e s t s o f c o m p a r i n g t h e o r e t i c a l c e n t r a l p l a c e s y s t e m s b u t t h e a u t h o r s do s u g g e s t ( i f a m b i g u o u s l y ) a p r a c t i c a l i n t e r p r e t a t i o n f o r t h e e x p o n e n t " a " . S i n c e t h e s l o p e i n (4.4) d e s i g n a t e s t h e r e l a t i v e n u m b e r o f s m a l l , m e d i u m , a n d l a r g e c e n t e r s , i t b e c o m e s a s a t i s f a c t o r y " i n d e x o f m e t r o p o l i z a t i o n " ( S i n g e r , 1936:254). When " a " i s s m a l l a n d t h e l e a s t s q u a r e s l i n e i s f l a t , t h e g r e a t e r b e c o m e s t h e p r o p o r t i o n o f l a r g e c i t i e s i n a g i v e n n u m b e r o f c i t i e s . I n o t h e r w o r d s , " a " i s a u s e f u l i n d e x f o r h i s t o r i c a l d e s c r i p t i o n a n d i n t e r n a t i o n a l c o m p a r i s o n s i n c e i t i l l u s t r a t e s how t o t a l u r b a n p o p u l a t i o n s a r e d i s t r i b u t e d among d i f f e r e n t s i z e d c e n t e r s . . . . (4.1) . . . (4.2) On t h e o t h e r h a n d , " h " may he g i v e n a r e l a t e d i n t e r p r e t a t i o n , s i n c e (4.1) a n d (4 .3) a r e c o i n c i d e n t w i t h i i a = b a n d A = B . A l s o , w h e n a = b = 1, t h e s i m p l e s t f o r m u l a t i o n o f t h e r a n k - s i z e ( A u e r b a c h , 1913) a n d P a r e t o r e l a t i o n s h i p s r e q u i r e s t h a t "A" a n d "B" a r e i d e n t i c a l c o n s t a n t s . H o w e v e r , t h e c o m p l e m e n t a r y a p p r o a c h e s b r i n g o u t a u n i q u e a n d r a t h e r f u n d a m e n t a l f e a t u r e o f t h e r a n k - s i z e r e l a t i o n s h i p . S i n c e i t i s s o l e l y c o n c e r n e d w i t h t h e m a n n e r i n w h i c h a l l c e n t e r s i n a n u r b a n s y s t e m a r e i n d i v i d u a l l y r e l a t e d t o t h e l a r g e s t c e n t e r t h e r e i n , t h e p o s s i b i l i t y o f e m p l o y i n g g r o u p e d d a t a i s o b v i a t e d . Y e t t h e s e e m p i r i c a l i n v e s t i g a t i o n s t h a t s u g g e s t d a t a f o l l o w a P a r e t o ( o r some o t h e r member o f t h e r e v e r s e - J f a m i l y ) d i s t r i b u t i o n a r e a l l c h a r a c t e r i z e d b y t h e u s e o f g r o u p e d d a t a w i t h a n o p e n c l a s s i n t e r v a l " * " f o r t h e l a r g e s t p o p u l a t i o n v a l u e s . I n t h i s c a s e , "R" i s r e a l l y i n t e r p r e t e d a s t h e n u m b e r o f c e n t e r s w i t h p o p u l a t i o n g r e a t e r t h a n o r e q u a l t o t h e s t i p u l a t e d c l a s s b o u n d a r y . Due t o t h e a f o r e m e n t i o n e d d a t a l i a b i l i t i e s , i n f o r m a n d n a t u r e , a n d t o f a c i l i t a t e a n a l y s i s we f r e q u e n t l y c o n s i d e r t h e d i s c r e t e s a m p l e s p a c e o f c i t y s i z e s a s b e i n g c o n t i n u o u s o r c o u n t a b l y i n f i n i t e . I n f a c t (4.1) a n d (4 .3) may b e t h o u g h t o f a s d i s c r e t e a n a l o g u e s o f s p e c i f i c C l a s s i n t e r v a l i s n o t m i s t a k e n f o r s i z e c l a s s w h i c h i s a d e v i c e a t t a c h e d t o c e n t r a l p l a c e t h e o r y . p r o b a b i l i t y d e n s i t y f u n c t i o n s . N e v e r t h e l e s s , i t i s h e l p f u l t o b e a w a r e o f w h a t we a r e p o s t u l a t i n g w h e n we e m p l o y a c o n t i n u o u s a p p r o a c h ( s i n c e a n a l y s i s d e a l s w i t h i n t e r v a l s a n d n o t p o i n t s ) . To b e g i n w i t h , t h e d i s c r e t e a p p r o a c h e s may h a v e c o n s i d e r a b l e d i s c r e p a n c i e s r e g a r d i n g t h e i r i n f o r m a t i o n i n t h e u p p e r t a i l s ( t h a t i s , l a r g e p o p u l a t i o n v a l u e s ) . T o b e m o r e s p e c i f i c , t h e p o p u l a t i o n o f t h e l a r g e s t c e n t e r may d e v i a t e n o t i c e a b l y f r o m t h e e s t i m a t e o f "A" g i v e n i n (4,4), U n f o r t u n a t e l y , i t i s i n t h i s r e a l m o f t h e s i z e d i s t r i b u t i o n t h a t we a r e f r e q u e n t l y m o s t i n t e r e s t e d . I t s e e m s t h a t i f we w i s h t o make i n f e r e n c e s f r o m t h e d i s c r e t e P a r e t o ( u s i n g g r o u p e d d a t a ) t o t h e r a n k s i z e r e l a t i o n s h i p , t h e n we m u s t make t w o b a s i c a s s u m p t i o n s : ( i ) T h a t t h e l o g a r i t h m s o f c i t y s i z e a r e r a t h e r e v e n l y d i s t r i b u t e d i n e a c h c l a s s i n t e r v a l ; t h i s s u p p o s i t i o n i s m o r e i m p o r t a n t i n t h e s i z e i n t e r v a l s o f t h e u p p e r t a i l w h e r e t h e s a m p l e d a t a b e c o m e i n c r e a s i n g l y s p a r s e ; ( i i ) T h a t pj ^ s A. The s e c o n d a s s u m p t i o n i s e s p e c i a l l y r e l e v a n t b e c a u s e s i n g l e c i t y p r i m a c y may b e i n a d v e r t e d l y c l o u d e d h y u s e o f t h e o p e n c l a s s i n t e r v a l . W i t h t h e s e p o s t u l a t e s , we may s u g g e s t how t h e i n d e x o f m e t r o p o l i z a t i o n i s r e l a t e d t o t h e u s u a l i n d e x C l a s s i n t e r v a l s a l s o r e f e r t o t h e r a n g e a v a r i a b l e a s s u m e s o v e r a f u n c t i o n . o f u r b a n i z a t i o n ? t h e l a t t e r b e i n g , o f c o u r s e , t h e p r o -p o r t i o n o f t o t a l p o p u l a t i o n i n a n a r e a t h a t i s c l a s s i f i e d a s u r b a n . P o p u l a t i o n s f o r t h e s y s t e m o f c i t i e s may be c a l c u l a t e d t h r o u g h (4.1) a n d (4.3) w h e n we a r e c o n f i d e n t o f a g o o d e m p i r i c a l f i t . F o r i n s t a n c e , i f a > 1, b ^ 1, a n d " P m a x " i s t h e p o p u l a t i o n o f t h e t h r e s h o l d ( s m a l l e s t ) p l a c e ("R" i s m a x i m i z e d ) , (4.3) i m p l i e s t h a t t o t a l u r b a n p o p u l a t i o n i s : finaxx d ( ? ) - ** ( p m a x x ~a d x aA = 1-a \ A a - 1 p a - l max . . . (4.5) W h i c h c o n v e r g e s ( a s A 0) a t : l i m A - * «° \ = aA p 1 " ) p s = r m a x .-a •max * - A . . . (4.6) o r . 1 1 b-1 B b ' p m a x . . . (4.7) w h i c h a r e c o r r e c t e d f o r m s o f t h e B e c k m a n n (1958:247) d e r i v a t i o n . On t h e o t h e r h a n d i f a 4 1, b > 1, (4.1) s u g g e s t s t h a t t o t a l u r b a n p o p u l a t i o n i s : 95 H dy = B (4.8) which converges (M 0) at« M 00 l i m B b-1 (4.9) 1 1-a (4.10) Unfortunately, a s i m i l a r harmonic s e r i e s d e r i v a t i o n i s not p o s s i b l e f o r a = b = 1, but t o t a l urban population i n t h i s case i s always l e s s than A ( l o g g R + 1). J. Q. Stewart (1947) provides convenient approximations to f i n i t e sum-mations f o r a l l three v a r i t i e s . I t i s i n t e r e s t i n g to note t h a t when convergence does occur i t may be a t t r i b u t e d to e i t h e r s i z e (4,6) or rank (4,9) depending on the nature of the index of m e t r o p o l i z a t i o n . In any event, when a rank-size or Pareto r e l a t i o n -s h i p holds f o r a system of c i t i e s we can formulate an index of u r b a n i z a t i o n f o r the nodal region through ( i ) the index of m e t r o p o l i z a t i o n "a", ( i i ) the constant "A", ( i i i ) the s i z e of the thre s h o l d center (a > 1), and ( i v ) t o t a l population. I t i s suggested that t h i s frequency parameter (metropolization) complements an aggregate u r b a n i z a t i o n index that may be b l u r r e d by va r y i n g i n t e r n a t i o n a l concep-t i o n s of r u r a l - u r b a n d i s t i n c t i o n . 1 96 Steady-state D i s t r i b u t i o n s Some very i n t r i g u i n g e f f o r t s t r e a t c i t y - s i z e patterns as e q u i l i b r i u m s t a t e s of an u n d e r l y i n g s t o c h a s t i c process. The approach t h e r e f o r e , i s t o t a l l y a p r i o r i : we are developing some r e a l world i n t e r p r e t a t i o n to an a b s t r a c t c a l c u l u s i n the hope that the s t r u c t u r e of a r e l a t e d theory may be i m p lied. For t h i s reason i t i s not s u r p r i s i n g that d i v e r s e phenomena (word frequencies i n prose samples, income d i s t r i b u t i o n s , etc.) are g i v e n i d e n t i c a l p r o b a b i l i t y i n t e r p r e t a t i o n s . The arguement may be i l l u s t r a t e d by c o n s i d e r i n g a system of c i t i e s at time " t ^ " d i v i d e d i n t o c l a s s . i t e r v a l s of equal proportionate width (that i s , the logarithms of c l a s s boundaries are evenly spaced). Now, as f o r c e s ( f o r i nstance: migration, investment, technology, e n t r e -preneurship) operate on the i n d i v i d u a l communities through time (consider t g , t ^ , . . ., t R ) , the i n i t i a l c l a s s i n t e r v a l s l i k e l y assume varying proportions of the t o t a l population i n the system. The r e d i s t r i b u t i o n of centers among the c l a s s i n t e r v a l s may be described by t r a n s i t i o n p r o b a b i l i t i e s i n a r e g u l a r matrix (Adelman, 1958). I f we assume: ( i ) That the d i s t r i b u t i o n of percentage changes over a time i n t e r v a l i s the same i n each c l a s s i n t e r v a l ; ( i i ) That these changes remain i n v a r i a n t over a l l time i n t e r v a l s ; then any i n i t i a l d i s t r i b u t i o n of c i t y s i z e s approaches a unique e q u i l i b r i u m state as t n >^ When we d e p i c t the f i r s t p o stulate as a normal d i s t r i b u t i o n (that i s , the frequency d i s t r i b u t i o n s of percentage changes i n pop-u l a t i o n s i z e of small, medium, and large communities a l l approach normal d i s t r i b u t i o n s with the same parameters), then we are e s s e n t i a l l y assuming the law of proportionate e f f e c t t A v a r i a t e subject to a process of change i s s a i d to obey the law of proportionate e f f e c t i f the change i n the v a r i a t e at any step of the process i s a random pr o p o r t i o n of the previous value of the v a r i a t e . ( A i t c h i s o n and Brown, 1957:22). The i m p l i c a t i o n here, of course, i s that population s i z e s are lognormally d i s t r i b u t e d i n the steady s t a t e , when the average number of centers e n t e r i n g each c l a s s i n t e r v a l per time p e r i o d equals the average number departing. The second assumption allows a crude e s t i m a t i o n of the t o t a l p opulation i n the system to be given. For instance, i f we think of a l l the centers having n e a r l y i d e n t i c a l propulations at time " t ^ " , then the spreading out of i n d i v i d u a l populations through the s t o c h a s t i c matrix accounts f o r a growth i n t o t a l population (Adelman, 1958, assumes that the mean value i n each c l a s s i n t e r v a l i s the same a t " t " as i n " t , " ) . We should r e a l i z e , however, n -L that t h i s estimate i s based on rather i n f l e x i b l e p ostulates since ( i ) f u r t h e r entry into the lowest c l a s s and ( i i ) extension beyond the highest c l a s s are disallowed. Simon (1955) avoids the f i r s t r e s t r i c t i o n by a l l o w i n g a steady i n t r o d u c t i o n of new communities over the 98 t h r e s h o l d s i z e . The t h r u s t of h i s argument i s to provide an i n t e r p r e t a t i o n f o r the p r o b a b i l i t y d e n s i t y f u n c t i o n : f ( x ) = f o r x £ T f o r x 2- T . . . (4.11) where £(x,f+l) i s the Beta f u n c t i o n of "x" (a random v a r i a b l e comparable to "p^" i n ( 4 . 3 ) ; that i s , urban s i z e ) and (where "f" > 1 i s a constant that determines the weighting of the new centers but i s r e a l l y employed anala-gously to " a " ) , " f ( x ) " i s the number of c i t i e s of s i z e "x" and "C" i s a normalizing constant. "T" may be i n t e r p r e t e d as the lowest value of "x" f o r which " f ( x ) " exceeds zero. However, where x 0, Simon demonstrates t h a t : f ( x ) = C P ( / + l ) x - ( ^ + 1 ) . . . (4.12) where " P " represents the gamma f u n c t i o n . (4.12) i s c l e a r l y the d e n s i t y f u n c t i o n of the Pareto law. Therefore, by making the conceptual leap to the continuous case once again (where the p r o b a b i l i t y that the v a r i a b l e " x " assumes a given value "x^" i s zero but the p r o b a b i l i t y that i t assumes a value between "x." and "x." may be determined), we may i n t e g r a t e " f ( x ) " between "1" and i n f i n i t y to e s t a b l i s h the d i s t r i b u t i o n f u n c t i o n "F(x)" where: f 0 f o r x < T F(x) = ) l l - T A " ^ f o r x y T . . . (4.13) This i n d i c a t e s the p r o b a b i l i t y that the continuous v a r i a b l e "x" assumes a value i n a s p e c i f i e d c l a s s i n t e r v a l , a 99 c o n d i t i o n that we may employ to determine the number *'N(x)" of centers having populations greater than or equal to "x" (x 7 T): N(x) = . . . (4.14) where "X" i s the t o t a l number of communities i n the system. This d e r i v a t i o n i s c e r t a i n l y analagous to the Pareto r e l a t i o n -ship given i n ( 4 .3 ) . Besides, t h i s i n t e r p r e t a t i o n demon-s t r a t e s p r e c i s e l y how the simple Berry-Garrison (1958a:88-89) a p p l i c a t i o n may be considered a v a l i d d i s c r e t e representation of Simon's sto c h a s t i c model (despite the apparent conceptual flaw i n t h e i r e f f o r t : see footnote 37 of page 88 of Berry and Garrison a r t i c l e ) . The lognormal d i s t r i b u t i o n truncated at point "T" ( A i t c h i s o n and Brown, 1957'87-99) was given i l l u s t r a t i o n i n matrix form above and Adelman (1958:894) provides a l i n k to the Yule i n t e r p r e t a t i o n i n a Markov process v i a ". , . a r e s e v o i r of p o t e n t i a l entrants." While i t i s d i f f i c u l t to s p e c i f i c a l l y compare the p r o b a b i l i t y d e n s i t i e s of the two d i s t r i b u t i o n s through t h e i r parameters, i t appears that the Yule approach gives a superior f i t near the point of truncation. In f a c t , by generating a perfect hypothetical rank-size d i s t r i b u t i o n i t i s simple to demon-st r a t e by graphical method that the best f i t t i n g lognormal t a i l overpredicts the number of communities i n the f i r s t c l a s s i n t e r v a l . 100 W i t h o u t l a b o r i n g t h i s p o i n t , we s h o u l d c o n c l u d e t h a t t h e d i f f e r e n c e s b e t w e e n t h e t w o d i s t r i b u t i o n s may b e n e g l e c t e d i f o u r a n a l y s i s i s r e l a t i v e l y i m p r e c i s e . F o r i n s t a n c e , i n B e r r y ' s (1961) s t u d y o f c i t y s i z e d i s t r i -b u t i o n s many o f t h e s u p p o s e d l o g n o r m a l d i s t r i b u t i o n s a r e c o n c a v e t o t h e s i z e a x i s w h e n p l o t t i n g i s p r o p e r l y s c r u t i n i z e d . W h i l e t h i s s u g g e s t s t h a t t h e Y u l e f r a m e w o r k a c h i e v e s a n i m p r o v e d f i t , c e r t a i n t y i s o b s c u r e d b y f a c t o r s l i k e ( i ) d a t a r e l i a b i l i t y , ( i i ) t h e e f f e c t s o f u n e q u a l ( i n a p r o p o r t i o n a t e s e n s e ) s i z e i n t e r v a l s o n s m a l l s a m p l e s p a c e s , ( i i i ) t h e o p e n c l a s s i n t e r v a l f o r t h e v e r y l a r g e s t c o m m u n i t i e s , a n d ( i v ) t h e n a t u r e o f t h e d i s t r i b u t i o n p a r a m e t e r s . ( I t s e e m s t h a t t h e l o g n o r m a l p r o v i d e s a s u p e r i o r f i t f o r t h e h y p o t h e t i c a l r a n k - s i z e d i s t r i b u t i o n w h e n t h e e x p o n e n t " b " i n (4,1) i s l e s s t h a n u n i t y ) . C o m p a r i s o n o f D i s t r i b u t i o n s T o a c e r t a i n e x t e n t , t h i s t o p i c h a s b e e n c o v e r e d i n v a r i o u s p a r t s o f t h e p r e v i o u s d i s c u s s i o n . H o w e v e r , t h e r e do a p p e a r t o e x i s t a d d i t i o n a l m i s c o n c e p t i o n s i n t h e l i t e r a t u r e c o n c e r n i n g t h e s i m i l a r i t y o f f r e q u e n c y d i s t r i -b u t i o n s . T o b e g i n w i t h , i t s e e m s we m u s t q u a l i f y a s u p e r -f l u o u s d i c h o t o m y o f t h e p r i m a c y a n d r a n k - s i z e t e r m s t h a t i s w i d e l y h e l d . The d i s t i n c t i o n , t h o u g h , s e e m s t o a r i s e f r o m J e f f e r s o n ' s (1939) a p p e a l f o r - a u n i q u e n e s s t h e s i s . M o r e r e c e n t s o c i a l s c i e n t i s t s e m p h a s i z e r e g u l a r i t i e s i n s t e a d , i n h o p e o f r e a l i z i n g g e n e r a l l a w s t a t e m e n t s o f human 101 behaviour. In any case, adherence to the idea of a unique primate c i t y precludes r a t i o n a l explanation of i t s economic a t t r i b u t e s . The persistence of t h i s d i v i s i o n seems r e l a t e d to the n o t i o n that primate c i t i e s are". . . a s s o c i a t e d with overurbanization and superimposed c o l o n i a l economies i n underdeveloped cou n t r i e s or with p o l i t i c a l - a d m i n i s t r a t i v e c o n t r o l s i n indigenous subsistence and peasant s o c i e t i e s . . . and rank-size r e l a t i o n s d e p i c t complex c i t y systems i n more advanced nations (Berry, 1961:574). Surely, no g e n e r a l i t y i s l o s t by t r e a t i n g primacy as but a d e v i a t i o n from the o v e r a l l p a t t e r n of c i t y s i z e s i n a system. I t may be accounted f o r i n two general ways$ ( i ) By the emergence of one great " c a p i t a l " as i n J e f f e r s o n ' s extreme conceptionj and, ( i i ) By a d e f i c i e n c y or v o i d of intermediate s i z e centers so t h a t a group of l a r g e centers are d i s t i n -guished from another group of small centers. The point i s that even i f we p r e f e r a s t r i c t l y i n d u c t i v e or e m p i r i c a l approach to the question of c i t y s i z e s , then the idea of primacy cannot be detached from the whole d i s t r i b u t i o n of community s i z e s . This i m p l i e s , of course, that any rigorous treatment of the problem must employ double-logarithmic p l o t t i n g ( i n e i t h e r s i z e versus rank or s i z e versus cumulative percentage form) with a t t e n t i o n devoted to c o r r e l a t i o n s and r e s i d u a l s of the l i n e a r regres-s i o n model. 102 Besides, inferences based upon l i m i t e d sample spaces (C . T. Stewart, 1958; Berry, 1961; Mehta, 1964; Linsky, 1965* Rosing, 1966) must be treated with considerable reserve. Unfortunately, Berry appears to be alone i n recognizing the importance of t h i s issue. Investigators also seem hesitant about accepting the s i g n i f i c a n c e of the exponent "b" i n the rank-size formulation. Since the exponent v a r i e s with the type of data c o l l e c t e d ( A l l e n , 1954) and i t i s given some i n t e r -p r e t a t i o n through the h i e r a r c h i a l and Simon models, i t would appear that the acceptance a p r i o r i of a "b" value of u n i t y (Vapnarsky, I969) i s rather questionable. Another most meaningful point should be r e i t e r a t e d at t h i s time. The use of grouped data f a c i l i t a t e s a n a l y s i s to a great degree but tends to obscure the more t r a d i t i o n a l form of primacy (that i s , the " c a p i t a l " c i t y ) through an open c l a s s i n t e r v a l f o r the la r g e s t population s i z e s . While we should agree with Lasuen, Lorca, and Oria (1967) that Berry's lognormal technique weights the e f f e c t of the large number of small centers i n the system, we cannot agree that i t adequately describes the deviation of the very greatest community. (Notice that the p l o t t i n g s f o r the sample of Spanish c i t i e s are done i n c o r r e c t l y i n both a r t i c l e s ) . A l l i n a l l , i t may be best to r e t a i n a consistent approach of di s p l a y i n g the frequency d i s t r i -butions throughout any single study. 103 I n c o n c l u s i o n we n o t e t h a t i n v e s t i g a t i o n o v e r l o o k s v a r i a t i o n ( i n t e r c e p t s a n d s l o p e s ) among s y s t e m s t h a t a r e a l l s u p p o s e d l y l o g n o r m a l l y c h a r a c t e r i s t i c . W h i l e A i t c h i s o n a n d B r o w n (1957:94) e m p h a s i z e t h e d i f f i c u l t i e s o f e s t i m a t i n g p a r a m e t e r s f r o m g r o u p e d d a t a , we may f o r f e i t t h i s o p e r a t i o n a n d s t i l l g a i n i n s i g h t "by s t u d y i n g t h e p r o p e r t i e s o f t h e g r a p h s a l o n e . I n f a c t , t h i s i s a m a j o r i s s u e i n t h e n e x t c h a p t e r o f t h e d i s c u s s i o n . I n t e r p r e t a t i o n a n d E x p l a n a t i o n A c c e p t a b l e e x p l a n a t i o n o f t h e s e e m p i r i c a l r e g u -l a r i t i e s i s a r a t h e r d e b a t a b l e t o p i c b u t t h e g e n e r a l a r g u m e n t s e e m s t o r e v o l v e a b o u t t h e r o l e a s s i g n e d t o t h e o r y i n e x p l a i n i n g r e a l i t y . T h e o b j e c t i v e s o f t h i s s e c t i o n a r e ( i ) t o e v a l u a t e t h e n a t u r e o f e x p l a n a t i o n f o u n d i n s t o c h a s t i c r e a s o n i n g , ( i i ) t o c o m p l e m e n t t h i s l i n e o f t h i n k i n g w i t h c e r t a i n n o t i o n s f r o m g e n e r a l s y s t e m s t h e o r y , a n d ( i i i ) t o h o p e f u l l y c l a r i f y t h e r e l a t i o n b e t w e e n t h e s e p r o b a b i l i s t i c u n i - s i z e a r g u m e n t s a n d t h e p r o p e r t i e s o f d e t e r m i n i s t i c h i e r a r c h i a l m o d e l s . S t o c h a s t i c A p p r o a c h e s B e r r y a n d G a r r i s o n (1958ai90) s u m m a r i z e t h e c a s e o f s t o c h a s t i c a r g u m e n t i n t h e c l o s i n g p a g e s o f t h e i r o f t e n c i t e d a r t i c l e a b o u t r a n k - s i z e r e l a t i o n s h i p s ! " F o r o n e t h i n g , a p r o b a b i l i s t i c e x p l a n a t i o n i n some s e n s e r e f e r s t o t h e p r e s e n c e o f a n i n f i n i t e n u m b e r o f c a u s e s a n d t h e a b i l i t y t o p r e d i c t i n t h e s e t e r m s i s n o t e n o u g h ; 104 we w i s h e x p l a n a t i o n s v i a b l e i n e x p l i c i t w a y s w i t h i n a b r o a d t h e o r e t i c a l c o n t e x t . " M a p p i n g t h e p r o p e r t i e s o f t h e r e a l w o r l d i n t o t h e l a w o f p r o p o r t i o n a t e e f f e c t i s a n i n t e r e s t i n g e x e r c i s e , b u t t h e t h e o r y t h a t i s p o r t r a y e d b y t h e a p r i o r i m o d e l i s n o t c l e a r a t a l l . H o w e v e r , a n i n t e r p r e t a t i o n o f t h e s t a t i s t i c a l s t a t e m e n t d o e s p r o v i d e some h e l p f u l i n s i g h t s . The f e e l i n g s o f S i m o n (1955:43?) a r e t h a t t h e Y u l e d i s t r i b u t i o n . . . w o u l d h o l d i f t h e g r o w t h o f p o p u l a t i o n w e r e d u e s o l e l y t o t h e n e t e x c e s s o f b i r t h s o v e r d e a t h s , a n d i f t h i s n e t g r o w t h w e r e p r o p o r t i o n a l t o p r e s e n t p o p u l a t i o n . T h i s a s s u m p t i o n i s c e r t a i n l y s a t i s f i e d a t l e a s t r o u g h l y . M o r e o v e r , i t n e e d n o t h o l d f o r e a c h c i t y , b u t o n l y f o r t h e a g g r e g a t e o f c i t i e s i n e a c h p o p u l a t i o n b a n d . F i n a l l y , t h e e q u a t i o n w o u l d s t i l l b e s a t i s f i e d i f t h e r e w e r e n e t m i g r a t i o n t o o r f r o m c i t i e s o f p a r t i c u l a r r e g i o n s p r o v i d e d t h e n e t a d d i t i o n o r l o s s o f p o p u l a t i o n o f i n d i v i d u a l c i t i e s w i t h i n a n y r e g i o n w e r e p r o p o r t i o n a l t o c i t y s i z e . " W a r d (1962) e x t e n d s t h i s t h e s i s i n a t y p i c a l P r e d i a n ( i n f o r m a t i o n a n d a b i l i t y t o a c t ) d i s c u s s i o n o f a g g r e g a t e m a r k e t e x p a n s i o n o p p o r t u n i t i e s t h a t a r e r e a l i z e d i n t h e l o n g r u n . T h e P a r e t i a n d i s t r i b u t i o n i s a t t a i n e d w h e n t h e r e l a t i v e f r e q u e n c y o f o c c u r i n g o p p o r t u n i t i e s i s r a n d o m l y p r o p o r t i o n a t e t o t h e s i z e o f u r b a n m a r k e t s . He a l s o p r o v i d e s s e v e r a l q u a l i f i c a t i o n s o f S i m o n ' s m o d e l s ( i ) T h e s t o c h a s t i c b a s i s r e q u i r e s t h a t c i t i e s i n a p a r t i c u l a r c l a s s i n t e r v a l t h a t a r e b e c o m i n g p r o p o r -t i o n a t e l y l a r g e r m u s t be m a t c h e d b y a s i m i l a r n u m b e r b e c o m i n g p r o p o r t i o n a t e l y s m a l l e r ; ( i i ) M i g r a t i o n f r o m r u r a l a r e a s o r a b r o a d i s n o t 105 accomodated (more than l i k e l y t h i s i s d i r e c t e d toward the l a r g e s t c enters); ( i i i ) The usefulness of the approach i s hindered by the nature of the data: c i t y s i z e s i n metropolitan or conurbation form have c h a r a c t e r i s t i c a l l y lower exponents than i n corporate form - the a p p l i c a b i l i t y of the Yule d i s t r i b u t i o n , though, i s r e s t r i c t e d to those values greater than or equal to u n i t y . Unfortunately, there appears to be at best very sketchy support f o r the type of growth postulated i n the s t o c h a s t i c matrix: Madden's (1956) study of the s t a b i l i t y of urban growth i n the United States i n d i c a t e s that the law of proportionate e f f e c t may w e l l be approximated i n the r e a l world when we consider the growth of a l l centers (no c l a s s i n t e r v a l s ) through equal time periods. Obviously, however, the i n v e s t i g a t i o n refutes q u a l i f i c a t i o n ( i ) above: t h i s points to a fundamental d i s t i n c t i o n between the f r e -quency d i s t r i b u t i o n s of percentage changes i n size of communities and, say, firms (Simon and B o n i n i , 1958) i n an industry. There i s c l e a r l y l e s s tendency f o r urban centers to take proportionate losses i n population (espec-i a l l y the large centers) than f o r firms to take s i m i l a r losses i n employees, value added, etc, Simon's model seems to i n d i c a t e a better d e s c r i p t i o n of the data than of the processes involved. In any case, the essence of the problem from a s c i e n t i f i c viewpoint i s that the p r o b a b i l i s t i c scheme 106 avoids introducing those f a c t o r s that supposedly cause the s t o c h a s t i c mechanism to operate. The most c r i t i c a l of these i s c e r t a i n l y the notion of distance or separation, f o r only through study of t h i s v a r i a b l e can we understand why and how urban communities are f u n c t i o n a l l y r e l a t e d . Or, to phrase the point d i f f e r e n t l y , the stoc h a s t i c argument i s a t o t a l l y aggregated one i n the sense that i t cannot a l l o c a t e weightings f o r d i f f e r e n t orders of opportunities. Without knowledge of how i n d i v i d u a l elements are r e l a t e d i t becomes impossible to make pr e d i c t i o n s about t h e i r a t t r i b u t e s at a l a t e r time. However, such discussions that simply weigh the d i f f e r e n t i a l merits of p o s i t i v i s t (or r a t i o n a l i s t ) and more conventional (or symbolic) viewpoints toward theory-r e a l i t y i n t e r r e l a t i o n s (Lukermann, 1961) tend to obscure an unnoticed d i s t i n c t i o n between the s t a t i s t i c a l and s p a t i a l economic arguments. To i l l u s t r a t e , the law of proportionate e f f e c t t y p i f i e s a model dealing with i n d i v -i d u a l population members ( r e c a l l Simon's 1955 i n t e r p r e -t a t i o n that the p r o b a b i l i t y the "k+l"st person i s found i n a center of si z e "x u i s proportional to "x f * ( x ) " , the t o t a l population of communities of t h i s s i z e ) , while the C h r i s t a l l e r or Loschian models deal with population groups (the C h r i s t a l l e r model i s depicted by population subsets associated with baskets of goods). In terms of systems, the Yule d i s t r i b u t i o n portrays random behaviour at a high r e s o l u t i o n l e v e l while c e n t r a l place theory provides p r e d i c t i o n at a lower l e v e l (Burton, I 9 6 3 discusses t h i s general point with regard to q u a n t i f i c a t i o n ) . In modeling terms, the former approach i s d e s c r i p t i v e , micro, and p r o b a b i l i s t i c while the l a t t e r i s a n a l y t i c a l , macro, and d e t e r m i n i s t i c . While t h i s scale dichotomy does r e l a x p h i l o s o p h i c a l debate i t cannot r e a l l y refute the superior value of the economic argument as a methodological device. General Systems Theory General systems t h i n k i n g e s s e n t i a l l y suggests a methodology or viewpoint focussing on properties common to a l l types of systems. I t i s an approach that favors studying the t o t a l i t y of r e l a t i o n s among elements and emphasizes q u a l i t i e s l i k e wholeness and organization (Rapoport, 1 9 6 6 ) . H a l l and Fagen (1956) s t i p u l a t e two macroscopic properties of systems that appear relevant to the present discussion. I f we r e c a l l our simple i l l u s t r a t i v e model of how an expanding space-economy induces an urban s t r u c t u r e , then t h i s i s c a l l e d progressive systematization, since we witness: ( i ) Strengthening of p r e - e x i s t i n g interdepend-encies; ( i i ) Development of r e l a t i o n s among members previously unrelated; ( i i i ) A ddition of parts and r e l a t i o n s to the e x i s t i n g system. 108 Besides, c e n t r a l i z a t i o n or dominance by a l e a d i n g member (the primate center) i s a common t r a i t that may accompany an increase i n the sum of r e l a t i o n s . Beckmann (1958) suggests that coincidence of allometry and the Fareto d i s t r i b u t i o n (see c o r r e c t e d form i n (4 .6 ) above) may mean some optimal a s s o c i a t i o n e x i s t s between po p u l a t i o n i n the l e a d i n g part and population throughout the e n t i r e c i t y system. In our r a t h e r f u n c t i o n a l view of the urban system, we s t r e s s i t s behaviour (flows, responses, etc.) both i n t e r n a l l y and through t r a n s a c t i o n s with i t s environment (Harvey, 1969*456 ) . We simply consider that environment as a higher order system (a socio-economy of i n d i v i d u a l consumers) from which the c i t y system i s conveniently, yet n e c e s s a r i l y , a bstracted. From our s t r i c t l y economic i n t e r p r e t a t i o n , we may determine (at l e a s t q u a l i t a t i v e l y ) the s p a t i a l extent of the environment v i a c o n s i d e r a t i o n of the p r o p o r t i o n of demands exercised through l o c a l or f o r e i g n markets. Needless to say, as progressive system-a t i z a t i o n occurs i n the urban system, the l o c a l or domestic p o r t i o n of the environment becomes r e l a t i v e l y more important. Systems are c l a s s i f i e d into c l o s e d ^ or open types according to the nature of energy exchange (information, commodities, innovations, c a p i t a l , etc.) with t h i s -^Note that closure i n t h i s i n t e r p r e t a t i o n i s s i g n i f -i c a n t l y d i f f e r e n t from i t s meaning - e a r l i e r i n the discussion? there, i t r e f e r r e d to s p a t i a l i s o l a t i o n which accounts f o r only a part of the t o t a l environment. 109 e n v i r o n m e n t . O b v i o u s l y t h e h i e r a r c h i a l s t r u c t u r e o f a c i t y s y s t e m d e n o t e s a n o p e n c o n d i t i o n , s i n c e i t i s d e r i v e d f r o m a c o m p e t i t i v e p r o c e s s d i r e c t l y r e l a t e d t o t h e w a n t s a n d n e e d s o f i n d i v i d u a l c o n s u m e r s . B e r r y (196?:?6-78) p r e s e n t s some i n t e r e s t i n g i d e a s c o n c e r n i n g t h i s n o t i o n . M o s t i m p o r -t a n t , t h o u g h , i s t h e c o n t e n t i o n t h a t : " . . . l i v i n g s y s t e m s c a n b e d e f i n e d a s h i e r a r c h i a l l y o r g a n i z e d o p e n s y s t e m s , m a i n t a i n i n g t h e m s e l v e s , o r d e v e l o p i n g t o w a r d a s t e a d y s t a t e . " ( v o n B e r t a l a n f f y , 1962^7). S p a t i a l c o m p e t i t i o n b y u r b a n c o m m u n i t i e s i s a g o o d e x a m p l e o f a p r o c e s s t h a t t e n d s t o m a i n t a i n a n e q u i l i b r i u m s t a t e ( h o m e o s t a s i s ) d e s p i t e e n v i r o n m e n t a l d i s t u r b a n c e s . H o w e v e r , s u c h a n a r g u m e n t i s r e s t r i c t e d t o o n l y c u l t u r e s t h a t v a l u e s u c h c o m p e t i t i o n a n d t o t h o s e t y p e s o f a c t i v i t i e s t h a t r e l a t e t o a d o m e s t i c e n v i r o n m e n t ( t h a t i s , w h e r e i n p u t p r i c e s v a r y r e l a t i v e l y l i t t l e i n s p a c e ) . U r b a n g r o w t h a t t r i b u t e d m o s t l y t o d e m a n d s p l a c e d i n a m o r e d i s t a n t p a r t o f t h e e n v i r o n m e n t ( f o r e i g n m a r k e t s ) i s d i r e c t e d b y l i n k a g e s t h a t a r e r a t h e r i n d e p e n d e n t o f t h o s e b e t w e e n e l e m e n t s i n t h e s y s t e m . When a l a r g e p r o p o r t i o n o f t h e t o t a l m a r k e t i s n o t i n t h e c i t y s y s t e m i t s e l f , c e n t r a l -i z a t i o n o r p r i m a c y ( a t t h e n a t i o n a l a n d r e g i o n a l l e v e l s ) i s a n a t u r a l o c c u r r e n c e . I n o t h e r w o r d s , n e g a t i v e f e e d b a c k s a n d t h e u r b a n h i e r a r c h y e v o l v e t o g e t h e r b u t w h e r e s u c h a h i e r a r c h y d o e s n o t e x i s t ( a s i n t h e m o s t p r i m i t i v e s p a c e -e c o n o m i e s ) , p o s i t i v e f e e d b a c k s a n d c o n c o m i t a n t c u m u l a t i v e -c a u s a t i o n e x p a n s i o n a r e p o s s i b l e i n p a r t i c u l a r s u b s y s t e m s 110 (Haruyama, 1963). Only as the l o c a l environment i n c r e a s i n g l y c o n t r o l s the t o t a l market, and planning favors r e g i o n a l convergence, may these p o s i t i v e feedbacks be checked. Information i s the term we employ to describe the o r g a n i z a t i o n of a system, I t s thermodynamic counterpart, entropy, i s s a i d to increase when a system becomes more randomized (information and negative entropy are analagous). Entropy i s maximized i n an urban system when community s i z e s d i f f e r only by chance. To c l a r i f y t h i s argument we consider the s t a t i s t i c a l d e f i n i t i o n of entropy ( K l i r and Valach, 1 9 6 ? i 6 l ) : I f out of "n" events, each can occur with the n p r o b a b i l i t i e s 0,, 0 9. . . . 0 , where 0. = 1 n i = l 1 ( i . e , some of the events do take p l a c e ) , then the n formulation H = - 27 0. l o g 0. i s c a l l e d entropy. i - l 1 a 1 I t should be obvious that when a l l "0^" are i d e n t i c a l , "H" i s maximized. Now we may i n t e r p r e t ( i n d e s c r i p t i v e terms at l e a s t ) the d i s t r i b u t i o n of the t o t a l urban population "P" amongst "n" centers i n entropy terms. By c o n s i d e r i n g "0^" as the r a t i o of a community's population "p^" to the t o t a l "P" we may i l l u s t r a t e t h a t i ( i ) Minimum entropy occurs when a l l "P" i s r e s i d e n t of one community and H = 0; ( i i ) Maximum entropy occurs when p^ = P/n and H = l o g a n . I l l However, i n a c i t y system maximum o r g a n i z a t i o n i s a t t a i n e d t h r o u g h the s p a t i a l h i e r a r c h y and the t r i v i a l case where H = 0 i s d i s m i s s e d . F o r i n s t a n c e , i f we know the p o p u l a t i o n o f a c e n t e r and i t s l e v e l i n the h i e r a r c h y , we c a n compute the p o p u l a t i o n s o f a l l the c e n t e r s i n the s y s t e m . I n the r e a l w o r l d , h i e r a r c h i e s are n e v e r p e r f e c t n o r complete h u t i t i s h y p o t h e s i z e d by B e r r y (196?i71) t h a t the r a n k - s i z e r u l e i s the s t e a d y s t a t e t h a t b a l a n c e s h i e r a r c h i a l o r g a n -i z a t i o n w i t h r a n d o m i z a t i o n due t o chance l o c a l v a r i a b i l i t i e s . O b v i o u s l y i n terms o f p o p u l a t i o n f i g u r e s a l o n e , a more o r g a n i z e d s t a t e may be a t t a i n e d t h e n t h r o u g h h i e r a r c h i a l c o n s t r a i n t b u t t h e o r y p r e c l u d e s such a c o n d i t i o n . T h i s d e m o n s t r a t e s p r e c i s e l y why we need a g u i d i n g t h e o r y i n e x p l a n a t i o n , f o r a s t a t e o f p e r f e c t p r i m a c y r e p r e s e n t s maximum o r g a n i z a t i o n i n a p o p u l a t i o n system b u t c e n t r a l p l a c e t h e o r y and i t s r e l a t e d models t e l l us t h a t t h i s i s i r r e l e v a n t i n an economic s y s t e m . O n l y t h r o u g h t h e o r y are we c e r t a i n o f e l i m i n a t i n g a b s u r d i t i e s . T h e r e f o r e i t seems more n a t u r a l t h a t we s h o u l d speak o f a c o n d i t i o n o f d e s i r a b l e e n t r o p y , but n o t o f minimum o r maximum e n t r o p y when we speak o f c i t y h i e r a r c h i e s and the r a n k - s i z e r u l e . However, i t may be u s e f u l t o p r o v i d e an e x p r e s s i o n o f e n t r o p y f o r a d i s c r e t e r a n k - s i z e d i s t r i -4 b u t i o n o f t o t a l p o p u l a t i o n " P " : Our d e r i v a t i o n c o n t r a d i c t s a f o r m u l a t i o n d e v i s e d by von F o e r s t e r ( I 9 6 6 ) , r e s t a t e d by C u r r y ( I 9 6 3 ) and i n c l u d e d i n r e v i e w s by B e r r y ( 1 9 6 4 ) and O l l s o n ( 1 9 6 6 ) . W h i l e 112 H = log l b + 4- log 2 b + 2DP + log n b log P i p F u l f i l l m e n t of t h i s steady state through time requires that as "P" and " n " grow, " H " remains r e l a t i v e l y s table . T h i s , i n f a c t , hypothesizes that the p r i n c i p l e s of e q u i f i n -a l i t y are met and population values are independent of central conditions ( r e c a l l that t h i s i s a natural r e s u l t of a stochastic interpreta t ion) . by considering the difference between independent settlements i n a backward economy and interdependent communities i n a progressive economy. The i n i t i a l case represents a our interpretations of entropy are rather d i f f e r e n t , the i n i t i a l e f f o r t suffers i n several respectsJ ( i ) the combinatorial r e d e f i n i t i o n of entropy means that the sum of the logarithms i n the second term of t h e i r expression must be minimized and not maximized, and ( i i ) correct use of Lagrangian m u l t i p l i e r s leads to this minimization. I t i s not c lear at a l l how entropy i s maximized through the rank-size r u l e . Note, too, that i n our formulation above "p " i s the same as " p „ 0 " employed e a r l i e r i n th is chapter and each i s not to be confused with the notation used i n the previous chapters! " p , " , of course, i s now the largest center i n the system. The meaning of the entropy approach may be enhanced 113 w e a k l y l i n k e d a n d r e l a t i v e l y c l o s e d s y s t e m t h a t p o s s e s s e s maximum e n t r o p y : a t t r i b u t e s o f c e n t e r s r e m a i n u n c o r r e l a t e d w i t h t h e i r s i z e . I n t e g r a t i o n o f t h e s p a c e e c o n o m y , o n t h e o t h e r h a n d , i s p e r s i s t e n t l y b r i n g i n g i n f o r m a t i o n i n t o t h e s y s t e m t h r o u g h t h e f r a m e w o r k o f e c o n o m i c c o m p e t i t i o n a n d s p e c i a l i z a t i o n . V a p n a r s k y (1969} p r e s e n t s a n i n t e r e s t i n g a r g u m e n t a l o n g t h e s e l i n e s , w h i l e f o c u s s i n g o n t h e e n v i r o n -m e n t o f t h e s y s t e m , t h a t d e p i c t s f o u r g e n e r a l s t a g e s i n t h i s i n t e g r a t i o n p r o c e s s . U n f o r t u n a t e l y , i t i s d i f f i c u l t t o o b j e c t i v e l y s p e c i f y t h e e f f e c t s o f a d d e d p o p u l a t i o n o n t h e e n t r o p y v a l u e . I f a l l c e n t e r s g r o w a t a b o u t t h e same p e r c e n t a g e r a t e t h e n e n t r o p y r e m a i n s c o n s t a n t i n t h e l o n g r u n . H o w e v e r , t h e v i g o r o u s g r o w t h o f s m a l l c e n t e r s r e l a t i v e t o l a r g e c e n t e r s may i n d i c a t e a n i n c r e a s e i n e n t r o p y a n d , v i c e v e r s a , t h e c o n c e n t r a t i o n o f p o p u l a t i o n i n l a r g e r c e n t e r s i m p l i e s a d e c r e a s e i n e n t r o p y . T h e A g g r e g a t e M o d e l R e c o n s i d e r e d I n d i f f e r e n t p a r t e o f t h e t h e s i s we h a v e e m p h a s i z e d t h a t u n i q u e s i z e c l a s s e s o f u r b a n c e n t e r s c a n n o t b e j u s t i f i e d o n a p o p u l a t i o n b a s i s a l o n e b u t o n l y t h r o u g h u n d e r s t a n d i n g how e c o n o m i c i n d i v i s i b i l i t i e s a n d p o p u l a t i o n b e c o m e i n t e r t w i n e d . I t i s n o t u n e x p e c t e d t h a t r a n d o m v a r i a t i o n o f i n d i v i d u a l c o m m u n i t y p o p u l a t i o n s may e x p r e s s a s i z e c o n t i n u u m w i t h i n a n u r b a n s y s t e m d e s p i t e s t r o n g h i e r a r c h i a l q u a l i t i e s f o r t h e e n t i r e s e t . To h o p e f u l l y i l l u s t r a t e how g r o w t h may be c o n s i d e r e d t o i n d u c e s u c h a c o n t i n u u m , l e t ' s r e c a l l 114 Beckmann's simple model. The c h a r a c t e r i s t i c feature of t h i s d e t e r m i n i s t i c " s " model i s the use of a basic progression component j—^ + ^ that relates populations on adjacent h i e r a r c h i a l l e v e l s . Within t h i s r i g i d framework, s p a t i a l v a r i a t i o n of population s i z e s i s severely r e s t r i c t e d . As a r e s u l t , we mentioned i n the previous chapter that i t i s h e l p f u l to view the " s " component as a random v a r i a b l e about the mean + 1 . In a growth context, l e t ' s begin by considering a vector 1^ that represents urban populations at time " t ^ " i n a c i t y system. I f r e l a t i v e growth i n the f i r s t time i n t e r v a l i s , then populations at time " t 2 " are expressed as: p2 = P l rfk + 1 * ' ' ( 4 ' 1 6 ) In t h i s manner, we can show: m-l . . . (4.1?) Pm ( r f k + ^ which c l e a r l y resembles (3.15)• Now, assuming that: ( i ) Values i n "p^" d i f f e r only by chance? ( i i ) "P m" 1 S lognormally d i s t r i b u t e d , a property that i s suggested by Beckmann (1958) and substantiated to a reasonable degree v i a p l o t t i n g ; ( i i i ) The time periods are reasonably comparable; " s " then the random v a r i a b l e ^ + 1 may also be lognormally d i s t r i b u t e d ( Aitchison and Brown, 1957:12). Unfortunately, t h i s discussion s u f f e r s i n two respects. I t i s d i f f i c u l t 115 to i n t e r p r e t independent growth i n the successive time i n t e r v a l s , a c o n d i t i o n that necessitates lognormal d i s t r i -bution of the component. Besides, the idea c o n f l i c t s with those notions of cross-time analysis that e s s e n t i a l l y " s " assume + 1 i s normally d i s t r i b u t e d . Nevertheless, the approach provides a more su i t a b l e framework f o r describing urban growth than can be attained v i a e q u i l i b r i u m adjustments alone. C l e a r l y , t h i s i s a t h e o r e t i c a l t o p i c that deserves increased a t t e n t i o n . While i t i s c e r t a i n that c e n t r a l place theory has a somewhat l i m i t e d domain i n the r e a l world, at t h i s time we have no other a n a l y t i c statements to d i r e c t explanation of the nature of urban systems. However, keen awareness of the fundamentals of that theory gives valuable i n s i g h t to a rigorous methodology f o r empirical research. In the f o l l o w i n g chapter we take a more s e l e c t i v e viewpoint and, a f t e r attaching various growth f a c t o r s to the c e n t r a l place framework, attempt to explain or refute c e r t a i n inductive generalizations concerning primacy and rank-size r e l a t i o n s h i p s . Chapter 5 CHANGING PATTERNS OF INTERURBAN STRUCTURE This chapter i s l a r g e l y devoted toward sketching the i n t e r r e l a t i o n s of various growth f a c t o r s and i n t e r -urban s t r u c t u r e , Beckmann's simple h i e r a r c h i a l model provides the framework f o r a type of comparative-statics a n a l y s i s , where optimal e q u i l i b r i u m conditions are assumed to p e r s i s t both before and a f t e r an impact i s introduced (Nourse, 1968:273). Besides, the concomitant e f f e c t of structure upon growth i s set w i t h i n a more d e s c r i p t i v e d i s c u s s i o n of the d i f f u s i o n process. A graphic i n t e r p r e t a t i o n of the simple model i s outl i n e d near the chapter's end. With t h i s idea i n mind, there are some attempts given toward a l i g n i n g or q u a l i f y i n g inductive g e n e r a l i z a t i o n s with the c e n t r a l place p r i n c i p l e s . (Burton, I963 emphasizes r e l a t i n g hypotheses to a developing body of theory). The methods used i n the argument, however, require considerable refinement before we can e s t a b l i s h strong statements r e l a t i n g d i s t r i b u t i o n patterns and independent f a c t o r s . 116 117 Growth i n a Theoretical Context The properties of the aggregate model lend i n s i g h t to the nature of change i n urban structure as fashioned by population and economic growth i n a region. The p a r t i c -u l a r drawbacks of t h i s t h e o r e t i c a l argument involve the assumption that development may be characterized i n a d e t e r m i n i s t i c model that precludes d i s e q u i l i b r i u m . Our procedure i s to study each f a c t o r i n i s o l a t i o n (that i s , holding a l l other f a c t o r s constant) and then attempt to conceptually integrate t h e i r diverse e f f e c t s . Population To begin with, l e t ' s consider a c i t y system that evolves on an i s o t r o p i c p l a i n so that population growth i s s p a t i a l l y d i r e c t e d by the e x i s t i n g structure. More p r e c i s e l y , we suppose that ( i ) centers a t t r a c t population increments proportional to t h e i r i n i t i a l s i z e s and ( i i ) s p a t i a l extension (or contraction) of the e n t i r e system proceeds symmetrically about the dominant center. Of course, our a n a l y s i s i s characterized by viewing func-t i o n a l d i f f e r e n t i a t i o n through bundles of commodities since we assume C h r i s t a l l e r i a n agglomeration. Now we can analyse the e f f e c t s of population growth by r e f e r r i n g back to Figure 1 i n the second chapter. Curve "D2" represents the aggregate demand facing a firm that i s earning normal p r o f i t s i n the competitive single good case. Under multiple good conditions, approximate 118 tangency of the average cost and demand curves may only characterize those firms providing h i e r a r c h i a l marginal goods (except, of course, where low order functions are given i n r e l a t i v e l y high l e v e l centers that possess r a t h e r s u b s t a n t i a l intraurban markets). In any case, f o r the remaining submarginal goods i n the same basket, threshold requirements are l e s s and excess p r o f i t s are l i k e l y greater f o r the r e l a t e d firms. A uniform increase i n population s h i f t s the demand curve "D2" to the r i g h t ( f o r i l l u s t r a t i v e purposes, say to "D^") and allows excess p r o f i t s to be attained f o r any p a r t i c u l a r commodity that i s i n i t i a l l y h i e r a r c h i a l marginal. Depending upon the amount of population increase, t h i s surplus of purchasing power may or may not be s u f f i c i e n t l y large enough to induce entry of a competing firm(s) o f f e r i n g t h i s same commodity at each center on t h i s l e v e l . However, c e r t a i n goods that were formerly supramarginal and only produced at a higher l e v e l probably f i n d a s u f f i c -i e n t threshold base i n these lower l e v e l places while other submarginal commodities surely stimulate a d d i t i o n a l intraurban competition.' 1' Considering a l l h i e r a r c h i a l l e v e l s together, we observe that bundles of goods become redefined according to the emergence of new h i e r a r c h i a l marginal goods due to increases of i n t e r s t i t i a l purchasing power. "*"The same good may be considered supramarginal or submarginal depending upon which endpoint of the basket we r e l a t e to. The most noticeable e f f e c t , then, of population growth i s a tendency toward a reduction of f u n c t i o n a l concentration i n the c i t y system. I f new functions are not added to the o r i g i n a l "M" baskets, density increases may lead to the spreading of these bundles over more than "M" l e v e l s . More s i g n i f i c a n t , perhaps, i s the r e s u l t of t h i s d e c e n t r a l i z a t i o n : ( i ) S i m i l a r numbers of functions may s h i f t downward i n each h i e r a r c h i a l l e v e l , but f i r m m u l t i p l i c a t i o n i s r e l a t i v e l y more ra p i d i n the lower l e v e l s (see Parr and Denike, 1 9 7 0 ) ; ( i i ) I d e n t i c a l bundles may become c h a r a c t e r i s t i c of the two smallest s i z e classes. Besides, the simple model does q u a l i f y our under-standing of how urban structures are transformed, at l e a s t i n the more advanced regions. By s t r e s s i n g that changes i n the s i z e and frequency d i s t r i b u t i o n of centers at higher l e v e l s depend upon the entry thresholds f o r places at a l l lower l e v e l s , i t eliminates the more i n t u i t i v e views such as: " I f the population should only double, there would a r i s e twice as many c i t i e s i n each rank order,. „ . " (Nourse, 1 9 6 8 : 2 1 0 ) . R e c a l l i n g that the correct sequence f o r s a t e l l i t e c i t i e s i s s(s + l ) n ~ 2 ( s i z e classes greater than the f i r s t ) , we can r e i n t e r p r e t "P^t-^"-, the t o t a l population served i n the c e n t r a l place system at time "t - ^ " p r i o r to population growth, i n urban and r u r a l compenents: 120 M w o PM <V = Pin ( V + s 2? (s + l ) n " 2 VSA.n¥l(t1) n=2 + ( s + I ) 1 ' 1 - 1 v± . . . (5 .1) Now i f other factors are constant, a r e l a t i v e l y small population increase leads to the beginning of a "M+l" l e v e l hierarchy. Hence "P^]^"^) " d 6 1 1 0 " ^ 3 population i n the same system a f t e r t h i s increase (Pj^+i ( ) = v PM ^ 1 ^ ' where "2P^(t-^)" simply means the o r i g i n a l population i s doubled) and the number " f " of centers at each l e v e l of the transformed system i s derived from: f<Pm + i ' V = Tiirisj- f ( W . . . (5.2) and f ( pl» V = ( s + 1 ) f ( p 2 ' t 2 ) • • ' where "k" i s the p r o p o r t i o n a l i t y f a c t o r and not a l l " f * s " are of integer form. In any case, population increases designate concom-i t a n t r u r a l density increases so that more c e n t r a l places emerge and centers of the same size move c l o s e r together. Parr and Denike ( 1 9 7 0 ) mention that t h i s character-i s t i c d e c e n t r a l i z a t i o n (import s u b s t i t u t i o n ) i s e s p e i c a l l y prevalent among the higher l e v e l s of the urban hierarchy i n the United States and i s due p a r t l y to increased regional demand f o r c e r t a i n s p e c i a l i z e d ( p r o f e s s i o n a l , f i n a n c i a l , etc.) services. The authors s t r e s s , on the other hand, the well-known case of decline i n r u r a l populations and how 1 2 1 t h i s causes threshold ranges to eventually become unattainable from the smallest centers. Unfortunately, our model i s not f l e x i b l e enough to simultaneously account f o r those f u n c t i o n a l t r a n s f e r s that converge at the intermediate s i z e c l a s s e s . Income Since we are now holding t o t a l population constant, an income increase amounts to a per c a p t i a income increase that i s the same f o r a l l consumers. Since we also assume resource use i s not handled more e f f i c i e n t l y v i a improved technology, t h i s income increase may a r i s e from an absolute increase i n the amount of productive resources used per head of population. (Lampard, 1 9 6 8 , emphasizes differences between growth and development). Therefore, just as i n the previous case, there i s increased purchasing power i n each a r e a l u n i t and the aggre-gate demand curve facing the various firms s h i f t s to the r i g h t i n a fashion c h a r a c t e r i s t i c of the p a r t i c u l a r commodity and the l e v e l on which i t i s being offered. Nourse ( 1 9 6 8 : 2 1 2 - 2 1 5 ) argues c o r r e c t l y that the income increase allows each i n d i v i d u a l to purchase more of a l l goods and that fewer people are needed to comprise a threshold size market f o r any p a r t i c u l a r c e n t r a l function. For purposes of a n a l y s i s , he s t i p u l a t e s that the income increase does not a f f e c t the supplying population needed f o r each bundle of f i r s t order goods 1 as a consequence, 122 the f a c t o r "k" r e l a t i n g community and market populations increases at the f i r s t l e v e l . In keeping with the properties of the simple model he extends the progression component " s " + 1 as w e l l and higher l e v e l places are r e l a t i v e l y greater than before. In general terms, the number of c e n t r a l places of the f i r s t l e v e l a f t e r the income increase becomes t f'l-k(t,) k ( t ? ) "V ^ J . . . (5.4) where p ^ ( t 2 ) = P^(t-^). Retaining the supposition of a constant number of s a t e l l i t e s , he proceeds to exhaust " P M ( t ^ ) " v i a a p p l i c a t i o n of the formula: B ( t ) - r s+i-k(t 2) i - k ( t ) \ m - 1 P m ( 2 ) " < l - k ( t 2 ) x s + i r a > P m ( t x ) I 2 X J ... (5.5) u n t i l he defines a l l the urban populations i n a system " P M*" ( M* ^ M ) . Nourse's e s s e n t i a l t h e s i s i s that a per c a p i t a increase of income extends the number of centers i n the system so that they become cl o s e r together. At the same time the hierarchy contracts to accommodate the expanded low l e v e l urban population t o t a l s . Unfortunately, i t seems that t h i s a nalysis s u f f e r s i n several respects. F i r s t of a l l , the l i n e s of i n t e r -dependencies among the d i f f e r e n t sized places are severed by considering such a truncated hierarchy. The system 123 cannot be considered i n equilibrium since ( i ) no s a t i s -f a c t o r y market e x i s t s f o r high order commodities and ( i i ) the s t a b i l i t y of r u r a l populations i s neglected. Furthermore, h i s employment of a "k" increase determined by reduced external markets i s a rather debatable feature even w i t h i n the confines of our crude model. A more thorough examin-a t i o n of the case i s c l e a r l y required. Let's concentrate e n t i r e l y upon the demand side. By using an approach that i s rather more appealing than Nourse's, we can avoid some of the conceptual flaws that mar h i s argument. R e c a l l i n g the fundamental assumptions (see Chapter 3) that l i n k urban and market populations we observe that supply c h a r a c t e r i s t i c s become at best an i m p l i c i t f a c t o r i n the modeling scheme. Table 2 of the t h i r d chapter i l l u s t r a t e s the case where a market threshold of 3000 i s needed to uphold the f i r s t bundle of goods and services. This f i g u r e may be considered halved, f o r example, i f per c a p i t a income i s doubled throughout the region. By keeping our reasoning more coincident with those notions of c e n t r a l place theory that advocate maximum s p a t i a l competition (see the purchasing power argument of Berry and Garrisons I958d) i t seems more pl a u s i b l e that t h i s new purchasing threshold i s met by populations drawn evenly out of e x i s t i n g centers and comple-mentary areas. In other words, two centers with "p-^ " equal to 500 and " r 1 " equal to 1000 replace the single center. I t i s also apparent that t h i s argument avoids 124 Nourse's l i m i t a t i o n to small income increments that do not induce r a p i d growth of the "k" f a c t o r . Now, the general e f f e c t of the increase i s described by: f ( P m . t 2 ) = v f C p ^ t . ^ . . . (5.6) where "v" denotes the proportion between per c a p i t a income a f t e r and before the increment. The r e s u l t of t h i s argument should be c l e a r . An income increase spread evenly over a l l consumers expands the number of places i n a c i t y system and lowers the populations of places on l e v e l s comparable to the i n i t i a l case. In f a c t , s u f f i c i e n t increments may induce two or more i d e n t i c a l and adjacent subsystems to replace the o r i g i n a l system. Obviously, though, our argument i s somewhat weaker than Nourse's on attacking the issue of supply. Nevertheless we may think of the process just outlined as being con-str a i n e d by some lower l i m i t of the supplying population at the f i r s t (and every other) l e v e l although the d e f i n i t i o n of t h i s bound l i e s outside our a p r i o r i s t r u c t u r e . In both of these cases, too, we assume away an added b e h a v i o u r i s t i c i m p l i c a t i o n . C e r t a i n l y as incomes r i s e , customers i n c r e a s i n g l y turn tov/ard income-elastic commodities (high s t y l e f u r n i t u r e , fashion c l o t h i n g , s p e c i a l i z e d medical services, etc.) rather than items l i k e a g r i c u l t u r a l staples and home f u e l s . Now i t i s exceedingly d i f f i c u l t to state just how t h i s new dimension 125 a f f e c t s the i n i t i a l system through an income increase but we i n t u i t i v e l y expect that ( i ) the t r a n s f e r of near-h i e r a r c h i a l marginal goods to lower l e v e l s and ( i i ) the mixture of fi r m m u l t i p l i c a t i o n i n e x i s t i n g centers with the entry of bundles i n new centers are s u i t a b l y pronounced. The tendency to agglomerate seems to go hand i n hand with i n c o m e - e l a s t i c i t y and may w e l l serve to su s t a i n the i n e r t i a ( h i e r a r c h i a l l e v e l s and number of communities on each l e v e l ) of the o r i g i n a l system. A l l i n a l l , i t i s impossible to present an accurate picture of progressive systematization i n a simple set of equations. While i t i s true that our views and Nourse's are at odds on c e r t a i n relevant points, i t i s s i g n i f i c a n t to note that e i t h e r approach suggests a constancy or contr a c t i o n i n both h i e r a r c h i a l l e v e l s and f u n c t i o n a l c e n t r a l i z a t i o n may be r e a l i z e d v i a per c a p i t a income increases. Innovations Innovations of knowledge and technique may be generalized as functions of i n t e r a c t i o n p r o b a b i l i t y or information exchange i n open systems (Berry and Horton, 1 9 7 0 ) . Economic development may then be viewed from the perspective of such innovations occurring i n the l a r g e s t centers and spreading through time to other communities i n the system. Lampard's ( 1 9 6 8 : 1 0 6 ) cybernetics framework emphasizes s t a b i l i t y of interurban structure since " . . . the trans-formation of human settlement patterns (the evolving 126 system of c i t i e s , f o r example) involves the emergence and g e n e r a l i z a t i o n of novelty w i t h i n the population system. . . ", although growth may appear deviation-amplifying i n the various smaller subsystems. In our present d i s c u s s i o n , four general cases appear to he of s p e c i a l i n t e r e s t : ( i ) Innovations i n t r a n s p o r t a t i o n technology, i n c l u d i n g both new means and route improvements: ( i i ) Innovations that strengthen r u r a l (farming) p r o d u c t i v i t y ; ( i i i ) Innovations i n marketing technology that permit the entry of scale economies i n t o c e r t a i n e x i s t i n g a c t i v i t i e s ; ( i v ) Innovations that bring e n t i r e l y new a c t i v i t i e s i n t o the d i f f e r e n t sized urban communities. Within our comparative-statics framework we may give more s p e c i f i c a t t e n t i o n to cases ( i ) and ( i i ) as they r e s t on the demand side l i k e the f a c t o r s just analysed above. Nevertheless, we can allude to s t r u c t u r a l transformations for the remaining cases; besides, here we emphasize how structure channels economic development as w e l l . In the f i r s t instance, a t r a n s p o r t a t i o n innovation c l e a r l y a f f e c t s only demand conditions w i t h i n the f.o.b. supposition. Generally, we may consider such an improvement as being s i m i l a r to an increase i n per c a p i t a income, although i t s benefits only accrue to the complementary area populations on each h i e r a r c h i a l l e v e l (since we assume 127 the supplying population to he located at the production s i t e ) . I n t u i t i v e l y , we expect t h i s impact to change the urban structure roughly along the l i n e s we hypothesized f o r an income increase. On the other hand, t h i s statement must be q u a l i f i e d according to v a r i a b l e s l i k e ( i ) the i n i t i a l value of the p r o p o r t i o n a l i t y f a c t o r "k", ( i i ) the very magnitude of t h i s b e n e f i t given to external consumers, and ( i i i ) changing behaviour ( f o r example, multipurpose t r i p s ) due to new means as opposed to simple decreases i n s p a t i a l f r i c t i o n . By i t s e l f , improved customer m o b i l i t y points to increased f u n c t i o n a l d e c e n t r a l i z a t i o n i n a c i t y system. The improvement of r u r a l p r o d u c t i v i t y through innovations of a g r i c u l t u r a l methods, mechanization, e t c , changes the urban structure i f concomitant r u r a l to urban migration i s assumed. By viewing t h i s r e d i s t r i b u t i o n process i n a v e i n s i m i l a r to that o u t l i n e d e a r l i e r f o r population growth, we expect a s l i g h t increase i n the "k" f a c t o r and the basic progression component to accompany the emergence of a new h i e r a r c h i a l l e v e l . This follows because as r u r a l d e n s i t i e s diminish, f i r s t l e v e l places become smaller and tend to lose functions (except the lowest order convenience goods) to second l e v e l places. Such an upward t r a n s f e r occurs throughout the e n t i r e hierarchy and e s s e n t i a l l y suggests increasing f u n c t i o n a l c e n t r a l i z a t i o n with a new dominant center entering the system. While the number of communities increases, t h e i r populations on l e v e l s comparable to the o r i g i n a l hierarchy tend to decrease. 128 Scale economies, allowing large plants to produce at lower marginal and average costs, place more emphasis on the supply side of the argument. Parr and Denike (1970) give l u c i d i l l u s t r a t i o n of how such scale changes permit functions to he transfered from lower to higher l e v e l s of the hierarchy. Considered i n i s o l a t i o n , the e f f e c t s of scale extensions are more noticeable among the lower h i e r a r c h i a l l e v e l s where ( i ) intraurban thresholds are not too s u b s t a n t i a l and ( i i ) convenience r e t a i l i n g , as characterized by minimal c a p t i a l outlays, dominates the basket items. The t r a n s f e r of functions suggests a contrac-t i o n of the urban hierarchy as many o f t h e i n e f f i c i e n t firms i n smaller communities are priced out of t h e i r markets. As the p o s s i b i l i t i e s of r e a l i z i n g scale economies f o r i n d i v i d u a l goods i n the same basket are not i d e n t i c a l , improved marketing technology may w e l l evoke increased f u n c t i o n a l concentration i n the c i t y system. Innovations that b r i n g t o t a l l y new commodities into the re g i o n a l market cannot be r e l a t e d to urban structure i n the same e x p l i c i t fashion as the previous f a c t o r s . Obviously, though, a high incidence of low order innovations r e l a t i v e to high order types may somewhat strengthen f u n c t i o n a l d e c e n t r a l i z a t i o n . On the other hand, the innovation idea provides a convenient means for discussing the other side of the co i n : that being, of course, how interurban s t r u c t u r i n g channels the course of economic growth. Let's consider 129 how items that are neither resource-oriented nor r e g i o n a l -s p e c i f i c spread from an innovation o r i g i n . I t appears that ( i ) the stronger the distance decay, the c l o s e r w i l l d i f f u s i o n follow the co n s t r a i n t s of distance while ( i i ) the weaker the distance decay, the c l o s e r w i l l d i f f u s i o n f o l l o w the s i z e d i s t r i b u t i o n of urban communities (Pederson, 1970). In any case, without c e r t a i n h i e r a r c h i a l aspects, an economic region may be t y p i f i e d by c u r t a i l e d d i f f u s i o n of both low and high order goods. The i n t e g r a t i v e r o l e of periphe r a l centers i s emphasized as a means to o f f s e t t h i s concentrated and frequently weak economic environment. Observers i n c r e a s i n g l y s t r e s s the focussing of regi o n a l p o l i c y upon the l o c a t i o n and f u n c t i o n a l c h a r a c t e r i s t i c s of growth centers (among others, Friedmann, I966, Lithwick and Paquet, 1968). Though investments are u s u a l l y associated with the p r o v i s i o n of high order goods, suggestions are forwarded that regional convergence may be sought via.low order goods as w e l l . P o l i c y i m p l i c a t i o n s here include ( i ) the increase of d i f f u s i o n sources, ( i i ) the improvement of a c c e s s i b i l i t y to the primary innovation center, and ( i i i ) the speeding up of the urban growth process i n low density areas (Pederson, 1970), The important point being s i g n i f i e d i s that the extent of urban amenities tends to determine the very economic health (growth, s t a b i l i t y , etc.) of the e n t i r e region. Despite considerable v a r i e t y i n how investment i s s p a t i a l l y a l l o c a t e d , the p r i n c i p l e s of comparative advantage I 130 evoke h i e r a r c h i a l symptoms at some l a t e r stage. Whether per i p h e r a l growth n a t u r a l l y follows a f r o n t i e r or diminishing returns s e t i n at the l a r g e s t centers or growth i s thought-f u l l y r e d i r e c t e d , the regi o n a l periphery i s eventually taken up by urban subsystems. Along the path toward hypothe-t i c a l r e g i o n a l convergense, " . . . the d i f f u s i o n of innova-t i o n s down the system of c i t y - s i z e s i s the means by which growth and change are transmitted throughout the economy and integrated n a t i o n a l development i s achieved and maintained." (Berry and Horton, 1970:6?) . D i f f e r e n t i a l urban growth may i t s e l f be considered some f u n c t i o n of the process of innovation d i f f u s i o n (Pred, I 9 6 6 ) , In various parts of t h i s t h e s i s we have stressed the manner i n which a w e l l defined hierarchy constrains the growth of d i f f e r e n t s i z e d communities. Our s e r i e s of equ i l i b r i u m adjustments are r e a l l y taken to i l l u s t r a t e a c i t y system a f t e r complete sa t u r a t i o n of the p a r t i c u l a r d i f f u s i o n process. Population members and innovations are the c r i t i c a l items s i g n i f i e d . Rationale f o r e q u i l i b r i u m tendencies depends upon the maximization of i n t r a r e g i o n a l flows of mobile f a c t o r s of production. In t h i s way employment (and, therefore, population) increments tend to be proportional to urban s i z e while economic a c t i v i t i e s f i l t e r down from innovating areas (Thompson, 1968:52-59 o u t l i n e s t h i s phenomenon). Unfortunately, not even a smoothly operating market process can promise f u l f i l l m e n t of t o t a l adjustment i n the long run. 131 A B r i e f Synthesis I t would be hazardous, indeed, to i n f e r anything but tendencies from the foregoing diverse impact arguments. Several general p o s s i b i l i t i e s seem to e x i s t : ( i ) Regions i n which population growth completely o u t s t r i p s t e c h n o l o g i c a l advance and income extension may w e l l be characterized by a m u l t i - l e v e l hierarchy and high r u r a l d e n s i t i e s ; besides, the "k" f a c t o r w i l l be r e l a t i v e l y low; ( i i ) Regions i n which population growth i s retarded but other growth v a r i a b l e s continue may show s t a b i l i t y or c o n t r a c t i o n of the h i e r a r c h i a l structure over time; the "k" f a c t o r l i k e l y increases, e s p e c i a l l y with s i g n i f i c a n t r u r a l -to-urban migration; ( i i i ) Regions d i s p l a y i n g more balanced growth of population and economic factors probably preserve the gross features of the i n i t i a l hierarchy to a great extent. On the other hand, i t becomes e s s e n t i a l to q u a l i f y these statements by considering that: ( i ) A f i n i t e set of community s i t e s would po s s i b l y a l t e r the various impacts i n a rather t y p i c a l fashion: f o r instance, population growth may be simply a t t r a c t e d to e x i s t i n g places so that "k" increases; ( i i ) Innovations may have c l e a r thresholds when proceeding independently but may act quite d i f f e r e n t l y with the simultaneous change of other f a c t o r s . Parr and Denike (1970) i l l u s t r a t e the case where a scale change i n marketing i s brought about by an improvement i n t r a n s p o r t a t i o n (or an increase of population or per c a p i t a income f o r that matter), a c o n d i t i o n that may w e l l lead to shrinking of the urban hierarchy. Of course, diverse forces that we cannot contain i n our t h e o r e t i c a l argument are p e r s i s t e n t l y a l t e r i n g r e a l world interurban structures as w e l l . We should be pleased, therefore, i f we approximately account f o r the d i r e c t i o n s of r e a l world changes alone. In conclusion, then, the confidence we place i n explaining these adjustments depends l a r g e l y upon our f e e l i n g s toward the merits of c e n t r a l place theory. Graphing the Aggregate Model The simplest i l l u s t r a t i o n of the size d i s t r i b u t i o n of places v i a the aggregate model i s given by p l o t t i n g values f o r a hypothetical system on logarithmic-normal p r o b a b i l i t y paper. Comparing d i f f e r e n t systems by t h i s method makes several graphic properties obvious: ( i ) A roughly s t r a i g h t l i n e that seemingly i n d i c a t e s a truncated lognormal d i s t r i b u t i o n i s , at c l o s e r inspection, s l i g h t l y convex to the si z e axis f o r few h i e r a r c h i a l l e v e l s and more concave to that axis f o r many le v e l s ? ( i i ) The slope of the apparently s t r a i g h t l i n e depends upon the number of h i e r a r c h i a l l e v e l s when the "k" f a c t o r remains constant; on the other hand, a reduction of "k" means a steeper l i n e , c e t e r i s paribus; 133 ( i i i ) A change i n geometry a f f e c t s the slope too; as "s" increases the l i n e becomes f l a t t e r . ( i v ) The a r b i t r a r y point of t r u n c a t i o n above minimum size d places a f f e c t s d i f f e r e n t systems i n various ways 5 a m u l t i - l e v e l e d hierarchy may give a f l a t l i n e i f "k" i s small (perhaps large r u r a l d e n s i t i e s ) since many centers l i e i n the f i r s t grouped i n t e r v a l . These notions are somewhat u s e f u l when r e l a t e d to Berry's (1961) empirical study. They suggest, f o r instance, why there i s considerable v a r i e t y i n these c i t y s i z e d i s t r i b -utions though many are nearly lognormal. I n v e s t i g a t i o n along t h i s path seems to be a l o g i c a l step toward strengthening c r o s s - c u l t u r a l comparisons of c i t y systems. Furthermore, the a p r i o r i statements suggest what may be the most relevant factors i n promoting cross-time s i m i l a r i t y of i n t e r n a t i o n a l data. Total population of the urban system, independent of the manner ( b i r t h r a t e s , rural-to-urban migration, etc.) in which i t i s devised, seems to be the one c r i t i c a l v a r i a b l e that expands the urban hierarchy. In cases where some h i e r a r c h i a l aspects are presented at a point i n time ( e i t h e r f o r economic, s o c i a l , or administrative reasons), these aspects are l i k e l y s o l i d i f i e d by the population growth and r e d i s t r i b u t i o n w i t h i n a maturing space-economy. Since no r e a l world . hierarchy approximates p e r f e c t i o n , the emergence of new l e v e l s simply makes the lognormal d i s t r i b u t i o n more p l a u s i b l e . To the extent that population t o t a l s and national areas 134 show some p o s i t i v e c o r r e l a t i o n , i t i s not s u r p r i s i n g that these two v a r i a b l e s are pointed to i n most em p i r i c a l studies as being conducive to low degrees of primacy (Berry, 1 9 6 l s Hehta, 1964s Linsky, I 9 6 5 ) . On the other hand, we should be somewhat h e s i t a n t , then, of b e l i e v i n g that income per c a p i t a alone always v a r i e s p o s i t i v e l y with the degree of rank-size shown by i n d i v i d u a l n a t i o n a l systems that unfold over time (Lasuen, Lorca, and Oria, I 9 6 7 , assume t h i s to be the case). C l e a r l y , added t h e o r e t i c a l arguments and more precise inductive approaches are needed before s u f f i c i e n t confidence may be placed i n the r o l e of disparate growth v a r i a b l e s . This promises to be an important and controver-s i a l t o p i c i n future interurban research. C h a p t e r 6 CONCLUDING REMARKS The p r o b l e m a r e a d i r e c t i n g t h e d i s c u s s i o n i n t h i s t h e s i s c o n c e r n s c i t y s i z e d i s t r i b u t i o n . C o n s i d e r a b l e a t t e n t i o n i s g i v e n t o v a r i o u s s u b t o p i c s on b o t h t h e o r e t i c a l a n d e m p i r i c a l f r o n t s w i t h i n a n e x p l i c i t s y s t e m s f r a m e w o r k . The c o n s c i o u s s u p p o r t o f t h i s framework t h r o u g h o u t t h e e n t i r e d i s c u s s i o n i s p e r h a p s t h e most s a l i e n t f e a t u r e o f t h e t h e s i s . The e s s e n t i a l p u r p o s e s o f t h i s t h e s i s a r e t o i m p r o v e t h e p r e v a l e n t m e t h o d o l o g i e s now i n use and t o a n c h o r t h e c i t y s i z e t o p i c more f i r m l y i n t o t h e g r o w i n g b o d y o f g e o g r a p h i c a l l i t e r a t u r e and t h e o r y . The f i n d i n g s o f t h e d i f f e r e n t a r guments a r e r a t h e r d i v e r s e . G e n e r a l l y , t h o u g h , t h e t o n e i s t h a t l o g i c a l a n a l y s i s s h o u l d r e p l a c e i n t u i t i o n a s g e o g r a p h i c e n d e a v o u r s p r o c e e d s c i e n t i f i c a l l y . U n f o r t u n a t e l y , i n t h i s p r o b l e m a r e a t h e r e i s a n atmo s p h e r e o f d a t a m a l l e a b i l i t y and w i s h f u l t h i n k i n g t h a t s u g g e s t s i n t u i t i v e t e n d e n c i e s a r e common. F u r t h e r m o r e , d i s r e g a r d o f r e s e a r c h m e t h o d o l o g y ( i n c l u d i n g p o i n t s s u c h as t h e d e f i n i t i o n o f s t u d y a r e a s and t e c h n i q u e s o f s t a t i s t i c a l a n a l y s i s ) s e v e r e l y c o n s t r a i n s t h e v a l u e o f i n f e r e n c e s t h a t many o b s e r v e r s p u t f o r w a r d . I f e f f o r t s a r e t a k e n t o c o n t i n u e s c i e n t i f i c r e s e a r c h on t h e c i t y s i z e t o p i c t h e n t h e y must be f u n n e l l e d a l o n g 135 f i 136 two r e l a t e d paths: ( i ) Toward improved i n t e r p r e t a t i v e schemes on the t h e o r e t i c a l side; an argument of t h i s t h e s i s i s that c e n t r a l place theory o f f e r s a strong "base f o r such c o n t r i -butions; and ( i i ) Toward c a r e f u l l y structured e m p i r i c a l studies on d i f f e r e n t scales that, i n conjunction with t h e o r e t i c a l extensions, w i l l suggest those f a c t o r s that p r i m a r i l y determine the form of the frequency d i s t r i b u t i o n of c i t y s i z e s i n a p a r t i c u l a r region. Besides, our review has revealed that several l e s s general subtopics deserve increased a t t e n t i o n as w e l l : ( i ) The scheme of h i e r a r c h i a l s e t s , which deals with one complete and many p a r t i a l h i e r a r c h i e s of an independent nature, suggests a more f l e x i b l e viewpoint (at l e a s t where r e s o l u t i o n l e v e l s are r e l a t i v e l y low) toward c o m p a t i b i l i t y of c e n t r a l place t h i n k i n g with the c h a r a c t e r i s t i c u n i - s i z e c l a s s d i s t r i b u t i o n s of empi r i c a l research; ( i i ) Concern over the Loschian model should enhance t h i s same c o m p a t i b i l i t y ; ( i i i ) Extensions of the economic base concept v i a c e n t r a l place theory may w e l l provide valuable feedback at both the i n t r a - and interurban l e v e l s ; ( i v ) Added e f f o r t s are needed i n the attempt to give c e n t r a l place arguments a reasonable temporal dimension (that i s , when a hierarchy i s already assumed to e x i s t ) ; 137 (v) Emphasis on general systems concepts should complement the d e t e r m i n i s t i c and s t o c h a s t i c arguments; perhaps, indeed, the entropy idea can a s s i s t i n d e s c r i b i n g non-equilibrium features w i t h i n a dynamic framework ( f o r example, i t may have promise as a device to describe i n t e r -urban s t r u c t u r i n g p r i o r to h i e r a r c h a l m a t u r i t y ) ; ( v i ) A crude a n a l y t i c a l base has been set f o r i n v e s t i g a t i n g the i n t e r r e l a t i o n s of growth and interurban s t r u c t u r e ; model b u i l d i n g w i t h i n t h i s subtopic may have i n t e r e s t i n g t h e o r e t i c a l and p r a c t i c a l i m p l i c a t i o n s ; ( v i i ) Emphasis placed upon the p e c u l i a r aspects {slopes, etc.) of i n d i v i d u a l d i s t r i b u t i o n s may prove f r u i t f u l i n the search f o r c r o s s - c u l t u r a l r e g u l a r i t i e s . 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