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Structure and process in the Christallerian system Mulligan, Gordon Fredrick 1976

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STRUCTURE AND PROCESS IN THE CHRISTALLERIAN SYSTEM by  GORDON FREDRICK MULLIGAN B . S c , The U n i v e r s i t y M. A . , The U n i v e r s i t y  of B r i t i s h of B r i t i s h  C o l u m b i a , 19&9 Columbia, 1972  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in the Department of GEOGRAPHY  We a c c e p t required  this  t h e s i s as conforming  to the  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1976  (c) Gordon Fredrick Mulligan  In p r e s e n t i n g t h i s  t h e s i s in p a r t i a l  an advanced degree at  further  agree  of  the  requirements  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make i t I  fulfilment  freely  available  for  this  thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department  of  this  thesis for  It  financial  or  i s understood that copying or p u b l i c a t i o n gain s h a l l not  written permission.  Depa rtment The U n i v e r s i t y of B r i t i s h Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  that  r e f e r e n c e and study.  t h a t p e r m i s s i o n for e x t e n s i v e copying o f  by h i s r e p r e s e n t a t i v e s .  for  be allowed without my  ABSTRACT  This d i s s e r t a t i o n deals with t h e o r e t i c a l central place systems of the C h r i s t a l l e r i a n type.  By employing a form-function-process methodology  the author attempts to embrace central place structure  and process  in a  c o n s i s t e n t and general manner. A t t e n t i o n i s f i r s t given to systemic structure as depicted by the general h i e r a r c h i a l model of c i t y s i z e .  Given t h i s  structural  framework, i n t e r e s t i s then turned to modelling within-systems adoptive processes (the issue of innovation i s not considered).  F i n a l l y , the  e f f e c t s of d i f f e r e n t types of parametric s h i f t s - both continuous ( i n s t a n taneous) and d i s c r e t e (long run) - are examined w i t h i n the context of the e s t a b l i s h e d models. By e l i c i t i n g a number of l a w - l i k e statements the author i s intending to l a y some of the foundations f o r a general theory of i n t e r urban growth and development.  The scope and content of the more relevant  a s s e r t i o n s are presently o u t l i n e d . It i s demonstrated that c e r t a i n a t t r i b u t e s of i n d i v i d u a l c e n t r a l places are i n t i m a t e l y r e l a t e d to o v e r a l l systemic p r o p e r t i e s .  For i n s t a n c e ,  the inverse of the b a s i c / n o n - b a s i c r a t i o of a system's l a r g e s t c i t y shown to be i d e n t i c a l to the urban/rural  ii  population balance f o r the  is  e n t i r e system.  In a d d i t i o n a novel type of input-output model i s i n t r o -  duced so as to i l l u s t r a t e  the economic base underpinnings of the  h i e r a r c h i a l model. Special concern i s given to the s e r v i c e m u l t i p l i e r s i n the s t r u c t u r a l argument:  these are shown to r e f l e c t employment and demand  r a t i o s for the various h i e r a r c h i a l a c t i v i t i e s .  Then the e f f e c t s of  s h i f t s i n these m u l t i p l i e r s upon central place properties are examined w i t h i n a comparative s t a t i c s  framework.  The p o l a r i z a t i o n of h i e r a r c h i a l and wave-like d i f f u s i o n a r y patterns i s e s t a b l i s h e d by showing the former ( l a t t e r ) to accompany: (i) (iii)  systemic openness ( c l o s u r e ) , ( i i )  area! ( l i n e a r )  dimensionality,  slow (rapid) d e c l i n e i n the service m u l t i p l i e r s , and ( i v )  f r i c t i o n a l c o n s t r a i n t s on s p a t i a l  low (high)  interaction.  F i n a l l y , temporal (long run population changes) and s p a t i a l ( a l l o c a t i o n of nonnodal a c t i v i t i e s ) v a r i a t i o n are shown to induce chara c t e r i s t i c changes in such d i f f u s i o n a r y  iii  patterns.  TABLE OF CONTENTS  Page ABSTRACT  ii  LIST OF TABLES  vii  LIST OF FIGURES  xi  ACKNOWLEDGEMENTS  xii  Chapter 1  2  INTRODUCTION  1  1.1  Background  1  1.2  General Intentions and Themes of the Thesis  7  1.3  Preview. . .  9  Footnotes to Chapter 1  16  THE GENERALIZED HIERARCHIAL MODEL  18  2.1  Introduction  18  2.2  Terms and Notation  2.3  Economic Base Considerations  21  2.4  Hierarchial  30  2.5  The Problem of Evolution  34  2.6  A Proposal of Stage-Like Development  35  2.7  Concluding Remarks  44  Input-Output Linkages  Footnotes to Chapter 2  .  18  45  Chapter 3  Page CENTRAL PLACE DIFFUSION.  47  3.1  Introduction  47  3.2  C r i t i c i s m s of the E x i s t i n g Models  49  3.3  An A l t e r n a t i v e Model of Central Place D i f f u s i o n  56  P r o p o s i t i o n s Based on the Model  57  3.4 3.5  3.6  A d d i t i o n a l Comments on the  Alternative Alternative  Model  89  Concluding Remarks  91  Footnotes to Chapter 3 4  93  PARAMETRIC INFLUENCES ON STRUCTURE AND PROCESS IN THE CENTRAL PLACE SYSTEM  5  97  4.1  Introduction  97  4.2  Population  100  4.3  Technology  125  4.4  Per Capita Income  136  4.5  Concluding Remarks  142  Footnotes to Chapter 4  143  A MODIFICATION OF THE STRICT HIERARCHIAL FORMAT  148  5.1  Introduction  148  5.2  Structure  148  5.3  Process  155  5.4  Concluding Remarks  . . .  Footnotes to Chapter 5  157 158  v  Chapter 6  Page SUMMARY AND CONCLUSIONS  159  6.1  Introduction  159  6.2  Structure  160  6.3  Process  163  6.4  D i r e c t i v e s for  Future Research  BIBLIOGRAPHY  164 168  APPENDICES A  STAGE MATURATION OF A FOUR LEVEL K=3 CENTRAL PLACE SYSTEM  174  B  THE NOTION OF EQUIVALENT CENTERS IN HUDSON'S STOCHASTIC TELLING PROCESS  180  C  THE ALTERNATIVE DIFFUSIONARY MODEL: PROPERTIES OF THE TABLES AND AN OUTLINE OF THE TELLING PROCESS  187  D  THE IMPACT OF EXOGENOUS EMPLOYMENT SHIFTS IN A SECOND LEVEL CENTRAL PLACE  195  E  THE DIVISION OF CENTRAL PLACE POPULATIONS ACCORDING TO ACTIVITY SECTORS  198  F  THE IMPACT OF AN EXOGENOUS SHIFT IN A SERVICE MULTIPLIER IN A SECOND LEVEL CENTRAL PLACE  201  A NOTE ON THE RELATION BETWEEN MONEY INCOME AND REAL INCOME IN THE HIERARHCIAL FORMAT  203  THE ALLOCATION OF NONNODAL ACTIVITIES  208  G  H  vi  LIST OF TABLES*  Table 3.1  Page Adoptive Times f o r Centers i n an Open K=3 Five Level Central Place System  58  3.2  Adoptive Times f o r Centers i n a Closed K=3 Five Level Central Place System  59  3.3  Adoptive Times f o r Centers i n an Open K=3 Five Level Central Place System  60  3.4  Adoptive Times f o r Centers i n a Closed K=3 Five Level Central Place System  61  3.5  Adoptive Times f o r Centers i n an Open K=3 Five Level Central Place System  62  3.6  Adoptive Times f o r Centers in a Closed K=3 Five Level Central Place System  63  3.7  Adoptive Times f o r Centers i n an Open K=3 Five Level Central Place System  64  3.8  Adoptive Times f o r Centers in a Closed K=3 Five Level Central Place System  65  3.9  Adoptive Times f o r Centers i n an Open K=3 Four Level Central Place System  66  than  their  T h e t a b l e h e a d i n g s p r e s e n t l y g i v e n a r e somewhat a p p r o p r i a t e c o u n t e r p a r t s in the t e x t .  vii  less  detailed  Table 3.10  3.11  3.12  3.13  3.14  3.15  3.16  3.17  Page Adoptive Times f o r Centers i n an Open K=3 Four Level Central Place System  66  Adoptive Times f o r Centers i n a Closed K=3 Four Level Central Place System  67  Adoptive Times f o r Centers in a Closed K=3 Four Level Central Place System  67  Adoptive Times f o r Centers i n an Open K=3 Four Level Central Place System  68  Adoptive Times f o r Centers in an Open K=3 Four Level Central Place System . . .  68  Adoptive Times f o r Centers in a Closed K=3 Four Level Central Place System  69  Adoptive Times f o r Centers i n a Closed K=3 Four Level Central Place System . . .  69  Adoptive Times for Centers in an Open Ki=3, K =4, K =3 Four Level Central Place System  70  Adoptive Times f o r Centers in an Open Ki=3, K =4, K =3 Four Level Central Place System  70  Adoptive Times f o r Centers i n a Closed Ki=3, K =4, K =3 Four Level Central Place System.  71  Adoptive Times for Centers in a Closed K =3, K =4, K = 3 Four Level Central Place System  71  Adoptive Times for Centers in an Open Ki=3, K =4, K =3 Four Level Central Place System  72  Adoptive Times f o r Centers i n an Open Ki=3, K =4, K =3 Four Level Central Place System  72  Adoptive Times f o r Centers i n a Closed Ki=3, K =4, K =3 Four Level Central Place System  73  2  3.18  2  3.19  2  3.20  2  3.22  2  3.23  3  3  x  2  3.21  3  2  3  3  3  3  viii  Table 3.24  Page Adoptive Times f o r Centers in a Closed Kx=3, K =4, K =3 Four Level Central Place System  73  3.25  Adoptive Times f o r Centers in an Open K=4 Four Level Central Place System  74  3.26  Adoptive Times for Centers in a Closed K=4 Four Level Central Place System  75  3.27  Adoptive Times f o r Centers in an Open K=2 Five Level Central Place System  76  Adoptive Times f o r Centers i n a Closed K=2 Five Level Central Place System  77  3.29  Adoptive Times f o r Centers in an Open K=2 Five Level Central Place System  78  3.30  Adoptive Times f o r Centers in a Closed K=2 Five Level Central Place System  79  3.31  Adoptive Times f o r Centers in an Open K=2 Four Level Central Place System  80  3.32  Adoptive Times f o r Centers i n a Closed K=2 Four Level Central Place System  80  4.1  A Proposal of a One Percent Growth and R e d i s t r i b u t i o n Scheme in a K=3 Four Level Central Place System.  116  An A n a l y s i s by A c t i v i t y Sectors of the Proposed One Percent Growth and R e d i s t r i b u t i o n Scheme in a K=3 Four Level Central Place System  117  The Impact of Population Growth on S p a t i a l Adoption in an Open K=3 Four Level Central Place System  123  2  3.28  4.2  4.3  3  ix  Table 4.4  5.1  5.2  A. l  B. l  Page The Impact of Population Growth on Spatial Adoption in a Closed K=3 Four Level Central Place System V a r i a t i o n i n Central Place Populations as a Consequence of a Localized A c t i v i t y i n a K=3 Five Level Central Place System  124  . . . 153  A Comparison of Adoptive Times (Standardized) Between ( i ) Centers in a Closed K=3 Five Level System and ( i i ) Centers in a Modified Version of that System  156  Populations of Rural Areas and Central Places i n a K=3 Four Level Central Place System which Matures in a Stage-Like Fashion  178  Types of Equivalent Centers in a K=3 Five Level Central Place System. . . .  181  x  LIST OF FIGURES  Figure 2.1  Page A c e n t r a l place system with nesting f a c t o r •  3.1  3.2  K  =  3  2  2  Adoptive times f o r centers in an open K=2 four l e v e l central place system  81  Adoptive times f o r centers in a closed K=2 four l e v e l central place system  82  xi  ACKNOWLEDGEMENTS  I should f i r s t of a l l l i k e to express my considerable indebtedness to my parents.  Without the pleasant working environment that they  provided f o r me i n t h e i r Squamish home I fear that even y e t I would be piecing together that a l l - i m p o r t a n t f i r s t d r a f t .  A quiet study, good  food and d r i n k , and winter sunsets on G a r i b a l d i made the absence of c e r t a i n urban amenities (ones that a bachelor can best appreciate) a b i t more t o l e r a b l e . Following that I must extend s p e c i a l gratitude to my advisor Ken Denike.  Ken f i r s t stimulated my i n t e r e s t in the research area of  present concern and was always helpful - both i n t e l l e c t u a l l y and f i n a n c i a l l y - during my graduate sojourn a t UBC. The inputs of the remaining members of my committee must a l s o be noted.  Curt Eaton's i n c i s i v e c r i t i c i s m s of the e a r l i e r d r a f t were  extremely useful and t h e i r consideration l e d , in my o p i n i o n , to a much tighter  final draft.  John Chapman and Gary Gates gave helpful o r g a n i z a -  t i o n a l comments as w e l l . More s i g n i f i c a n t , however, was the i n t e l l e c t u a l  freedom that .  t h a t committee - and indeed t h i s could be stated f o r the e n t i r e Geography  xii  Department at UBC - accorded me.  By never channeling my thoughts in any  one s p e c i f i c d i r e c t i o n and by never imposing s t r i c t guidelines f o r my t h e s i s scheduling, those i n d i v i d u a l s allowed the whole exercise to be as i t must be - a worthwhile l e a r n i n g experience. I must also thank the Central Mortgage and Housing Corporation (CMHC) f o r the f e l l o w s h i p funding that was extended to me i n the e a r l y 1970's and UBC as well f o r the stipend that was granted in 1974.  The  Urban Studies (thanks Walter) and Geography teaching a s s i s t a n t s h i p s were l i k e w i s e much appreciated during t h i s f i n a l y e a r . Also at t h i s time I should l i k e to commend Sharon H a l l e r f o r doing her usual f i n e typing job and acknowledge Nick Watkins f o r  drafting  up the three figures included w i t h i n the t e x t . Of course, I must also express more than a l i t t l e appreciation to a number of f i n e fellows that I met through the Geography Department a t UBC. With Warren, Mike, J . B . , Sparky, A r t h u r , and others I shared a number of Monday-to-Friday ups and downs (but in the good old y e a r s , n a t u r a l l y , they only r a r e l y ended on the F r i d a y ) .  F i n a l l y , my good  f r i e n d s from the "outside world" must also be thanked:  John and T r u d i ,  Tony and D i l s o n , Max, S t e n c i l , Thorns, Dink, M i l n e r , Pursewarden, P r e t t y Boy, and Susan - i t ' s been a l l  right.  xi i i  Chapter 1  INTRODUCTION 1.1  Background For some time now s o c i a l s c i e n t i s t s have expressed concern over  the s i z e d i s t r i b u t i o n of urban communities.  An evergrowing body of  litera-  t u r e , e x h i b i t i n g the nuances of various d i s c i p l i n e s — g e o g r a p h e r s , economists, h i s t o r i a n s , s o c i o l o g i s t s , among o t h e r s , have a l l commented on the t o p i c — has made the c i t y s i z e d i s t r i b u t i o n inter-urban perspective on human o r g a n i z a t i o n .  issue central to an  Quite n a t u r a l l y ,  however,  the i n t e n t i o n s of these observers have not always been c o i n c i d e n t :  as a  r e s u l t , somewhat d i s t i n c t pools of i n t e r e s t may be discerned w i t h i n the mainstream of  thought.  The theme was i n i t i a t e d on a s t r i c t l y (1913) and Lotka (1924, 1941).  empirical basis by Auerbach  More s p e c i f i c a l l y , t h i s e a r l y work was  s o l e l y d i r e c t e d toward r e l a t i n g the cumulative numbers of urban places (above a c e r t a i n population t h r e s h o l d , that i s ) in a designated region to t h e i r respective population s i z e s . space ( i . e . i n d i f f e r e n t  The r e p e t i t i o n of t h i s exercise over  regions) and through time eventually made i t  apparent that patterns of human settlement in d i f f e r e n t  parts of the world  could be characterized by a rather wide v a r i e t y of such d i s t r i b u t i o n s .  1  2  This deviation was e f f e c t i v e l y p o l a r i z e d i n the w r i t i n g s of Jefferson (1939) and Z i p f (1949) who i n i t i a t e d the now commonly-made d i s t i n c t i o n between primate and r a n k - s i z e properties of c i t y s i z e d i s t r i butions and who gave the e a r l i e s t r a t i o n a l i z a t i o n — the degree to which a region was " i n t e g r a t e d " — f o r t h i s d e v i a t i o n . ^ With the growing r e c o g n i t i o n that some f a c t o r s ( e . g . area! extent or t o t a l population of the region) seemed to induce the development of one pattern or another, i t was only natural that c e r t a i n models e x p l a i n i n g such d i s t r i b u t i o n a l  v a r i e t y should be engendered.  The i n t e r e s t e d reader  might r e f e r to Simon (1955), Berry (1961), Curry (1964), Thomas (1967), Fano (1969), Berry and Horton (1970), and Parr and Suzuki (1973) f o r some relevant examples. aspatial  What a l l these models have i n common i s a decidedly  perspective and a s t o c h a s t i c - e n t r o p i c mechanism so as to e l i c i t  d i s t r i b u t i o n a l change over time.  (i)  Models Having an E x p l i c i t Spatial-Economic Structure However, the c i t y s i z e topic has been approached from an e n t i r e l y  d i f f e r e n t vantage as w e l l .  Due to the enlightening (yet only p a r t i a l l y  s u c c e s s f u l ) t h e o r e t i c a l c o n t r i b u t i o n s of C h r i s t a l l e r (1966) and Losch (1954), 3 an a l t e r n a t i v e means of modelling patterns of c i t y s i z e has been r e a l i z e d . The proper modelling of t h e i r theories e n t a i l s f i r s t l y a r e c o g n i t i o n of the s p a t i a l (geometric) and economic properties of i n d i v i d u a l activities.  This has been — a n d s t i l l  remains — a somewhat d i f f i c u l t task:  l a r g e l y because the theories themselves have t h e i r own p a r t i c u l a r s h o r t comings in e x p l i c i t l y dealing with the notion of a spatial-economic e q u i l i b r i u m .  3  As a consequence, modelling f o r the most part has followed C h r i s t a l l e r i a n d i r e c t i v e s , since that theory i s much the simpler and more f l e x i b l e of the two and because i t does at l e a s t seem i n agreement with c e r t a i n domains of 4 reality. This seems an appropriate spot to emphasize that the upcoming t h e s i s deals s o l e l y with the C h r i s t a l l e r i a n case, although i n places geometries are u t i l i z e d which are more general than C h r i s t a l l e r himself o r i g i n a l l y suggested.  For that reason, then, the reader should confine h i s  t h i n k i n g to the realm of C h r i s t a l l e r i a n theory whenever the term central appears i n the upcoming t e x t .  place  Beckmann (1958) was r e a l l y the f i r s t to model c i t y s i z e s according 5 to c e n t r a l place p r i n c i p l e s .  His h i e r a r c h i a l s t r u c t u r e was t y p i f i e d by  two c h a r a c t e r i s t i c s : (i)  (ii)  c e n t r a l p l a c e p o p u l a t i o n s o f v a r y i n g s i z e remained a c o n s t a n t p r o p o r t i o n o f t h e market a r e a p o p u l a t i o n s which they s e r v i c e d ; and the r a t i o o f t h e populations of c e n t r a l places (market a r e a s ) on s u c c e s s i v e l e v e l s remained c o n s t a n t throughout t h e h i e r a r c h y .  Unfortunately, Beckmann at t h i s time erred i n his  interpretation  of the C h r i s t a l l e r i a n geometry — he s p e c i f i c a l l y mistook i t s p r i n c i p l e s of domination — and several of his a s s e r t i o n s became of questionable status.  This p o s t u l a t i o n a l e r r o r was eventually r e c t i f i e d by Beckmann (1968)  himself but the reader might well prefer the l u c i d d e t a i l i n g of the problem found i n the independent statement of Parr (1969). P a r r ' s a r t i c l e was e s p e c i a l l y important because there he challenged not only the assumptions of the seminal e f f o r t , but the deduced r e s u l t s of  4  that argument as w e l l .  In p a r t i c u l a r , Beckmann (1958) asserted that h i s  basic progression component model (see property  (ii)  above) s u c c e s s f u l l y  r e l a t e d C h r i s t a l l e r i a n theory to the r a n k - s i z e p r i n c i p l e ; P a r r , however, took great care i n demonstrating that such coincidence was not a n a l y t i c a l l y possible.  6  In the meantime, Dacey (1966) r e p l i e d with a somewhat more general model based upon h i s previous experiences with the c e n t r a l place topology (see Dacey (1965) f o r a comprehensive axiomatic treatment of both the C h r i s t a l l e r i a n and Loschian geometries).^  Concern here was e s p e c i a l l y  d i r e c t e d toward e l i m i n a t i n g the s i n g l e geographic c o e f f i c i e n t of proport i o n a l i t y (see property ( i ) above) by introducing a complete set of s e r v i c e multipliers — i n  t h i s way d i f f e r e n t production technologies f o r diverse  bundles of goods and services could be embraced w i t h i n the h i e r a r c h i a l scheme ( i . e . there would be an i n d i v i d u a l m u l t i p l i e r hierarchial  f o r each bundle or  level).  Now the central place populations were seen to be composed of h i e r a r c h i a l sectors and the supply technology c h a r a c t e r i s i z i n g each sector was interpreted to be i n v a r i a n t despite changes in the scale of o p e r a t i o n . To take an example, a l l central places i n a system would produce a set of f i r s t l e v e l (convenience) goods but the same technology would be employed i n the l a r g e s t center as in a l l the smaller centers of the system. Very r e c e n t l y , the present author (see Mulligan (upcoming 1976)) demonstrated that these m u l t i p l i e r s were a c t u a l l y inherent to or  implicit  i n the basic progression component model so that the two models could not r e a l l y be considered d i s t i n c t  (as o r i g i n a l l y surmised) and t h a t , in  5  p a r t i c u l a r , the o n e - m u l t i p l i e r model was j u s t a s p e c i a l case of the more general  formulation. Another s i g n i f i c a n t c o n t r i b u t i o n by Dacey involved the d e l i n e a -  t i o n of the aforementioned population sectors into basic and non-basic ( l o c a l i z e d ) components.  (export-oriented)  His i n t e r p r e t a t i o n , although  entirely  c o r r e c t , was extremely b r i e f and the reader might well be advised to consult Parr (1970) for c l a r i f i c a t i o n .  The l a t t e r i s s t i l l  the f i n e s t  critique  of the various central place models, in that i t covers t h e i r a n a l y t i c p r o p e r t i e s , underlying economic p o s t u l a t i o n s , and some relevant empirical qualification. In a contemporary a r t i c l e , Beckmann and McPherson (1970) phrased the general argument in somewhat d i f f e r e n t  terms.  P a r r , Denike, and  Mulligan (1975) have r e c e n t l y i l l u s t r a t e d , however, that the assumptive underpinnings of t h e i r model were c o i n c i d e n t with those of Dacey's, although the former did allow f o r a new dimension of f l e x i b i l i t y  i n the central place  topology (through a v a r i a b l e nesting f a c t o r on the same l a t t i c e ) .  In a d d i -  t i o n , t h i s author in Mulligan (upcoming 1976) has questioned other unsupported a l l e g a t i o n s by Beckmann and McPherson; i n f a c t , he has s p e c i f i c a l l y demonstrated how i t was unclear that a "constrained" version of model might be considered c o i n c i d e n t with ( i ) o r i g i n a l Beckmann (1958) model and ( i i )  their  the revised e d i t i o n of the  the r a n k - s i z e r u l e .  Besides t h i s , even f u r t h e r debate has been engendered by the structural  reformulations of Dacey (1970) and Dacey and Huff (1971).  For  the present time, though, i t should be s u f f i c i e n t to say that the P a r r , Denike, and Mulligan (1975) paper has hopefully c l a r i f i e d the idea that the general h i e r a r c h i a l model — i r r e s p e c t i v e of how i t  i s mathematically  6  presented — d e f i n i t e l y has an economic base i n t e r p r e t a t i o n .  It  follows  from the above, then, that any simpler model (such as the case with the basic progression component) which can be shown to be a particular  example  of that general model has a v a l i d economic base i n t e r p r e t a t i o n as w e l l . This in b r i e f has been an overview of those c o n t r i b u t i o n s which the author deems were most i n f l u e n t i a l i n t h i s s p e c i f i c research a r e a .  i n the development of h i s own thought  Now i t i s only f a i r to note that t h i s  f a m i l y of s t r u c t u r a l models based on C h r i s t a l l e r i a n p r i n c i p l e s has a t t r a c t e d a c e r t a i n amount of c r i t i c i s m — s e e Parr (1970), Henderson (1972), M i l l s (1972), and Richardson (1973), amongst o t h e r s . In t h i s author's o p i n i o n , however, these a s s e r t i o n s must properly be construed as being d i r e c t e d to the shortcomings (and i n c o n s i s t e n c i e s i n the Loschian case) of the relevant body (bodies) of theory.  As long as  such a p o s t e r i o r i models f a i t h f u l l y r e f l e c t the c r i t i c a l aspects of a v a i l able theory — a l b e i t such models might at times seem naive v i s - c l - v i s the real world — then they do not merit c r i t i c i s m in themselves:  e i t h e r an  improvement of or a s u b s t i t u t i o n f o r the relevant body (bodies) of theory is called for. N a t u r a l l y i f the reader i s acquainted with the C h r i s t a l l e r i a n and Loschian arguments then i t i s common knowledge that both c o n t r i b u t i o n s have been c r i t i c i z e d and q u a l i f i e d at numerous times in the p a s t .  The  author suggests Isard (1956), von Boventer (1963), and Parr and Denike (1970) as three good c r i t i c a l reviews of the a v a i l a b l e m a t e r i a l .  Neverthe-  l e s s , i t should be s t i l l more than evident that no other corpus of thought e x i s t s which can even s u g g e s t s challenge to e i t h e r as a " . the l o c a t i o n , s i z e , nature, and spacing o f .  . .theory of  . ." human economic a c t i v i t i e s  at the r e s o l u t i o n l e v e l (scale) which i n t e r e s t e d the two seminal researchers.  7  (ii)  Models of Process Based on a Given Structure Out of the mainstream of recent thought concerning the d i f f u s i o n  of items through space have arisen two models which s p e c i f i c a l l y deal with adoption in the central place s e t t i n g .  Hudson (1969) and Pederson (1970)  have devised s i m i l a r techniques — e a c h of which bears a generic resemblance to Hagerstrand's (1965) well-known model characterized by a mean informat i o n f i e l d — f o r representing adoptive patterns amongst communities the i n t r o d u c t i o n  given  (or innovation) of an item at the l a r g e s t center of a  central place system. By f i r s t postulating an appropriate geometry and then a p a r t i c u l a r c i t y s i z e d i s t r i b u t i o n as well — that i s , by f i r s t p o s t u l a t i n g a s p e c i f i c s t r u c t u r a l model — each was able to incorporate the notion of process or change (through the transmission of items) i n t o the otherwise s t a t i c setting.  The author f e e l s , however, that there are serious i n c o n s i s t e n c i e s  i n both of these c o n t r i b u t i o n s and that the source of these i n c o n s i s t e n c i e s must be examined before accurate modelling may proceed.  In a d d i t i o n ,  neither of the two authors was able to take advantage of the rather wide d i s t r i b u t i o n of c i t y s i z e s which may be represented by the general h i e r a r c h i a l model and, as a consequence, t h e i r deductive assertions would they were c o r r e c t ) be n e c e s s a r i l y confined by the rather  (if  inflexible  premises of t h e i r assumptive models.  1.2  General Intentions and Themes of the Thesis Given t h i s structure-process dichotomy, the author wishes now  to specify his general intentions i n or motivations f o r undertaking the  8  w r i t i n g of a t h e s i s in t h i s p a r t i c u l a r research a r e a .  In b r i e f form these  i n t e n t i o n s are as f o l l o w i n g : (i)  t o a r t i c u l a t e further the structural properties of t h e g e n e r a l h i e r a r c h i a l model and r e s o l v e , i f p o s s i b l e , new l a w - l i k e s t a t e m e n t s which have h i t h e r t o gone u n n o t i c e d b u t which a r e n e v e r t h e l e s s implicit t o t h a t model;  .(ii)  t o i n c o r p o r a t e t h e n o t i o n o f p r o c e s s (by d e v i s ing a new d i f f u s i o n a r y model) i n t o t h e s t a t i c c e n t r a l p l a c e s e t t i n g i n a manner which would appear both more c o n s i s t e n t and more g e n e r a l than p a s t a t t e m p t s have been; and  (iii)  t o i n t r o d u c e temporal and s p a t i a l v a r i a t i o n due t o d i f f e r e n t t y p e s o f s h i f t s which may o c c u r in the a c t i v i t i e s located a t the v a r i o u s c e n t r a l p l a c e s - i n t o t h e s t r u c t u r a l model and then d i s c e r n j u s t how such changes would be e x p e c t e d to redefine diffusionary patterns.  The  author f e e l s that while numerous p a r t i c u l a r statements of a  l a w - l i k e character do emerge here and there throughout the t e x t (see the f o l l o w i n g s e c t i o n f o r a preview), the i n t e g r a t i n g theme or idea of the t h e s i s i s that structure and process may now be e x p l i c i t l y r e l a t e d i n the central place context.  Inasmuch as the author requires a single  over-  r i d i n g theme to have — by d e f i n i t i o n — a t h e s i s , then the argument that those two research perspectives ( i . e . s t r u c t u r e and process) can now be a r t i c u l a t e d in a more c o n s i s t e n t and general manner than they have been i n the past is that t h e s i s . Since the author i s working at the i n t e r f a c e of two such general perspectives i t  i s only natural that the t e x t i s interspersed with con-  s i d e r a b l e methodological comment.  I t i s perhaps a p p r o p r i a t e , t h e n , to  close t h i s section by emphasizing that concern in t h i s t h e s i s i s more d i r e c t e d to modelling e x i s t i n g theory rather than to extending the underlying  9  theory per s e .  There i s , of course, considerable feedback between the  operations of devising theory and representing such theory i n model form; i n the present context, the author i s t r u s t i n g that by: (i)  (ii)  (iii)  devoting e x p l i c i t a t t e n t i o n t o the internal statements o f t h e e x i s t i n g t h e o r i e s o f s t r u c u t r e and p r o c e s s ( t h i s i s a c c o m p l i s h e d by r e p r e s e n t i n g more o f such s t a t e m e n t s i n t h e r e l e v a n t m o d e l l i n g f o r m a t ) ; and q u a l i f y i n g ( t o some e x t e n t a t l e a s t ) t h e domain of r e a l i t y which can be f a v o r a b l y covered by such t h e o r i e s ; as w e l l as e x t e n d i n g t h e f l e x i b i l i t y (now i n an added a p r i o r i sense) o f t h e models r e p r e s e n t i n g t h e t h e o r i e s so as t o s u g g e s t a p p r o p r i a t e new avenues f o r t h e o r y e x t e n s i o n  he i s playing a part — a l b e i t quite minor — in the f i n a l  articulation,  at some future time, of a general structure-process theory of  inter-urban  growth and development. Before c l o s i n g t h i s chapter the author wishes to give a preview of some of the more p a r t i c u l a r arguments which are taken up i n the t e x t . A cursory examination should make i t apparent that more of the a n a l y t i c a l working i s in f a c t devoted to the s t r u c t u r a l — rather than processual — side of the argument but t h i s n a t u r a l l y follows from the author's research i n t e r e s t s in the past as well as from the "form-function-process" methodology presently being advocated.  1.3  Preview The upcoming chapter of t h i s t h e s i s serves to o u t l i n e the presently  accepted h i e r a r c h i a l formulation.  The author, however, does not simply  summarize here the e x i s t i n g m a t e r i a l :  r a t h e r , he i s a l s o determined to  10  present a more general (accurate) picture of the i n d i v i d u a l urban economies by d i s t i n g u i s h i n g , amongst the general populations t h e r e i n , between those that are a c t i v e economic agents (employees) and those that are not q (dependents). As a consequence, i t becomes possible to demonstrate that any community's r a t i o of basic to non-basic a c t i v i t i e s (as expressed i n terms of employees) depends upon the s e r v i c e m u l t i p l i e r s t y p i f y i n g the economy of the e n t i r e system as well as the actual placement (ordering) of that center w i t h i n the o v e r a l l h i e r a r c h i a l scheme.  The reader might notice  that t h i s o u t l i n e bears a generic resemblance to the P a r r , Denike, and Mulligan (1975) statement and he i s advised to consult that a r t i c l e  if  some concern i s f e l t over r e l a t i n g t h i s new g e n e r a l i z a t i o n to some of the earlier  contributions. The author then r e i t e r a t e s his argument but takes a somewhat  d i f f e r e n t tack i n doing s o .  He complements the economic base i n t e r p r e t a -  t i o n by i d e n t i f y i n g the w i t h i n - l e v e l  and between-levels  linkages as r e f l e c t e d in the set of service m u l t i p l i e r s .  input-output This  input-output  perspective i s s i m i l a r to that t r a d i t i o n a l l y u t i l i z e d i n economics; now, however, attention i s s p e c i f i c a l l y given to the linkages amongst a c t i v i t e s which c o e x i s t in space ( i . e . occupy the same point) but are on d i f f e r e n t l e v e l s of the h i e r a r c h y . In the f i n a l part of that chapter, the author presents some rather s p e c u l a t i v e views on the s t a g e - l i k e properties which are to the h i e r a r c h i a l model. existing  Put simply, i t  implicit  i s demonstrated how various  sectors of the space-economy may be affected — given  initially  the l o c a t i o n and number of the centers of production, an a v a i l a b l e technology  11  with constant returns to scale i n a g r i c u l t u r e and i n d u s t r y , and the  ongoing  a l l o c a t i o n of bundles (with everincreasing thresholds) according to central place doctrines — as the system's h i e r a r c h i a l a t t r i b u t e s become defined over time.  This i s suggested as an extremely simple evolutionary exten-  sion of the s t a t i c model. An important accompaniment of t h i s discourse i s the t i o n that the r a t i o of urban to rural  illustra-  population f o r the e n t i r e system  (or any w e l l - d e f i n e d subsystem) i s i n t i m a t e l y r e l a t e d to the b a s i c / n o n basic r a t i o of the l a r g e s t center in the system (subsystem). The second chapter, then, may be construed as an extension of the a n a l y t i c debate in the l i t e r a t u r e . purpose as w e l l .  But that chapter serves an a d d i t i o n a l  In that the argument d e l i m i t s the s t r u c t u r a l  properties  of a family of central place systems, and these same systems (more accura t e l y , t h e i r r e a l world mappings) are commonly associated with process and change, then the second chapter a l s o serves to d e l i m i t the domain of structures i n which inter-urban processes may be modelled and analysed. While t h i s might seem a b i t f a s t i d i o u s to some observers — a n d perhaps even t r i t e to some "form fashions f u n c t i o n " adherents — i t i s a point which should be well taken.  As the t h i r d chapter i n d i c a t e s ,  its  neglect i n the past has been the source of numerous methodological problems i n the  literature.  That chapter embraces process and change under the general heading of d i f f u s i o n .  In t h i s t h e s i s , d i f f u s i o n i s simply construed as a  macro-adoptive process.  The author says macro because, in the present  c o n t e x t , communities themselves are the i n d i v i d u a l s which adopt a p a r t i c u l a r item — w h e t h e r that item i s a new consumer product, a new technique of  12  production, a d i s e a s e , some new opinion or piece of knowledge, or the l i k e — a l t h o u g h , in f a c t , i t  i s always groups ( i n d i v i d u a l  consumers, f i r m s ,  professional o r g a n i z a t i o n s , e t c . ) w i t h i n those same communities which perform the real act of adoption. The issue of immediate concern, however, i s the manner i n which t h i s macro process may be conceptualized.  The author f e e l s that researchers  i n the past — m o s t noticeably Hudson (1969) and Pederson (1970) — h a v e shown good i n s i g h t i n extending the Hagerstrand (1965) argument from the micro l e v e l to the macro l e v e l but he disagrees somewhat with a number of t h e i r e x p l i c i t a n a l y t i c a l statements.  The author begins by q u a l i f y i n g  these past contributions and then goes on to suggest r e v i s i o n s which are hopefully more consonant with — as well as being more general t i o n s of — the tenets of central place  interpreta-  thought.  The purpose of t h i s t h i r d chapter, then, i s to d e l i m i t f a c t o r s which would seem most c r i t i c a l  those  in e f f e c t i n g c h a r a c t e r i s t i c  and temporal adoptive patterns w i t h i n the central place s t r u c t u r e .  spatial The  author contends, f o r i n s t a n c e , that the r e l a t i v e i s o l a t i o n of the system ( v i s - 5 - v i s other systems), the nature of the s i z e d i s t r i b u t i o n  of the  system's communities ( r e f l e c t i n g the r e l a t i v e rate of d e c l i n e of the s e r v i c e multipliers),  and the e f f i c i e n c y of the system's t r a n s p o r t a t i o n  facilities  are a l l s i g n i f i c a n t i n determining whether inter-urban d i f f u s i o n would be wave-like (where the acceptance of an item would spread outward from the l a r g e s t center — t h i s  being the assumed source of d i f f u s i o n )  or whether  the adopted item would "jump around" down through the urban h i e r a r c h y . It should be emphasized that the hypotheses generated in t h i s chapter are achieved through the merging of thought i n two d i s t i n c t research  13  areas — central place theory and g r a v i t y - p o t e n t i a l  theory — and there  are n e c e s s a r i l y concomitant a n a l y t i c a l problems (not j u s t concerning eventual confirmation per se) involved i n such an o p e r a t i o n .  Most impor-  t a n t l y , the author wonders whether or not the two theories are completely independent of one another — improved future research may suggest, f o r i n s t a n c e , that the parameters of those models representing the two theories are i n t i m a t e l y r e l a t e d .  In the absence of such i n d i c a t i o n s , however, the  author i s content to assume independence at the present time. In a d d i t i o n , the parametrical v a r i e t y allowed by t h i s merging — Hudson and Pederson did not have t h i s advantage — i s exceedingly great so the author decided to opt for a h e u r i s t i c presentation which i s chara c t e r i z e d by a host of numerical examples. The fourth chapter p r i m a r i l y represents an attempt at i n c o r porating some s t r u c t u r a l parametric change i n t o the s t a t i c model.  In a  sense these comments form a lengthy c r i t i q u e and r e v i s i o n of the statements found i n Nourse (1968) — a p p a r e n t l y the sole observer who has endeavored to r e l a t e system-wide  "before and a f t e r " h i e r a r c h i a l  attributes.  However, i n t e r e s t i s presently deflected to the s i n g u l a r impacts of v a r i a t i o n s i n population, per c a p i t a income, and technology and transportation)  (marketing  at a point i n time (Nourse d i d not deal with such  instantaneous s h i f t s ) as well as over the long run (Nourse d e a l t with long run s h i f t s — although he c a l l e d t h i s comparative s t a t i c s a n a l y s i s — but the present author i s s k e p t i c a l towards many of those a s s e r t i o n s ) . contrast to the evolutionary argument of the second chapter,  In  inter-urban  migrations (of a c t i v i t i e s and productive f a c t o r s over the long run) amongst the centers of the system are now accommodated (even to the extent that the emergence of new centers becomes t h e o r e t i c a l l y f e a s i b l e ) .  14  In a d d i t i o n , t h i s chapter pays some a t t e n t i o n to r e s o l v i n g j u s t how independent  population growth amongst the communities of the central  place system would be expected to a f f e c t the o v e r a l l s p a t i a l and temporal properties of an ongoing d i f f u s i o n a r y p r o c e s s . ^ If there i s a p e r s i s t e n t underlying theme f o r the long run a n a l y s i s i n the fourth chapter i t  i s that numerous voids haunt the pre-  sently existing theoretical formulations. pictured as being a s o r t of t r a d e - o f f :  H i e r a r c h i a l modelling i s r e a l l y  a t the r e s o l u t i o n l e v e l of between-  c i t i e s issues the aggregation inherent to the s e r v i c e m u l t i p l i e r approach i s a valuable a n a l y t i c a l convenience ( i f  not a necessity) yet when i n d i c a t i o n s  of long run change i n those same m u l t i p l i e r s are d e s i r e d , a much more thorough understanding of t h e i r actual composition (in terms of i n d i v i d u a l and not bundles of — goods and s e r v i c e s ) i s r e q u i r e d . ^ The f i f t h chapter introduces a s t r u c t u r a l d i s t o r t i o n i n t o the usual symmetry of the central place scheme.  After i n s t i t u t i n g a localized  (found at a central place s i t e ) incremental s h i f t in an a c t i v i t y of a noncentral place type, the author analyses the subsequent a l l o c a t i o n of ' ;  the total  population which would be needed to support (as in the c l a s s i c a l  argument) t h i s new body. dent upon the specific  This a l l o c a t i o n i s observed to be highly depen-  l o c a t i o n of the new a c t i v i t y .  For i n s t a n c e , the  arrangement (in e q u i l i b r i u m ) of the extra s e r v i c i n g population would be remarkably d i f f e r e n t i f the new a c t i v i t y were to locate in a small center near the system'sdominant (M th l e v e l ) center rather than in a center of comparable s i z e c l o s e r to the system's boundary (endpoints). In a d d i t i o n , t h i s s t r u c t u r a l  asymmetry i s viewed as a d i r e c t i v e  to any d i f f u s i o n a r y process which would unfold in that system:  instead of  —  15  proceeding i n a symmetric f a s h i o n , central place adoption would now e x h i b i t leading and lagging sectors r a d i a t i n g outwards from the system's dominant center.  FOOTNOTES TO CHAPTER 1  J e f f e r s o n (1939:231) evoked the p r i n c i p l e of the primate c i t y where: "A c o u n t r y ' s leading 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 large and e x c e p t i o n a l l y expressive of national c a p a c i t y and f e e l i n g . " Z i p f , l i k e the c i t e d e a r l i e r observers, was more concerned with the s i z e r e l a t i o n s amongst many centers in a r e g i o n : hence the e v o l u t i o n of the r a n k - s i z e rule: p(l)=R p(R) b  where R i s the rank of the c i t y , p(R) i s the population of the c i t y of rank R, p ( l ) i s the population of the l a r g e s t c i t y (of rank 1 ) , and b i s an e m p i r i c a l l y derived constant. It has been widely stated that as a regional system becomes more complex and the communities more i n t e r r e l a t e d that i s , more integrated s o c i a l l y , p o l i t i c a l l y , and economically — there occurs a s h i f t from primate to r a n k - s i z e features in the c i t y s i z e distribution.  —  The author says aspatial because space i s only recognized as being a " c o n t a i n e r " of a set of c i t i e s in the s t o c h a s t i c approaches w h i l e , on the other hand, in the c e n t r a l place perspective space i s seen as an element which organizes (constrains) the r e l a t i v e a t t r i b u t e s of any community v i s - c i - v i s i t s neighboring communities. These theories have been only " p a r t i a l l y s u c c e s s f u l " because they are not (yet) w e l l - a r t i c u l a t e d in terms of both t h e i r i n t e r n a l s t a t e ments and the domain(s) of r e a l i t y which they purport to cover. ^Parr (1973) has recently suggested a means of modelling the Loschian argument. Losch (1954) had an i n i t i a l model which was fundamentally d i f f e r e n t from the models discussed here; in that o r i g i n a l case, the v a r i e t y amongst central place populations rested upon the s i z e of the nesting f a c t o r per s e .  16  17  Mulligan (upcoming 1976) has demonstrated, however, that C h r i s t a l l e r i a n theory and the r a n k - s i z e rule are indeed compatible under somewhat d i f f e r e n t c o n d i t i o n s . It i s only f a i r to note that Dacey, in f a c t , was r e a l l y the f i r s t to embrace a model w i t h i n the c o r r e c t geometry. See Berry (1967:3). Dacey (1966) e x p l i c i t l y suggested t h i s d i s t i n c t i o n but f a i l e d to u t i l i z e i t in his subsequent formulations. By independent growth the author means that the population expansion (contraction) of the system occurs independently of the d i f f u sionary process; the growth of any one center i s dependent, however, on the growth of the e n t i r e system. The composition of the service m u l t i p l i e r s might be understood by introducing the concept of threshold (as a surrogate f o r supply and demand a n a l y s i s ) into the modelling argument. Some knowledge of t h i s composition becomes c r i t i c a l f o r long run a n a l y s i s because that i s the appropriate scale for t e s t i n g hypotheses of s t r u c t u r a l change in central place systems. As the fourth chapter demonstrates, the composition of each bundle i s less of a problem f o r comparative s t a t i c s a n a l y s i s .  Chapter 2  THE GENERALIZED HIERARCHIAL MODEL  2.1  Introduction In t h i s chapter a t t e n t i o n i s devoted to the terms, notation and  conceptual form of the t r a d i t i o n a l  modelling procedure.  The argument i s  i n i t i a l l y concerned with e l i c i t i n g a h i e r a r c h i a l inter-urban compatible with economic base theory.  structure  From that vantage, the urban employ-  ment linkages of the space-economy may be t y p i f i e d by a p a r t i c u l a r of input-output  approach.  sort  The d i s c u s s i o n then turns to the problem of  g i v i n g a s a t i s f a c t o r y evolutionary i n t e r p r e t a t i o n  f o r the central place  format and a naive s t a g e - l i k e development, i m p l i c i t to the c l a s s i c a l argument, i s proposed as a basis f o r future e x t e n s i o n s .  2.2  Terms and Notation A center that provides the m th  bundle  s e r v i c e s i s said to have the composite function that center does not provide  lf  m +  -j)  Since the center provides the m th  {f > m  of goods and  m (1 < m < M) and i f  i t i s s a i d to have order  m as w e l l .  basket f o r a complementary a r e a , i t  i s said to m - dominate the e n t i r e population i n that surrounding a r e a .  18  19  The notational format adopted here i s a synthesis of those other formats already e x i s t e n t in the l i t e r a t u r e .  Since several new  concepts are to be u t i l i z e d (and hence symbolized), i t was f e l t that a l l concepts in the argument should be e x p l i c i t l y symbolized before proceeding: m:  the h i e r a r c h i a l l e v e l w i t h i n the system (m = 1,2,**«,M); a l s o r e f e r s to the bundle of goods and services associated with that l e v e l (see Figure 2 . 1 ) ;  M:  the t o t a l number of h i e r a r c h i a l l e v e l s w i t h i n the system; a l s o r e f e r s to the t o t a l number of bundles offered w i t h i n the system;  e^: e^.:  the t o t a l employment i n a center on the m th l e v e l ( i . e . in a center o f f e r i n g bundles l , 2 , « » « , m ) ; the sector of e^ engaged in o f f e r i n g the i th bundle ( i . e . engaged in o f f e r i n g {f.} in a center which i s i t s e l f on the mth l e v e l ) ; note that 1 < i < m and t h a t : T T T T e = e , + e \ + ••• + e m ml m2 mm  g e : m  e  B  the export-oriented (basic) body of employment i n a center on the m th l e v e l ; T the sector of e . engaged i n o f f e r i n g the i th bundle to households r e s i d i n g outside of the center on the m th l e v e l ; i . e . the basic sector providing the i th bundle; note t h a t : e = e + e + ••• + e m ml m2 mm B  Pi  e : m  B  B  the non export-oriented (non-basic) body of employment in a center on the m th l e v e l ; note t h a t : e  e  N m i  :  B  T  m  = e  B  m  +  e  N  m  T the sector of e . engaged in o f f e r i n g the i th bundle to a l l households r e s i d i n g w i t h i n a center on the m th l e v e l ; i . e . the non-basic sector providing the i th bundle; note t h a t :  e  m  = e + e + * * ml m2 1  ,  +  0  e mm  and: e . = e . mi mi T  e  N -j  the sector of  :  m i  e . mi  B  T e^  N  +  engaged i n o f f e r i n g the  bundle to the basic component  j th  e . and to a l l m mi n sectors ( i n c l u d i n g i t s e l f ) which s e r v i c e e . with ' mi the bundles 1 , 2 , • • • , i , j , • • • , m ; note t h a t : B  a  e - = e , . + e „ . + * mi mil m2i  ,  ,  + e  mmi  eo:  the number of employees i n each basic r u r a l a r e a ;  E :  the t o t a l employment on the m th l e v e l ; includes both the employment i n the m th l e v e l center and the employment i n the (complementary) area which that center s e r v e s ; note t h a t :  m  c ~T , mm E = e + -j— m m k m e  d.:  the number of dependents f o r each member (employee) T  1  of the sector e . ; mi d :  the number of dependents f o r each rural employee;  P :  the t o t a l population of a center on the  0  m  P .j: m  the population sector of a center on the m th l e v e l engaged i n o f f e r i n g the i th bundle; note that: p . = (1 + d.) e . mi i mi T  T  H  and: p  B p -: m1  m th l e v e l ;  v  m ~ ml p  +  p  1  m2  +  *"  +  mm  p  T the export-oriented portion of p ^ ; note t h a t :  pi, = (1 + d.) mi i mi K  v  21  p : m  the t o t a l export-oriented population of a center on the m th l e v e l ; note t h a t : B p  p^:  m  =  B p  B  +  ml  p  m2  . . .  +  'B  +  ***  mm  p  the non export-oriented (non-basic) population of a center on the m th l e v e l ; note t h a t : r  m  r  m  r  m  r :  the population of the complementary area of a center on the m th l e v e l ; the population of the center i t s e l f i s not i n c l u d e d ; ri i s the population of each basic rural a r e a ;  P :  the t o t a l (market area) population served by a center on the m th l e v e l ; note t h a t :  m  m  P = p + r m m m H  k : m  a s e r v i c e m u l t i p l i e r f o r the 0 < k < 1 m  K  -j:  and  m th l e v e l ;^ note t h a t : M  Y k < 1 , m m= 1 L  m  a nesting f a c t o r , representing the geometry of the c i t y system, which s p e c i f i e s the (equivalent) number of market areas of l e v e l m-l which are contained i n a market area of l e v e l m (m > 1 ) ; i n a C h r i s t a l l e r type central place system note t h a t : K , = K for a l l m > 1 m-l  K _i - 1:  2.3  the (equivalent) number of s a t e l l i t e centers of level m-l which are served by a center on the m th l e v e l (see Figure 2 . 1 ) .  Economic Base Considerations Suppose that employment i n a, r u r a l complementary area i s  that each employee has  d  0  dependents.  Then the t o t a l  eo  ( i . e . household)  r u r a l population i n each of these fundamental s p a t i a l units i s :  and  SITE  MARKET AREA  O  (m-l)  o  (m-2)  Figure 2 . 1 .  BOUNDARY  st level centre nd level centre  A central place system with nesting f a c t o r K , = K = 3; Source: Parr (1969: 241). "' m  23  n  = e  Q  + e ( d ) = (1 + do) e 0  0  (2.1)  0  g Each f i r s t l e v e l center has a basic sector of cerned with o f f e r i n g a commodity bundle If,  e n employees s o l e l y con-  { f i > to these rural households.  f o r a s p e c i f i c service mix and production technology,  k  x  f i r s t level  employees are required for each r u r a l employee (household), then: e?i = ki e  (2.2)  0  g In a d d i t i o n , however, these  en  households must be serviced by a non-  basic sector (which provides f o r i t s e l f as w e l l ) of e n = en  ki + ( k j  k e " 1 - k  2  e ^ i employees where:  + •  B  The t o t a l employment  e[  (2.3) x  i n a f i r s t l e v e l center i s then: T T B . N ei = e n = e n + e n B en 1 - k.  If each employee has g Pi  di  (2.4)  dependents the export-oriented  population  of a f i r s t l e v e l center i s : P? = p ? i = (1 +  and the t o t a l population  p  e?i  (2.5)  is: P i = (1 + d J e L  (2.6)  24  The e x e r c i s e i s simply repeated f o r second level c e n t e r s . t h i s case the employment ponents:  e  In  in the basic sector has two d i s t i n c t com-  2  B e i concerned with the provision of  with the p r o v i s i o n of the new bundle  and  (fi)  2  {f }.  B e  2 2  concerned  According to the geometrical  2  c o n s t r a i n t s of central place theory (see Figure 2 . 1 ) :  e  e  M  Now  e n 2  2 2  B  = k  2 1  :  B . = e n = ki  Ki e  0  e  0  (2.7)  + (Kx - 1) el  M  and  e  employees are required to provide  2 i 2  r e s p e c t i v e l y to the component eSu + e  2 1 2  e i  (and to one another);  2  = e?i i ( k  {fi>  x  + k ) + (ki + k ) 2  2  2  and  {f } 2  i.e.:  +  (2.8)  where: P  N  _  e 11 2  ki efx 1 - ki - k  :  and P  N  e 12 2  N Likewise component  e  2 2  e where:  k e^ 1 - ki - k  (2.9)  2  2  N and  2 2 i  e  _  e  2 2 2  employees are needed to s e r v i c e the second  :  2 2  i + e  2 2 2  - e  2 2  ( k i + k ) + (ki + k ) 2  2  2  +  (2.10)  25  and:  „ N  e 2 2 2  _  _  B k e r1T -T kT T- Tk: 2  , . (2.11)  2 2  x  Total employment  eli  engaged in f i r s t l e v e l a c t i v i t i e s i s :  eli =  e^i  = e  =  and t o t a l employment  el  2  J e  e  2 i  B  2 1  + e n  +  2  e 2i 2  k^Je|j^eLl  i +  ( 2  devoted to providing second order goods i s :  J  2 2  + e  J  4.  = e _ B - e  2 2  2 2  + e , N + e i  2 2  2  It follows that the o v e r a l l employment  2  N + e  2 2 2  el  T T ^ T e = e i + e B „B _ e i + e k1 - k 2  2  Q  in a second l e v e l place i s  2 2  A  2  2 2  2  If the households of employees engaged i n the two a c t i v i t i e s are of s i z e P 1 + di  and  1 + d  2  r e s p e c t i v e l y , then the export-oriented population  of a second l e v e l center i s :  p  2  .  1 2  )  26  B , B P21 + P22  B P2  (1 + d ) e?i + (1 + d ) x  In l i k e manner the t o t a l population  T P2 = p  eL  2  p  (2.15)  of t h i s same center i s :  2  T + p  2 1  2 2  (1 + d J e L + (1 + d ) e l 2  (2.16)  2  By the same procedure i t follows that the components of the basic sector  e of a center on the m th m f ^ p r o v i s i o n of the set --Cf-> I i = l , 2 , * " , m - I J m  1  e  ml  =  k l  e  e  m2  =  k z  {  e!. = k 'm3  of commodity bundles, are:  °  K i e  °  +  ( K l  "  1 }  KiKzeo + ( K i K  3  rm-1  B e „ = k„ 4 n K.e + mm m i= l m-1  n K. -  1=2  1=3  - K ) e j + ( K - 1) e l ^  1=1  K.  2  2  n K. -  m-1  n  2  f-m-1  0  where:  l e v e l , concerned with the  m-1  n K.  1=2  K , -1 m-1  J 1 'm-1  (2.17)  27  e  m  ml  e  Mo  +  m2  e  I k.  i=2  +  Now  e  J  mil  j=l  a-1 n  k  j=i  5 5 1  ^  m  )  K  i-i-  1  a-1 K. -  n  (2.18)  K.  j = i+l  J  J  employees are needed to provide the  to the component  e^.  (and to one another); that i s :  , N B + e . = e . mim mi  mi 2  k. '  i=2  J  i=l  e  I  0  i' «  e  'j2'**"' mim {f..}  m  K.e +  n  T  m  elements of  1  a-2  I a=3  1  i-l  m  m  + e mm  m3  (kj + k  +  2  +  U nr  + (ki + k  2  + ••• + k )  2  m  where:  (2.19)  +  rk. eB .  "mij  However, t o t a l employment  e ..  e . = e . mi mi T  _j  1 - I'.  B  mi  engaged i n  + e  mmi  k. ( e + e + + e ) i - ml m2 mm 1 - ki - k - ••• - k m B  B  B  v  n  m  l e v e l a c t i v i t i e s i s then:  e . mi  A  e  i th  N  +  m 9  and o v e r a l l employment  (2.20)  m  B . N , N e • + e ,. + e . + mi ml i m2i B mi  (1 < j < m)  +  2  becomes  (2.21)  28  T T T e = e , + e' + m ml m2  + e" mm m  m  1x  B , mi  ,  B  L  -  k  X  I=I  i  le . + e + .-. + e ml m2 mm B  m  R  m  B  2  1  L  R  m  B  L  D  i=l  n  Y emi. + k . ' I- . me i . + . . . + km . I, emi . , i=i i=i i=l  k  1  B  B  m  -  1  +  B e  ml  ,h  <  k  + ... + e m2 mm  B e  B  m  "  1  k  ,1  i  'm 1  - 1  1  where  m  1 - Y iii  (2.22) k  i  represents the export base m u l t i p l i e r . k. 1  A g a i n , i f the households of employees engaged in the size lation  1 + d^ ( i = 1 , 2 , . " p m r  of a  m th  m  ,m)  m a c t i v i t i e s are of  r e s p e c t i v e l y , then the e x p o r t - o r i e n t e d popu-  l e v e l center i s :  B  Pml  x +  B .  m2  p  = (1 + d j  +  *'  mm  e ^ + (1 + d ) e + -m2 B  2  + (1 + d ) e' m' mm m  9  (2.23)  29  and the t o t a l population  p  p = p\ m ml K  K  of t h i s same center i s  m  + p^" + • • • + m2 mm K  H  0  These formulations are simply refinements of a recent statement i n the l i t e r a t u r e by P a r r , Denike, and Mulligan (1975).  The c r i t i c a l new assump-  t i o n i s that household s i z e varies according to commodity bundles (due, f o r i n s t a n c e , to d i f f e r e n c e s i n wage l e v e l s ) but independently of where those bundles are o f f e r e d .  The refinement i s s o l e l y intended to i s o l a t e  the d i s t i n c t e f f e c t s of s e r v i c e mix and family s i z e m u l t i p l i e r s in c r e a t i n g the s i z e d i s t r i b u t i o n of urban communities. In a d d i t i o n , several i n t e r e s t i n g properties of central place systems are revealed i n the above arguments. N e of a m m  m th  Since the non-basic a c t i v i t y  l e v e l place i s : r  N N N N e = e , + e o + ••• + e m ml m2 mm c  m  i=l  B  1  < m  m  1  - I i =l  (2-25)  Ic, 1  i t follows that the b a s i c / n o n - b a s i c r a t i o (for employment)  b  m  is:  30 m i = l  (2.26)  m  ,1  Besides, the r a t i o  b>*  m  of the export-oriented population (supported by the  basic employees) to the l o c a l s e r v i c e population (supported by the nonbasic employees) i s :  (2.27)  The l a s t two equations are s i g n i f i c a n t because they make i t apparent t h a t : (i)  (ii)  the b a s i c / n o n - b a s i c r a t i o d e c r e a s e s as c e n t e r s i n c r e a s e in s i z e ; ' the b a s i c p o p u l a t i o n / n o n - b a s i c p o p u l a t i o n d e c r e a s e s ( a s s u m i n g , of c o u r s e , t h a t a l l are s i m i l a r and  (iii)  2.4  ratio dj  in s i z e ) as c e n t e r s i n c r e a s e in  size;  both r a t i o s are independent of the s y s t e m ' s topology.  H i e r a r c h i a l Input-Output Linkages Closer examination of the economic base i n t e r p r e t a t i o n suggests  that employment constrained i n a h i e r a r c h i a l manner permits the urban share of the space-economy to be amenable to a type of input-output a n a l y s i s .  The  format used here to describe i n d i v i d u a l urban economies bears a generic resemblance to the t r a d i t i o n a l  sectoral ( s e r v i c e s , manufacturing, transpor-  t a t i o n , e t c . ) input-output model of spaceless economics but, i n s t e a d , d e l i m i t s employment linkages between the d i f f e r e n t h i e r a r c h i a l sectors of central p l a c e s .  31  Considering the s e r v i c e mix m u l t i p l i e r s as surrogates f o r technical c o e f f i c i e n t s , r e c a l l (2.4) where, rearranging terms:  T T ei = e n = k  x  T B en + en  (2.4)  Now consider the second l e v e l case and create a d i s t i n c t i o n between vectors J  of gross employment  e  and net employment  2  T e i  ki k  el  k  x  2  where the matrix  2  2  k  2  T e i 2  e  e  D  :  2  e  +  B 2 i  (2.28)  T  e  2 2  B 2 2  k  of s e r v i c e mix m u l t i p l i e r s s p e c i f i e s direct 3 quirements f o r each bundle. Solving by matrix a l g e b r a : T e i  1 - k  2  e  1 1 - ki - k  T  B e i  x  2  (2.29) 2  1 - k  x  e  2 2  2 2  _*  The new c o e f f i c i e n t matrix indirect)  k  2  labor re-  k  simply determines the t o t a l ( d i r e c t +  labor requirements f o r each a c t i v i t y .  (2.29) may be s i m p l i f i e d  to:  e  e _  T 2 1  T 2 2  e  e  B  ki k  2 i  B  B e i  :  2  (2.30)  1 - k i - k;  2 2  k  2  k  _  which i s i d e n t i c a l to (2.12) and (2.13)  together.  2  e  B 2 2  32  In a d d i t i o n , the population vector  T  p  2  may be s p e c i f i e d as:  1 + di  P21  el.  (2.31) T  e^  then:  ki ki ki •••  kj  J  k  2  k  2  k  2  •••  k  2  e  J  k  3  k  3  k  3  •••  k  3  e  "m3  e2 2  Considering the general case for.  "ml  T  1 + dj  P22  T m2 T m3  +  • •  J  k k k m m m  "mm  m  e  (2.32)  m3 • •  T  mm  e  mm  I t may be demonstrated that t h i s i m p l i e s :  e  m3  =  e  B ml  which i s i d e n t i c a l to  1  m3  m  • •  •  mm  ki ki k  e  k  2  k  2  k  2  •••  k  2  k  3  k  3  k  3  •••  k  3  km km km  B  mm  m  p^  m  m  'ml 2  m2| B  "m31 m 'mm I  (2.21).  Now the population vector  ••• ki  x  is:  (2.33)  33  T ml  1+dx  T Pm2  0  p  p  T m3  =  l+d  0  2  0  0  0  0  l+d  •  0 3  'ml T  J "m3  (2.34)  •  T mm  p  0  0  1+d. m  J 'mm  Readers f a m i l i a r with comments on the r e l a t i o n between the economic base and t r a d i t i o n a l  input-output  approaches w i l l notice that the b a s i c /  non-basic dichotomy i s given a very s p e c i f i c i n t e r p r e t a t i o n by central place t h e o r i s t s . In both empirical and a n a l y t i c a l studies controversy has always e x i s t e d over d e l i m i t i n g a basic industry i n an open economy.  Recently, f o r  i n s t a n c e , Romanoff (1974) has directed considerable a t t e n t i o n to the problem of c l o s i n g the input-output model f o r basic a c t i v i t i e s ( i . e . moving the productive requirements of basic i n d u s t r i e s from f i n a l to demand).  intermediate  His remarks deserve the concern of those i n t e r e s t e d i n the short  run estimating properties of the regional input-output  format.  However, in a pure c e n t r a l place system, goods and s e r v i c e s may not be exported from the e n t i r e system (a large economic region) but they c e r t a i n l y are exported from centers of one l e v e l to surrounding rural areas and 4 smaller c e n t e r s .  It i s not unreasonable, then, to consider an a c t i v i t y  basic and non-basic simultaneously with reference to the s p e c i f i c market area of a central p l a c e . s p e c i f i e d as above.  The two p r o p o r t i o n s , basic and non-basic, may be  34  2.5  The Problem of Evolution The d i s c u s s i o n now turns to a p a r t i c u l a r l y d i s t u r b i n g aspect of  h i e r a r c h i a l modelling:  the problem of how to r a t i o n a l i z e the  of a central place system with h i e r a r c h i a l p r o p e r t i e s .  evolution  Parr (1970, 1973)  has pointed out that i n the seminal e f f o r t s by C h r i s t a l l e r (1966) and Losch (1954) i t was not e n t i r e l y c l e a r whether the authors were concerned with d e s c r i b i n g economic transformations i n an i d e a l i z e d region or whether they were presenting competing theories f o r e q u i l i b r i u m patterns of supply points and market areas at a c e r t a i n moment in time.  In any c a s e , the derived models  (see Parr (1970) f o r some a l t e r n a t i v e s ) , which are more explicit  in t h e i r  concern f o r income l e v e l s , population f i g u r e s , t r a n s p o r t a t i o n c o s t s , e t c . , are best interpreted i n a s t a t i c sense. Recently there have been attempts to i s o l a t e the d i s t i n c t  effects  of parametric changes in population, technology, e t c . on the functional composition of the hierarchy as i t e x i s t s at one point in time (see Parr and Denike (1970)). to the individual  The a n a l y s i s was p a r t i a l  ( i n that concern was d i r e c t e d  goods that make up commodity bundles) but provided  e s t i n g p o s s i b i l i t i e s f o r future a n a l y t i c extension.  inter-  Nourse (1968), on the  other hand, observed such parametric influence in a much more aggregate and long run f a s h i o n .  By focussing upon the bundles of goods and s e r v i c e s  themselves - as opposed to t h e i r i n d i v i d u a l components - he was able to phrase his argument i n a system-wide manner and not j u s t confine his d i s cussion to change w i t h i n d i s t i n c t central p l a c e s .  Unfortunately, Nourse  was unable to e x p l i c i t l y define the system's s e r v i c e m u l t i p l i e r s at points in time.  different  35  H o p e f u l l y , a d d i t i o n a l i n v e s t i g a t i o n s along these l i n e s w i l l about a much improved understanding of how a t t r i b u t e establishments, e t c . ) changes within (i)  (ii)  bring  (numbers of employees,  central places may be r e l a t e d t o :  t h e somewhat permanent s t r u c t u r a l p r o p e r t i e s ( n e s t i n g f a c t o r s , d i s t a n c e s between c e n t r a l p l a c e s , e t c . ) o f t h e c e n t r a l p l a c e systems t h e m s e l v e s ; and t h e more ephemeral p r o c e s s e s ( t h e i n n o v a t i o n and a d o p t i o n o f new t e c h n i q u e s , a c t i v i t i e s , e t c . ) which o c c u r amongst t h e s e t ( s ) o f c e n t r a l pI a c e s .  In f a c t , the remaining d i s c u s s i o n of t h i s t h e s i s i s l a r g e l y devoted toward c r e a t i n g a set of a n a l y t i c a l perspectives for dealing with such temporal change w i t h i n central place systems.  It i s only f a i r to note,  however, that nowhere i s a comprehensive evolutionary argument being o u t l i n e d i t i s t h i s author's opinion that such an argument i s s t i l l  quite a distance  o f f - but rather a basis i s being a r t i c u l a t e d f o r j u s t d i s c u s s i n g systemic c e n t r a l place changes i n a space-time framework. As a r e s u l t , i t might be best to begin with a s t a g e - l i k e i n t e r p r e t a t i o n of a developing system which has been already b u i l t into the s t a t i c model.  While t h i s only serves as an i d e a l i z e d explanation of how h i e r a r c h i a l  properties could a r i s e , the d e l i b e r a t e m o d i f i c a t i o n of i t s more i m p l i c i t assumptions could h i g h l i g h t c e r t a i n f a c t o r s i n the theory of  inter-urban  systems that have been neglected to date.  2.6  A Proposal of Stage-Like Development Suppose that a geometrically closed central place "system" matures  ( i . e . central places take on a d d i t i o n a l a t t r i b u t e s  i n the form of  activity  36  bundles, employees, e t c . and inter-urban r e l a t i o n s are redefined over time) 5 i n a s e r i e s of stages as suggested i n Section 2 . 3 .  By geometric c l o s u r e ,  the author i s e f f e c t i v e l y avoiding the problem of determining the various central place s i t e s v i s - S - v i s a r u r a l population base:  he i s assuming  that these s i t e s have already been selected ( i . e . that a geometry already e x i s t s ) but that the a c t i v i t i e s (concerned with providing goods and s e r v i c e s ) which are f i r s t housed there are of a minimal ( f i r s t l e v e l or convenience) nature. Suppose, in a d d i t i o n , that new bundles of a c t i v i t i e s are i n t r o duced to t h i s s t r u c t u r e , e i t h e r through innovation or importation of t e c h nique; i n any case, assume that such new a c t i v i t i e s : (i) (ii)  (iii)  a r e i n t r o d u c e d one bundle a t a t i m e ; are introduced in accordance with t h r e s h o l d o r d e r i n g and l o c a t e a c c o r d i n g t o c e n t r a l place p r i n c i p l e s ; a r e c h a r a c t e r i z e d by s e r v i c e m u l t i p l i e r s which do n o t change a f t e r t h o s e f u n c t i o n s have been i n t r o d u c e d . 6  Then, i t may be demonstrated that a central place system, f e a t u r i n g a well-ordered s i z e d i s t r i b u t i o n of communities ( i . e . arranged according to d i s t i n c t s i z e c l a s s e s ; see Parr (1969) f o r the rank ordering of centers according to these s i z e c l a s s e s ) , would develop over time.  Furthermore,  t h i s process of systemic development would r e a l l y be c h a r a c t e r i z e d by the successive i n t e g r a t i o n of i d e n t i c a l subsystems i n t o one large system.  During  t h i s s t a g e - l i k e maturation, h i e r a r c h i a l a t t r i b u t e s of the subsystems would be modified - although h i e r a r c h i a l domination would be retained - while the rural population base would increase in density so as to accommodate  37  the increments of population in the urban centers themselves.  The system,  then, i s viewed as being the f i n a l r e s u l t of processes of i n c r e a s i n g complexity w i t h i n and i n t e g r a t i o n amongst a set of i d e n t i c a l subsystems. The postulate of a rural base ( s e l f - s u f f i c i e n t or trading with an outside region) having a constant density i s retained from c l a s s i c a l theory.  It i s f u r t h e r postulated that density increases would be spread  evenly as development proceeded.^ For purposes of s i m p l i c i t y the argument deals with a standard K  m  -j = K C h r i s t a l l e r i a n geometry and u t i l i z e s the more t r a d i t i o n a l  nota-  t i o n f o r h i e r a r c h i a l models ( i . e . the d i s c u s s i o n i s phrased in terms of populations and not employees and t h e i r dependents).  These s i m p l i f i c a t i o n s  are useful f o r the sake of notation although the more general cases could be discussed i n the same manner. Before proceeding i t might well be best to introduce the new notation which i s necessary because of the temporal element i n the argument:  k : 0  the r u r a l service m u l t i p l i e r , cultural  technology (k  0  < 1 -  representing a g r i m  £  k-, where  i=l  ki, k2,*** k 5  m  are the usual urban s e r v i c e m u l t i -  p l i e r s of the s t a t i c argument); s t i p u l a t e s , in r a t i o form, how many rural residents are required to serve a given body of urban residents (and other rural residents throughout subsequent rounds of p r o d u c t i o n ) ; t : 0  the i n i t i a l point of time; time at which the rural households are s e l f - s u f f i c i e n t and no central place sites exist; the f i r s t point in the m th time i n t e r v a l (m = 1 , 2 , « " , M ) ; the population engaged in providing the m th bundle of goods and s e r v i c e s f i r s t appears at  38  time  t  while i t s supporting population i s subsequently  formed during at time P  L"t ] :  m  ti  A t ; the central place geometry i s defined m  when the f i r s t l e v e l functions appear;  the basic sector f o r the m th bundle; i t i s introduced at those s i t e s which are on the m t h , (m+1) s t , « « « , and M th l e v e l s when the system has c l o s e d ;  C  P  D  the total supporting population needed f o r p [ t ] ; t h i s population i s d i s t r i b u t e d throughout the r u r a l areas and smaller centers dominated by the m th l e v e l place as well as being a l l o c a t e d to the center i t s e l f (note again that t h i s m th l e v e l center may or may not take on a d d i t i o n a l higher order functions at l a t e r times t , ^+2'"*'^'  [At ]:  m  m  m  m  n  p  S mi  t  S P° ° f P t m-' 9 9 i n providing a g r i c u l t u r a l , f i r s t l e v e l , second l e v e l , a n d m th l e v e l goods and 9 m _ s e r v i c e s ( i = 0 , 1 , 2 , • • • ,m); pjj [ A t J = I p ^ [ A t J ; h  e  r t l o n s  At  e n  a  e d  m  i =0  [At ]:  f*i  p  p  S mij  L  m 'm L  t  ' m-'  :  m-'  :  At  +At  c  the portion of area;  m  p  [At ]  mQ  a l l o c a t e d to  each  basic r u r a l  S P° P - [ ] (i 1,2,• • • ,m-l) a l l o c a t e d to each central place on the i t h , (i+1) s t , * * * , and m th l e v e l s (j = 1 , 2 , " « , m ) ;  t h e  r t i o n  o  f  At  mn  t  h  e  t  o  t  a  direct  l  a l l o c a t i o n to the  P^ t] ^m [ t m + A m L  I f there are  =  m  m  M-1 K  = m p  B  J  m th  [tm ] +mm p S  K  L  J  K  L  level place;  [At m ] J  points of supply introduced at a time  ti,  then  M-1 i t may be argued that  K  r i [ t ] , " e x i s t e d " at time 0  provides  {fi>  rural market areas, each of population s i z e t  0  (ti  > t ). 0  The basic sector  to a rural area) introduced at each  involves a population of:  p T [ t i ] (which  of these supply points  39  Pi [ t l ]  In the time i n t e r v a l formed throughout  [AtJ  = kiTi  (2.35)  [to]  a supporting body of population  p\ [ A t ] x  is  each f i r s t l e v e l market area (both at the point of pro-  duction and i n the complementary area) where:  Pi [Atx] = p? [ t ]  +(k  x  (ko + ki) 1  Pi [Ati]  _  k o  _  k l  + kj  0  + (k  + k )  0  +  z  x  B -, Pi L t d  (2.36)  f  has two components:  P  ?o [ A t j  -  _ i:  1  _  k i  P  ?  P  l  (2.37)  [tj  and:  Pu  [AtJ  1  k1 - ko - k!  B  n  r. [  t  l  ]  (2.38)  which imply that the population in each rural complementary area at time t  x  + Ati  is:  r i [ti  + Ati] = r j [t ] 0  + pfo  [Ati]  r> l 4- k p k ^ i  and the population i n each f i r s t l e v e l place i s :  [tp]  (2.39).  40  Pi [ t i  + A t J = p? [ t j ]  +  P  l  [AtJ  l  ki r 1 - k  [t ] - ki  2  = kiTi [ t ]  +  0  At time  t  (t  2  > ti + A t J  2  (2.40)  0  l 0  basic sectors are introduced at  M-2 K  points i n order to provide  services  K  rural areas and  {f }.  Each of these second l e v e l s i t e s  2  f i r s t l e v e l places (one center i s  K  "itself":  the f i r s t l e v e l population occupying the same p o i n t ; the geometric arguments by Dacey (1965, 1966) should e l u c i d a t e t h i s p o i n t ) .  The number of  persons comprising each of these basic sectors i s : P2 [ t ]  - k  2  ^K  2  [t  Pl  + Atx] + K r i [ t  x  Once a g a i n , now in the time i n t e r v a l  + AtJl  :  (2.41)  A t , a supporting population i s 2  required:  S p [At ] 2  2  B = p [t ] 2  =  ' j-(k + k i + k z)  2  + (k  0  (k + k i + k ) 1 - k - ki - k Q  P2  2  0  0  + k  + k )  x  2  2  + •••  (2.42)  [t ] 2  2  which has three components:  P20 [ A t ] 2  P«  =  tAt ] = 2  1 - k  ]  .  k Q  - k i - k P2  0  [t ] 2  (2.43)  2  \  .  k z  p? [ta]  (2.44)  41  It follows that each r u r a l area has an incremental gain of:  r i [At ] 2  S = Vf- [ A t ]  (2.46)  2  T h i s , of course, constrains the population growth i n the  K-l  centers M-2  which do not take on second l e v e l a c t i v i t i e s (there are ( K - l ) K these in the e n t i r e system). In each of these c e n t e r s , which w i l l  of remain  f i r s t l e v e l centers during the closure of the system, incremental growth i s :  pin [At ] = T - r ^ - r i [At ] 2  The remainder  p  [At ]  2 1 2  2  of incremental f i r s t l e v e l a c t i v i t y i s a l l o c a t e d  to the second l e v e l center  P212 [At ] 2  (2.47)  2  itself:  = pf! [ A t ] 2  - (K-l) p L i [At ] 2  In a d d i t i o n , a l l persons d i r e c t l y providing  {f } 2  (2.48)  are residents of the  second l e v e l center:  P2 [ t  2  + A t ] = p? [ t ] 2  2  + pf  2  [At ]  (2.49)  2  Hence, considering the system as a whole, there a r e : (i)  M-l K Ti [ t  rural a r e a s , each of s i z e : 2  + A t ] = rx [ t i 2  + At ] + n x  [At ] 2  (2.50)  42  M2 ( K - l ) K " f i r s t l e v e l c e n t e r s , each of s i z e :  (ii)  Pi [ t  and  2  + Atx] + p  x  [At ]  2 1 1  + A t ] = px [tx + Atx] + p f [ A t ] + p  2  (2.51)  2  M-2 K " second l e v e l c e n t e r s , each of s i z e :  (iii)  Pa [ t  + A t ] = p! [ t  2  2  1 2  2  [t  2  + At ]  2  (2.52)  2  The argument i s repeated for subsequent time periods u n t i l geometric closure at time  t  + At^.  M  The a l l o c a t i o n process becomes a b i t  more complicated but follows the above i n a symmetric f a s h i o n .  The basic  sectors introduced during subsequent time i n t e r v a l s are of the form:  P? [ t ]  = k  3  |K  3  P  2  [t  + A t ] + K(K-l)  2  2  + K r  [t  2  p? [ti»] = U -j Kp  + K (K-1)  3  [t  PM  [ t  M  ]  =  V  ! <  P l  P -1  [ t  M  ,M-2 + K"-'(K-1) + K " M  ] r i  [t  + At ]  2  2  3  + A t ] + K rx [ t 3  3  3  M-l  P l  M - ]  +  [t  A t  M-l  M - 1  + At  ]  + At M - 1  ]'  2  + At ] 2  }  + A t ] + K(K-l) p  3  [t  2  x  [t  P l  3  2  [t  3  + At ] 3  + At ] 3  +  M - ]  ] (2.53)  43  A careful s c r u t i n y of the case leads to the conclusion that r u r a l population f o r the e n t i r e system i s :  m (l-ko)O R  m ^m  +  A t  m^  =  ^  ^ ^  at time  t  + At .  ^  "  1  I  I  k  i=0  k.)  i  (2-54)  In addition t o t a l population (rural and urban) f o r the  e n t i r e system i s :  m K  T  +  At  m] '  r  . Ct«3  '  1  -  1  °  k m  I i=0  at time  t  m  + At -  This implies that the r u r a l / t o t a l  m  m  R  (2-55) k  i ' population r a t i o  =p- a t time t + A t i s simply 1 - £ k. (where the r a t i o s d e c l i n e m i=l according to the sequence; 1, l - k 1 - k i - k , e t c . as the system matures). U I t follows that the urban/rural population r a t i o ^ at t h i s time i s : m m  l 5  2  m  U m  T - R m m  r  —R—  =  m where  b  m  =  m  T m  •,  i  R~ ~ m  k  J,  i  ,  =l  to ac\  m— 1  r  ,  =  B-  ( 2  -56)  m  i s the b a s i c / n o n - b a s i c r a t i o defined i n Section 2 . 3 . This l a s t r e s u l t serves to i l l u s t r a t e a very i n t e r e s t i n g feature  of the s t a g e - l i k e perspective on development in central place systems. As long as a l l s e r v i c e m u l t i p l i e r s are assumed to remain constant over time, then the urban/rural population balance of the system (or any subsystem) i s  44  (i)  independent of the r u r a l service m u l t i p l i e r and the system's ( s u b - /  system's) topology and ( i i )  equals the inverse of the b a s i c / n o n - b a s i c r a t i o  of the l a r g e s t center in the system (subsystem).  This r e s u l t , of course,  holds true f o r the s t a t i c model o u t l i n e d e a r l i e r in Sections 2.3 and 2.4 ( i . e . the s t a t i c model represents the f i n a l stage at time t  2.7  + At ).  o  Concluding Remarks The i n t e n t i o n of t h i s chapter was to introduce the reader to the  elementary s t r u c t u r a l  properties of the general h i e r a r c h i a l model.  s o , several additions were made to the e x i s t i n g l i t e r a t u r e : basic r a t i o was given a simple i n t e r p r e t a t i o n , ology was embraced in a h i e r a r c h i a l input-output  In doing  the b a s i c / n o n -  the economic base methodscheme, and systemic e v o l u -  t i o n was given a naive s t a g e - l i k e p e r s p e c t i v e . The author advocates use of the h i e r a r c h i a l model because of flexibility  its  - t h i s i s i m p l i c i t to the arguments i n Sections 2.3 and 2.4  and i s stressed throughout the remainder of the t h e s i s - and i t s both i n a s t r u c t u r a l  and functional  symmetry,  sense ( e . g . the economy of the l a r g e s t  center i n the system was demonstrated to be c l o s e l y t i e d to the urban/ r u r a l population balance of the e n t i r e system).  FOOTNOTES TO CHAPTER 2  'The m u l t i p l i e r k may be interpreted as the number of households w i t h i n a c i t y of level m (or higher) that are needed to provide m th l e v e l goods and services ( i . e . the m th l e v e l bundle), as a proportion of the number of households served ( i . e . throughout the e n t i r e m th l e v e l market area of the c i t y ) . When a l l households are of the same s i z e , then "populat i o n " may be substituted f o r "households" i n the above. m  Becker (1956:509) has stated that t h e : . . . traditional view, based usually on simple correlations, has been that an increase in income leads to a reduction in the number of children per family. If, however, birth-control knowledge and other variables were held constant economic theory suggests a positive relation between family size and income, and therefore that the traditional negative correlation resulted from positive correlations between income, knowledge, and some other variables.  The vector  e  T  leli  identifies e  involved i n f i r s t This i s analagous output a n a l y s i s sector, including  the t o t a l  employment  2 2 2  and second level a c t i v i t i e s i n a second l e v e l center. to the term of gross output used i n t r a d i t i o n a l inputthat i s , the t o t a l output produced by an industry or a the portion consumed by the industry (sector) i t s e l f . |e?i  The vector  identifies  e5 |e?  the portion of  e  2  which i s not involved  2  in s e r v i c i n g itsel_f_ with goods and services - that i s , i t represents the basic sector of  e l . The analogous term i n the usual input-output a n a l y s i s  45  46  i s net output, which represents the portion of gross output w i t h i n an industry (sector) not consumed by the various production i n d u s t r i e s ( s e c t o r s ) ; i n other words, i t represents f i n a l demand. The matrix k i n d i c a t e s the production technology a v a i l a b l e at the second l e v e l place f o r u t i l i z i n g employees (labor i n p u t s ) ; the analagous term for sectoral input-output a n a l y s i s i s the matrix of technical c o e f f i c i e n t s . By examining the general formulations of equations (2.32) and (2.33) the reader should see that t h i s technology i s i n v a r i a n t despite changes i n the s i z e s of central places ( i . e . there are no scale economies to be r e a l i z e d by performing the same a c t i v i t y in a l a r g e r c e n t e r ) .  By r e l a x i n g assumptions, commodities may be exported from the system (as Parr (1970) has suggested), but the b a s i c / n o n - b a s i c r a t i o would not be conceptualized any d i f f e r e n t l y .  The author says "system" because, to be e n t i r e l y c o r r e c t , a l l the central place s i t e s are not elements of a system u n t i l a well-ordered s i z e d i s t r i b u t i o n i s formed - that i s , not u n t i l the various independent subsystems have merged together into one large system.  It might be argued that t h i s i s a d e t e r m i n i s t i c counterpart to the ergodic p r i n c i p l e of s t a t i s t i c s . In s t a t i c central place modelling i t i s assumed that s e r v i c e m u l t i p l i e r s are the same i n d i f f e r e n t - s i z e d centers (perhaps t h i s i s a b i t of a sore p o i n t ) ; now, i t i s a d d i t i o n a l l y assumed that these m u l t i p l i e r s remain constant over time.  S t r i c t l y speaking, the argument assumes that there are no diminishing returns to a g r i c u l t u r a l p r o d u c t i v i t y as the r u r a l base i n c r e a s e s . This simply complements the s u p p o s i t i o n , mentioned above, that the urban s e r v i c e m u l t i p l i e r s have f i x e d labour-output r a t i o s .  It may be demonstrated that t h i s r e s u l t holds t r u e , as w e l l , f o r the more general argument with a v a r i a b l e nesting f a c t o r .  Chapter 3  CENTRAL PLACE DIFFUSION  3.1  Introduction Over the past f o r t y years a number of empirical studies have  focussed on d i f f u s i o n i n an inter-urban context.^  From t h i s body of  l i t e r a t u r e has emerged a corpus of inductive laws r e l a t i n g  diffusionary  a t t r i b u t e s f o r d i v e r s e items ( i n n o v a t i o n s , o p i n i o n s , e t c . ) to v a r i a b l e s such as the a v a i l a b l e transport technology, the s o c i a l structure of communication, the distances from adopting centers to e a r l i e r adopters or nearest l a r g e s t neighbors, e t c . However, i t  i s only in the very recent past that c e r t a i n  observers have attempted to phrase more systematic statements about the inter-urban adoptive process.  Hudson (1969) was the f i r s t to advocate use  of a p o s t u l a t i o n a l format - embracing the geometry of central place thinking with g r a v i t y - p o t e n t i a l  p r i n c i p l e s - which allowed f u r t h e r s t a t e -  ments to be i n f e r r e d (and eventually t e s t e d ) .  Pederson (1970), in a con-  temporary a r t i c l e , discussed a wide v a r i e t y of issues which should s t i l l be of i n t e r e s t to t h e o r e t i c i a n s and e m p i r i c i s t s a l i k e ;  unfortunately,  his more a n a l y t i c a l statements, based as they were on a misunderstanding of the C h r i s t a l l e r i a n topology, must be greeted with some s k e p t i c i s m .  47  48  The ideas expressed i n these two papers served as invaluable methodological guidelines for the present author when he f i r s t began to r e f l e c t about d i f f u s i o n in a central place context.  While i t  i s presently  argued that there are s i g n i f i c a n t i n c o n s i s t e n c i e s i n e i t h e r of these contributions  - hopefully these are eliminated in the upcoming a s s e r t i o n s -  the d i f f u s i o n a r y model which i s o u t l i n e d in t h i s chapter i s r e a l l y fashioned from the more promising (in t h i s author's opinion of course) features of Hudson's and Pederson's proposals. More to the p o i n t , t h i s author f i r m l y advocates t h e i r deductive approach as being the proper means of u n i t i n g , in a n a l y t i c a l form, and process in central place systems.  structure  Eichenbaum and Gale (1971:541) would  perhaps agree that t h i s represents a s p e c i f i c attempt at the macro scale of making " . . .  the t r a n s i t i o n to form-function-process. . . " from a  s t r i c t l y form-function methodology. The i n i t i a l  task of t h i s chapter i s to examine c l o s e l y the r e l e -  vant properties of the two models. f i r s t and his s t r u c t u r a l  Since Hudson's proposal d i d appear  ( i . e . geometrical) axioms were c o r r e c t l y stated  ( r e c a l l that Pederson's were n o t ) , much of the upcoming d i s c u s s i o n i s a p p r o p r i a t e l y d i r e c t e d towards " h i s " model. t i o n , however, that these s t r u c t u r a l  It  i s t h i s author's conten-  axioms require a second look when  they are placed in a structure-process nexus. On the other hand, t h i s author advocates Pederson's conception of the time element ( i . e . continuous versus d i s c r e t e ) i n the d i f f u s i o n a r y argument and a l s o favors his preference towards parametric i n the g r a v i t y - p o t e n t i a l  assertion.  flexibility  49  Besides, t h i s author has several refinements of his own to add. Perhaps the most s i g n i f i c a n t stems from his ongoing experiences with h i e r a r c h i a l c i t y s i z e modelling (as i n the previous chapter):  he i s able  to employ a much wider v a r i e t y of community s i z e d i s t r i b u t i o n s  in his  argument than e i t h e r of his predecessors. In contrast to the previous chapter, though, the d i s c u s s i o n proceeds in a very h e u r i s t i c (more s p e c i f i c a l l y , numerical) f a s h i o n .  Even  i n the i d e a l i z e d central place s e t t i n g the reader should already appreciate that the i n t e r p l a y of such diverse f a c t o r s might well create an assortment of acceptance patterns.  It was decided, t h e r e f o r e , that a n o n - a n a l y t i c a l  approach, characterized by a s e r i e s of tables showing generated would be most helpful  data,  i n suggesting how i n d i v i d u a l parametric values and  s p e c i f i c d i f f u s i o n a r y schemes might be r e l a t e d . The author had both p h i l o s o p h i c a l and p r a c t i c a l reasons f o r advocating the h e u r i s t i c approach.  F i r s t l y , he shares the opinion with  many others (see Bergmann (1957) and Rudner (1966)) that o v e r - f o r m a l i s a t i o n i s not n e c e s s a r i l y useful in the e a r l y stages of an e x e r c i s e .  Secondly,  t h i s author i s not e n t i r e l y c e r t a i n whether the argument may in f a c t be stated i n a s u i t a b l e a n a l y t i c fashion - perhaps an a l g o r i t h m i c format i s the most rigorous p o s s i b l e .  3.2  C r i t i c i s m s of the E x i s t i n g Models Hudson and Pederson shared the common i n t e n t i o n of  to r e l a t e some rather disparate concepts i n d i f f u s i o n theory.  attempting The " n e i g h -  borhood e f f e c t " of c l u s t e r e d or contagious growth, innovation appearance  50  according to the s i z e d i s t r i b u t i o n  of c i t i e s , and the use of the l o g i s t i c  curve i n d e p i c t i n g cumulative adoption were topics of special concern. They were each able to devise a Hagerstrand-1ike t e l l i n g process f o r i n i t i a l community adoption and lend some credence to the hypothesis t h a t the above phenomena are not mutually e x c l u s i v e , but s y s t e m a t i c a l l y r e l a t e d , properties of d i f f u s i o n a r y processes at the macro l e v e l . reader should soon see, however, that the degree to which t h i s  The  hypothesis  has been confirmed i s somewhat tenuous. To begin w i t h , each of t h e i r models i s l i m i t e d due to the shared assumption that c e n t r a l place populations would increase according to the system's nesting f a c t o r f o r market areas ( i . e . in a munity populations might increase in the sequence 27000,•••)•  K = 3  system com-  1000, 3000, 9000,  In Parr (1970) may be found a comprehensive summary of com-  peting h i e r a r c h i a l models where the s t r u c t u r a l  p  m  1 - ki  t y p i f y i n g t h e i r assumption i s included.  equation:  K  ( 3  '  By a rather simple modification  of the statement in Mulligan (upcoming 1976), t h i s model -  incidentally,  f i r s t suggested by Losch - may be shown to be but a special case of the general model o u t l i n e d in the previous chapter of the t h e s i s . Secondly, i t was an enlightening idea f o r each to employ demographic force as the impetus f o r inter-urban d i f f u s i o n but, Hudson r e s t r i c t e d his argument to the c l a s s i c a l form:  unfortunately,  i . e . , where the  distance exponent has a value of two (see Stewart and Warntz (1958)). Pederson, however, opted f o r the increased g e n e r a l i t y which could be achieved by applying the potential  formula:  51 Pj P !  where  TH  =  G  H  ~J~ TH  ( 3  i s the p o t e n t i a l expressed at a center of population  a center of population  p  T  located  d.^  p^  units d i s t a n t , the constant  i s simply a s c a l i n g f a c t o r , and the exponent  B represents the  by G  friction  of d i s t a n c e . At one time considerable debate revolved about the meaning, and therefore magnitude, of t h i s distance exponent (see Isard (I960)) but more r e c e n t l y t h e o r e t i c i a n s (see Niedercorn and Bechdolt (1969)) and e m p i r i c i s t s a l i k e have shunned the s t r i c t Newtonian approach.  S u f f i c e i t to say  t h a t i n r e s t r i c t i n g the distance exponent to a value of two, an observer i s making an e n t i r e l y a p r i o r i assumption - one which i s e s p e c i a l l y untenable i n the central place s e t t i n g where i n t e r a c t i o n i s s o l e l y d i r e c t e d by distribution-consumption motives ( i . e . the domain of t r i p purposes i s 2 limited).  With c l o s e r s c r u t i n y of Hudson's a r t i c l e , the reader w i l l  doubtless agree that t h i s r e s t r i c t i o n was employed because i t had a "cancel 1ing-out e f f e c t " when united with the h i e r a r c h i a l model mentioned above. More s i g n i f i c a n t l y , however, i t  i s d i f f i c u l t to agree with  various aspects of the author's t e l l i n g - h e a r i n g procedures.  For i n s t a n c e ,  one-way domination by c i t y s i z e , while being a s t r u c t u r a l tenet of C h r i s t a l l e r i a n theory, i s d i f f i c u l t to j u s t i f y as a necessary c o n d i t i o n f o r central place d i f f u s i o n .  Since in theory two centers create the same  p o t e n t i a l on one another ( a l b e i t one place may be much l a r g e r ) i t would be more natural f o r adoption to proceed i n e i t h e r d i r e c t i o n although in accordance with the previous h i s t o r y of the d i f f u s i o n a r y process. Perhaps  52  a v i a b l e way of looking at t h i s point i s to d i s t i n g u i s h between the short run and the long run: s i d e r a b l y more secondary  from the f a c t that a large center may have conadoptions (due to second, t h i r d , e t c . hearings)  than a small c e n t e r , i t does not follow that the large center i n i t i a l l y adopted (for the f i r s t time) p r i o r to the small c e n t e r . This seems an appropriate time to emphasize that the Hudson and Pederson c o n t r i b u t i o n s and the model to be outlined below are a l l confined to primary  adoption; however, t h i s does not preclude t h e i r value as a  basis f o r developing more s o p h i s t i c a t e d models which could incorporate secondary adoption. In a d d i t i o n , the actual generative structure of Hudson's t e l l i n g procedure leaves something to be d e s i r e d . emanate from t e l l i n g centers  p  T  He has postulated that messages  to hearing centers  p^  during d i s c r e t e  time i n t e r v a l s and that these messages are only e f f e c t i v e when the t e l l ing center has i t s e l f already adopted.  Unfortunately one i s l e f t with  the impression that a l l e f f e c t i v e t e l l i n g performed w i t h i n the same time interval  i s characterized by the same demographic force ( p o t e n t i a l ) .  As  Appendix B demonstrates, t h i s i s not r e a l l y the case at a l l . But perhaps the real issue concerns the usefulness of d i s c r e t e time i n t e r v a l s f o r conceptualizing message generation at the macro l e v e l where the i n t e r a c t i o n between communities tends to be e s p e c i a l l y continuous. Hagerstrand (1965) himself was even a b i t s k e p t i c a l about employing t h i s assumption at the micro ( i n d i v i d u a l )  level.  While the d i s c r e t e i n t e r p r e t a t i o n  i s e s p e c i a l l y useful f o r s t o -  c h a s t i c modelling, the continuous perspective has other advantages. F i r s t l y , i t allows process time to be r e a l i z e d :  i . e . an o b j e c t i v e measuring  53  of time in terms of a system's process (parameters).  S p e c i f i e d periods  of time may then be r e l a t e d on a r a t i o scale (perhaps, as Pederson suggested and i s the case below, with respect to the threshold between " i n v e n t i o n " in a where).  time i n t e r v a l  M th l e v e l place and i t s f i r s t appearance e l s e -  In a d d i t i o n , continuous time ( a l b e i t conditions are i d e a l i z e d )  represents an improved basis f o r d i s c u s s i n g p r e d i c t i o n of process a t t r i butes.  The use of d i s c r e t e time i n t e r v a l s , while being valuable f o r d e s c r i p -  t i o n with h i n d s i g h t , i s a b i t tenuous f o r p r e d i c t i o n when no r u l e s e x i s t f o r s t i p u l a t i n g the appropriate lengths of those i n t e r v a l s . F i n a l l y , in order to c l a r i f y what Hudson's c h a i n - l i k e adoptive procedure a c t u a l l y e n t a i l s , i t becomes necessary to delve i n t o a somewhat unique feature of central place t h i n k i n g : centers.  the existence of  equivalent  The d i s c u s s i o n w i l l disgress f o r a short time and then return  to the Hudson model. As a consequence of t h e i r symmetric c o n f i g u r a t i o n s , a l l place systems are characterized by an ordering p r i n c i p l e whereby  central m th  l e v e l places m - dominate the urban (as well as r u r a l ) population i n t h e i r market areas.  However, they directly  dominate only shares of the s u r -  rounding places on the ( m - l ) s t , (m-2)nd, e t c . l e v e l s . f o r i n s t a n c e , an  In a  K=3  system,  m th l e v e l place m - dominates two places of the ( m - l ) s t  o r d e r , s i x places of the (m-2)nd o r d e r , e t c . but i t d i r e c t l y dominates only two places (one-third of the s i x surrounding centers) of each lower order. It  i s in t h i s sense that centers dominate equivalent centers in the central  place scheme. For c e r t a i n o p e r a t i o n s - f o r i n s t a n c e , simply counting up the number of places (points) belonging to a M l e v e l system - t h i s poses only a  54  minor problem.  Since most of the centers are obviously i n t e r i o r  to the  system an assignment p r i n c i p l e need only be applied to those centers located along the boundary of the e n t i r e system ( o r , in the l i n e a r case, a t the endpoints of the system). On the other hand, when a t t e n t i o n  turns to domination as a  d i r e c t i v e for a s p e c i f i c process (as f o r d i f f u s i o n in Hudson's argument), equivalence cannot j u s t i f i a b l y  be overlooked.  In the f i r s t p l a c e , i t s  neglect gives a rather d i s t o r t e d s p a t i a l perspective of a process which is  i d e a l l y symmetric (even i n a s t o c h a s t i c sense; see Appendix B ) .  The  tendency to ignore equivalence - even though t h i s may be an expedient approach f o r i l l u s t r a t i n g market nesting ( e . g . see Berry and Pred (1961))  -  is e s p e c i a l l y dangerous, however, because i t s k i r t s the important issue 3  of closure  f o r the e n t i r e system. The geometric argument f o r central place theory  by an assumption of perfect adjustment) statements may r e f e r (i) (ii)  general that  to:  one i s o l a t e d several  system o f m l e v e l s ; o r t o  a d j a c e n t systems o f  The f i r s t i n t e r p r e t a t i o n second i n t e r p r e t a t i o n ,  is sufficiently  (characterized  m  levels.  simply r e f e r s to a p e r f e c t l y closed  to the extent that d i f f e r e n t  system.  The  systems may share  centers on t h e i r common boundaries (at t h e i r common endpoints), refers an open system.^  to  A system's state of openness (closure) i s maximized  (minimized) i n the geometrical sense ( i . e . the system may be considered 5 p e r f e c t l y open) when that system i s completely enveloped by other systems. However, real world systems are a l l open: the property of closure in t h e i r a b s t r a c t representations i s simply the r e s u l t of an i n t e l l e c t u a l  55  operation - whether t h i s be due to the r e s e a r c h e r ' s judgement, convenience i n gathering d a t a , or some other reason.  In any case, i t remains exceedingly  d i f f i c u l t to speculate about how the actual degree of closure ( i . e .  the  r e s i d u a l openness) in real world systems may influence the a t t r i b u t e s of processes unfolding t h e r e i n .  At l e a s t by employing the concepts of  perfect  c l o s u r e and openness, the l i m i t s of a t t r i b u t e v a r i a t i o n can be e s t a b l i s h e d i n the i d e a l i z e d representations ( i . e . abstract models) of those real world systems.  This i s hardly a problem which i s confined to central place  systems and the interested reader might wish to consult Harvey (1969:419-420) f o r a few general comments on the i s s u e . In a l l honesty, the author i s not at a l l suggesting a neat s o l u t i o n to t h i s dilemma but only wishes to emphasize i t s e x i s t e n c e .  By at  l e a s t recognizing the problem and e s t a b l i s h i n g bounds to i t s s i g n i f i c a n c e , the empirical q u a l i f i c a t i o n of t h e o r e t i c a l  statements may proceed i n an  atmosphere of improved confidence. Returning to Hudson's c o n t r i b u t i o n , then, i t appears that he has postulated the existence of a p e r f e c t l y open system since a "demonstration" was given to i l l u s t r a t e  that equal amounts of demographic force were being  expressed at a l l centers d i r e c t l y dominated by the  M th l e v e l p l a c e .  However, according to the above argument, he has then neglected the potential expressed on the ( M - l ) s t l e v e l places on (at) the system's boundary (endpoints) by tive interpretation  M l e v e l centers in adjacent systems.  The a l t e r n a -  - that of perfect closure - i s precluded by the over-  emphasis of the populations of these same ( M - l ) s t l e v e l  places.  6  In the upcoming section a t t e n t i o n i s directed to t h i s s p e c i f i c problem as well as to the other q u a l i f i c a t i o n s mentioned before i t .  When  the reader becomes acquainted with the new model and sees t h a t i t  is a  m o d i f i c a t i o n of Hudson's and Pederson's o r i g i n a l endeavors, then, hopefully  he should see t h i s lengthy c r i t i q u e i n the r i g h t l i g h t :  not as an  o v e r f a s t i d i o u s attack but r a t h e r as a c l a r i f i c a t i o n of methodology.  3.3  An A l t e r n a t i v e Model of Central Place D i f f u s i o n It  i s now postulated t h a t : (i)  (ii)  (iii)  a c e n t r a l p l a c e system e x i s t s which may be a r e a I o r l i n e a r , c l o s e d o r open, b u t whose urban popul a t i o n s may be e x p r e s s e d a c c o r d i n g t o t h e g e n e r a l h i e r a r c h i a l model o f t h e second c h a p t e r ; a d i f f u s i o n a r y p r o c e s s may be c o n c e p t u a l i z e d t e l l i n g - h e a r i n g manner;  in a  the mechanism f o r t h i s t e l l i n g procedure i s i n t e r c i t y p o t e n t i a l (as e x p r e s s e d i n e q u a t i o n (3.2) above) which i s a r t i c u l a t e d a c c o r d i n g t o t h e principles o f m - d o m i n a t i o n and which i s e f f e c t i v e o n l y when one o f t h e c i t i e s has i n f a c t adopted; t e l l i n g commences j u s t as soon as a c e n t e r does adopt (knows) and i t may proceed i n e i t h e r d i r e c t i o n between t h e two c e n t e r s ; ^  (iv)  potential i s e x p r e s s e d by t h e entire populations of r e l a t e d c e n t e r s a l t h o u g h , as was argued e a r l i e r , o n l y s h a r e s o f ( m - l ) s t ( n e a r e s t (m-2)nd, (m-3)rd, e t c . ) l e v e l p l a c e s a r e d i r e c t l y dominated by each m t h l e v e l p l a c e ; f o r i n s t a n c e , i n a K=3 system, an M t h l e v e l p l a c e e x p r e s s e s p o t e n t i a l a t six s u r r o u n d i n g c e n t e r s housing (M-I)st level a c t i v i t i e s but t h e p o p u l a t i o n s o f t h o s e ( M - I ) s t l e v e l p l a c e s a r e dependent on t h e system's c l o s u r e ;  (v)  complete h e a r i n g ( i . e . a d o p t i o n ) o n l y o c c u r s when a s u f f i c i e n t amount o f r e s i s t a n c e has been overcome; f o r primary ( i n i t i a l ) a d o p t i o n t h i s threshold is the same f o r a l l c e n t e r s i n t h e system.8  If  the modelling procedure i s s t i l l  a bit  unclear the reader  should turn to Appendix C where adoptive times are c a l c u l a t e d i n a  57  step-by-step manner f o r the open and closed cases, r e s p e c t i v e l y , of a four l e v e l central place system.  K=3  This same procedure underlies the gen-  erated data given i n the f o l l o w i n g s e r i e s of tables - t h i r t y - t w o in a l l were selected by the author (from a set of about f i f t y o r i g i n a l l y handcomputed) in order to i l l u s t r a t e j u s t how parametric v a r i a t i o n may drama9 t i c a l l y modify adoptive patterns in the central place scheme. I f the notation in the table headings appears a b i t e s o t e r i c i s common in c i t y s i z e arguments), then the reader should again r e f e r  (it to  Appendix C where the s p e c i f i c population f i g u r e s are presented f o r each case.  An acquaintance with the l i t e r a t u r e on h i e r a r c h i a l modelling and  gravity-potential  theory should i n d i c a t e that the chosen magnitudes of  the parameters are w i t h i n p l a u s i b l e bounds. In a d d i t i o n , Figures 3.1 and 3.2 are included to i l l u s t r a t e adoptive times (for both the open and closed cases) in a  K=2  the  four l e v e l  context. 3.4  Propositions Based on the A l t e r n a t i v e Model Scrutiny of the aforementioned tables should i n d i c a t e that adoptive  patterns may indeed be permuted by a v a r i e t y of f a c t o r s . i n the absence of a great number of such t a b l e s , i t  Unfortunately,  i s possible to specu-  l a t e only l o o s e l y about the s i n g u l a r e f f e c t s of parameters: the  formulation  of c e t e r i s paribus statements i s a b i t r i s k y when parametric values are so i n t i m a t e l y r e l a t e d .  On the other hand, some general trends are apparent  and these are considered i n the upcoming d i s c u s s i o n . I t should be emphasized again ( i f the reader missed the point i n Appendix C) that the adoptive patterns presented in the tables are  58  Table 3.1 Adoptive Times for Centers in an Open Five Level Central Place System; K = 3, p! = 1000, p /p , = 4 (1 < m < 5 ) , b = 1.5 ^ m ^m-1 K  Cumulative Adopti ng Center  Equiv. No.  Centers %  Adopti ng Time  % Total Time  5  1  1.23  0.00  0.00  4  3  3.70  1.00  1.93  3  9  11.11  3.84  7.45  15  18.52  8.64  16.76  21  25.92  11.87  23.02  27  33.33  13.68  26.53  33  40.74  15.98  30.99  ,2  39  48.15  37.37  72.48  1  6  51  62.96  43.10  83.59  1  3  63  77.77  43.88  85.10  69  85.18  50.57  98.08  81  100.00  51.56  100.00  2 2  1  3  2  2  i  1  1  1  5  k  59  Table 3.2 Adoptive Times f o r Centers in a.Closed Five Level Central Place System; K = 3,  P l  = 1000, p / p m  m  , = 4 (1 < m < 5 ) , b = 1.5  Cumulative Adopti ng Center  Equiv. No.  Centers %  Adopting Time  % Total Time  5  1  1.23  0.00  0.00  3  7  8.64  1.00  7.04  4  9  11.11  1 .55  10.92  15  18.52  1.67  11.76  21  25.92  3.04  21.38  27  33.33  3.57  25.14  2 I  1  1  2  2  I  2  33  40.74  7.14  50.27  I  3  45  55.55  9.42  66.31  3  51  62.96  9.79  68.94  1*  63  77.77  11.19  78.79  I  s  69  85.18  13.87  97.65  I  6  81  100.00  14.20  100.00  2  60  Table 3.3 Adoptive Times for Centers in an Open Five Level Central Place System; K = 3, pi = 1000, P / P _ - | = 4 (1 < m < 5 ) , b = 2 m  m  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  5  1  1.23  0.00  0.00  4  3  3.70  1.00  2.97  3  9  11 .11  3.00  8.93  15  18.52  5.07  15.08  .21  25.92  7.05  20.98  27  33.33  9.02  26.83  33  40.74  9.53  28.35  39  48.15  21.69  64.53  51  62.96  22.92  68.19  63  77.77  27.77  82.62  75  92.59  32.81  97.63  81  100.00  33.61  100.00  2 I 2  1  1  3  2  2  ,2  1  G  1  3  5  61  Table 3.4 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 3, p  = 1000, P / P _  x  m  m  1  = 4 (1 < m < 5 ) , b = 2  Cumulative Adopting Center  Equiv. No.  Centers %  Adopti ng Time  % Total Time  5  1  1.23  0.00  0.00  3  7  8.64  1.00  7.70  2  1  13  16.05  1.30  9.98  1  1  19  23.46  1.76  13.57  21  25.92  1.91  14.70  27  33.33  3.53  27.17  4 2  2  l  2  33  40.74  5.47  42.13  l  3  45  55.55  7.80  60.06  3  51  62.96  10.01  77.04  1"  63  77.77  10.96  84.39  1  69  85.18  12.42  95.62  81  100.00  12.99  100.00  2  l  5  6  62  Table 3.5 Adoptive Times for Centers in an Open Five Level Central Place System; K = 3,  = 1000, p (R ) = p  P l  5  (1 < m < 5 ) , b = 1.5  Cumul a t i ve Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  5  1  1.23  0.00  0.00  4  3  3.70  1.00  8.36  3  9  11.11  2.35  19.65  2  1  15  18.52  3.82  31.94  2  3  21  25.92  4.60  38.46  2  2  27  33.33  5.08  42.47  33  40.74  5.36  44.82  39  48.15  10.23  85.53  51  62.96  10.43  87.21  63  77.77  11.00  91.97  69  85.18  11 .43  95.57  81  100.00  11.96  100.00  1I  •I  2  1  6  13 1  •, 4  5  63  Table 3.6 Adoptive Times f o r Centers in a Closed Five Level Central Place System; K = 3, pi = 1000, p (R ) = p  5  (1 < m < 5 ) , b = 1.5  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  5  1  1.23  0.00  0.00  3  7  8.64  1.00  15.09  13  16.05  1.19  17.96 '  19  23.46  1.57  23.73  21  25.92  1.96  29.55  27  33.33  2.29  34.52  33  40.74  3.08  46.49  3  45  55.55  3.84  57.93  1"  57  70.37  4.95  74.64  I  63  77.77  5.54  83.67  69  85.18  5.56  83.85  81  100.00  6.63  100.00  2 I  1  1  4 2 l  2  1  l  2  5  6  Table 3.7 Adoptive Times for Centers in an Open Five Level Central Place System; K = 3, pi = 1000, p . ( R ) = p m  5  (1 < m < 5 ) , b = 2  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  5  1  1.23  0.00  0.00  4  3  3.70  1.00  13.70  3  9  11.11  1.89  25.89  15  18.52  2.33  31.92  21  25.92  2.41  33.01  2 l  1  1  2  3  27  33.33  3.47  46.71  2  2  33  40.74  3.60  49.31  45  55.55  5.71  78.22  51  62.96  5.79  79.31  63  77.77  6.84  93.70  . 1"  75  92.59  7.28  99.73  I  81  100.00  7.30  100.00  I  6  1 I  2  3  s  65  Table 3.8 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 3, pi = 1000, p ( R ) = p  5  (1 < m < 5 ) , b = 2  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  5  1  1.23  0.00  0.00  I  7  8.64  1.00  16.58  13  16.05  1.02  16.85  19  23.46  1.06  17.56  25  30.86  2.41  39.96  27  33.33  2.43  40.36  40.74  2.48  41.12  1  2  1  3 2  2  4 1 I  2  33  •  3  45  55.55  3.29  54.55  1*  57  70.37  4.32  71 .63  I  63  77.77  4.93  81.83  5  2  3  69  85.18  5.76  95.56  1  6  81  100.00  6.03  100.00  66  Tables 3 . 9 , 3.10 Adoptive Times f o r Centers in an Open Four Level Central Place System; K = 3, pi  = 1000,  P /P _-j m  m  =  4 (1 < m <  4);  T 3.9: b = 1.5, T 3.10: b = 2 Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  4  1  3.70  0.00  0.00  3  3  11.11  1.00  7.31  9  33.33  3.84  28.08  15  55.55  8.64  63.14  2 I  1  1  1  3  21  77.77  11.87  86.77  1  2  27  100.00  13.68  100.00  4  1  3.70  0.00  0.00  3  3  11.11  1.00  10.49  9  33.33  3.00  31.47  2  1  l  1  15  55.55  5.07  53.23  l  3  21  77.77  9.02  94.60  l  2  27  100.00  9.53  100.00  67  Table 3.11, 3.12 Adoptive Times for Centers in a Closed Four Level Central Place System; K = 3, pi = 1000, P / P _ - , = 4 (1 < m < 4 ) ; m  m  T 3.11: b = 1.5, T 3.12: b = 2  Cumulative Adopting Center 4  Equiv. No.  Centers %  Adopting Time  % Total Time  1  3.70  0.00  0.00  7  25.93  1.00  10.22  9  33.33  1.55  15.85  15  55.55  1.67  17.07  21  77.77  3.58  36.53  27  100.00  9.79  100.00  1  3.70  0.00  0.00  7  25.93  1.00  10.00  1  13  48.15  1.30  12.95  3  15  55.55  1 .91  19.08  21  77.77  3.53  35.27  27  100.00  10.00  100.00  2  1  3 l  1  1 I  2  3  4 2 I  1  1 I  3  2  68  Table 3.13, 3.14 Adoptive Times f o r Centers in an Open Four Level Central Place System; K = 3, pi = 1000, p  m  (R ) = p m  4  (1 < m < 4 ) ;  T 3.13: b = 15. T 3.14: b = 2  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  4  1  3.70  0.00  0.00  3  3  11.11  1.00  19.68  i  9  33.33  2.34  46.14  15  55.55  3.82  75.15  2  I  1  l  3  21  77.77  4.60  90.54  l  2  27  100.00  5.08  100.00  4  1  3.70  0.00  0.00  3  3  11 .11  1 .00  27.76  9  33.33  1.89  52.47  2  1  l  1  15  55.55  2.33  64.74  l  3  21  77.77  3.47  96.42  l  2  27  100.00  3.60  100.00  I  69  Tables 3.15, 3.16 Adoptive Times for Centers in a Closed Four Level Central Place System; K = 3, pi = 1000, p  m  (R ) = p, (1 < m < 4 ) ; m  T 3.15: b = 1.5, T 3.16: b = 2  Cumulative Adopting Center 4  Equiv. No.  Centers %  Adopting Time  % Total Time  1  3.70  0.00  0.00  7  25.93  1.00  17.83  1  13  48.15  1.19  21.22  3  15  55.55  1.96  34.92  I  21  11.11  2.29  40.79  27  100.00  5.61  100.00  4  1  3.70  0.00  0.00  l  7  25.93  1.00  17.34  13  48.15  1.04  18.05  2  19  70.37  2.37  41.05  3  21  77.77  2.39  41.47  I  27  100.00  5.77  100.00  2 I  l  2  3  1  2 I  1  3  1  70  Tables 3.17, 3.18 Adoptive Times for Centers in an Open Ki = 3, K = 4, K = 3, 2  3  P l  Four Level Central Place System; = 1000, P / P _ - | = 4; m  m  T 3.17: b = 1.5, T 3.18: b = 2  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  4  1  2.78  0.00  0.00  3  3  8.33  1.00  7.49  2  1  9  25.00  3.16  23.67  2  2  12  33.33  3.44  25.77  18  50.00  6.97  52.21  I  1  l  2  24  66.67  11 .47  85.92  l  3  36  100.00  13.35  100.00  4  1  2.78  0.00  0.00  3  3  8.33  1.00'  11.26  2  1  9  25.00  2.60  29.28  2  2  12  33.33  3.00  33.78  l  1  18  50.00  3.84  43.24  l  2  24  66.67  8.24  92.79  36  100.00  8.88  100.00  I  3  71  Tables 3.19, 3.20 Adoptive Times for Centers in a Closed Four Level Central Place System; Ki  =  3, K = 4, K = 3, 2  3  P l  = 1000, p ^ p ^  = 4;  T 3.19: b = 1.5, T 3.20: b = 2  Cumulative Adopting Center 4  Equiv. No.  Centers %  Adopting Time  % Total Time  1  2.78  0.00  0.00  7  19.44  1.00  19.49  13  36.11  1.67  32.55  15  41.67  1.95  38.01  1  2  21  58.33  3.71  72.32  2  2  24  66.67  3.83  74.66  36  100.00  5.13  100.00  1  2.78  0.00  0.00  2  1  7  19.44  1.00  18.16  1  1  13  36.11  1.30  23.59  3  15  41.67  2.60  47.22  I  21  58.33  3.76  68.29  24  66.67  4.77  85.76  36  100.00  5.51  100.00  2 l  1  1  3  I  3  4  2  2 I  3  2  72  Tables 3.21, 3.22 Adoptive Times for Centers in an Open Four Level Central Place System; Kx = 3, K = 4, K = 3, 2  3  P l  = 1000, p  m  (Rj  = p„;  T 3.21: b = 1.5, T 3.22: b = 2  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  4  1  2.78  0.00  0.00  3  3  8.33  1.00  15.68  2  2  6  16.67  2.50  39.21  2  1  12  33.33  2.71  42.48  18  50.00  4.19  65.69  24  66.67  5.77  90.41  36  100.00  6.78  100.00  4  1  2.78  0.00  0.00  3  3  8.33  1.00  23.51  9  25.00  2.00  47.01  15  41.67  2.16  50.77  18  50.00  2.18  51.29  l  1  1 l  3  2 I  1  1  2  2  2  l  2  24  66.67  4.15  97.53  l  3  36  100.00  4.25  100.00  73  Tables 3.23, 3.24 Adoptive Times f o r Centers in a Closed Four Level Central Place System; K i = 3, K  = 4, K  2  3  = 3,  P l  = 1000, p  m  (R ) = p , ; m  T 3.23: b = 1.5, T 3.24: b = 2  Cumulative Adopting Center 4 2 l  1  1  3  Equiv. No.  Centers %  Adopting Time  % Total Time  1  2.78  0.00  0.00  7  19.44  1.00  27.72  13  36.11  1.27  35.34  15  41.67  2.22  61.56  I  2  21  58.33  2.55  70.62  2  2  24  66.67  3.55  98.34  1  3  36  100.00  3.61  100.00  1  2.78  0.00  0.00  4 2  1  7  19.44  1.00  24.29  I  1  13  36.11  1.02  24.68  l  2  19  52.78  2.54  61.72  3  21  58.33  2.79  67.82  l  33  91.67  3.78  91.91  36  100.00  4.12  100.00  2  3  2  74  Table 3.25 Adoptive Times f o r Centers in an Open Four Level Central Place System; K = 4,  P l  = 1000, p (R ) = p„, b = 2 m  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Ti me  % Total Time  4  1  1.56  0.00  0.00  3  4  6.25  1.00  17.90  10  15.62  1.56  27.98  16  25.00  1.68  30.02  22  34.37  2.42  43.27  28  43.75  3.79  67.85  1"  34  53.12  4.02  71.96  I  5  46  71.87  4.23  75.80  I  3  58  90.62  4.90  87.72  64  100.00  5.59  100.00  2 l  1  2 I  l  1  2  2  6  75  Table 3.26 Adoptive Times for Centers in a Closed Four Level Central Place System; K = 4 , pi = 1000, p ( R j = p , , b = 2 m  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  4  1  1.56  0.00  0.00  I  7  10.94  1.00  11.49  13  20.31  1.03  11.88  3  16  25.00  2.00  23.01  l  22  34.37  2.28  26.26  28  43.75  2.56  29.39  I  40  62.50  3.74  42.95  r  46  71 .87  3.84  44.14  1  2  1  2  2 3  2  i  5  58  90.62  4.65  53.51  i  6  64  100.00  8.70  100.00  76  Table 3.27 Adoptive Times f o r Centers in an Open Five Level Central Place System; K = 2,  P l  = 1000, p / p _ m  m  1  = 4, b = 2  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  5  1  6.25  0.00  0.00  4  2  12.50  1 .00  6.55  3  4  25.00  1.80  11.80  6  37.50  1.99  13.02  8  50.00  2.00  13.10  10  62.50  5.58  36.53  V  12  75.00  8.17  53.52  l  14  87.50  11.33  74.20  16  100.00  15.27  100.00  2 I  1  2  I  1  2  2  3  77  Table 3.28 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 2, pi = 1000, p / p _ m  m  1  = 4, b = 2  Cumulative Adopting Center  5  4 2  2  Equiv. No.  Centers %  Adopting Time  % Total Time  1  6.25  0.00  0.00  7  43.75  1.00  9.40  8  50.00  5.00  47.00  10  62.50  5.65  53.12  l  2  12  75.00  5.70  53.55  l  3  14  87.50  9.24  86.88  16  100.00  10.64  100.00  1-  78  Table 3 . 2 9 Adoptive Times f o r Centers i n an Open Five Level Central Place System; K =  2,  P  l  =  1000,  p  (R  )  = p , b = 5  2  Cumulative Adopting Center  Equiv. No.  Centers %  Adopting Time  % Total Time  1  6.25  0.00  0.00  3  18.75  1.00  16.16  5  31.25  1.82  29.38  3  7  43.75  3.47  56.02  I  9  56.25  4.12  66.62  10  62.50  4.52  73.04  12  75.00  5.88  95.02  5  l  1  2  1  2  4  l  3  2  2  14  87.50  5.93  95.78  1"  16  100.00  6.19  100.00  79  Table 3.30 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 2,  P l  = 1000, p ( R j = p , b = 2 m  5  Cumulative Adopting Center  5  Equi v. No.  Centers %  Adopting Time  % Total Time  1  6.25  0.00  0.00  3  18.75  1.00  9.95  5  31.25  1.82  18.09  7  43.75  3.47  34.50  9  56.25  4.12  41.03  11  68.75  6.11  60.82  13  81.25  7.01  69.78  4  14  87.50  8.73  86.94  1"  16  100.00  10.05  100.00  l  1  2  1  3 1 I  2  3  2  2  80  Tables 3.31, 3.32 Adoptive Times f o r Centers in a Four Level Central Place System; K = 2,  P l  = 1000, p / p _ m  m  1  = 4, b = 2,  T 3.31: open, T 3.32: closed  Cumulative Adopting Center  Equi v. No.  Centers %  Adopting Time  % Total Time  4  1  12.50  0.00  0.00  3  2  25.00  1.00  37.71  2  4  50.00  1.80  67.87  l  6  75.00  1.99  74.96  2  8  100.00  2.65  100.00  4  1  12.50  0.00  0.00  I ,2  5  62.50  1.00  23.19  3  6  75.00  1.80  41.74  I  8  100.00  4.31  100.00  1  I  1  2  O  o  O  o  (a)  3  I  2  I  1  (b)  16000  1000  4000  (c)  1.00  2.65  (d)  2  5  2  o  O  o  O  4  I  1  2  I  2  3  (a)  1000  64000  1000  4000  1000  16000  (b)  1.80  1.99  0  1.99  1.80  2.65  1.00  (c)  3  4  I  4  3  5  Q  2 ( d )  ( a ) type of centre ( b ) population of centre ( c ) standardized adoptive times ( d ) order of adoption  Figure 3 . 1 .  Adoptive times for centres in an open four level central place system; K = 2 , {p |1 < m < 4} = {1000, 4000, 16000, 64000}; Source: Table 3.31. CO  O  o — O  o  2  I  1  o  0  o  4  I  1  2  I  Q  Q  ( a)  3  I  (b)  8000  1000  4000  1000  64000  1000  4000  1000  8000  (b)  (c)  1.80  4.31  1.00  1.00  0  1.00  1.00  4.31  1.80  (c)  ( d)  4  5  2  2  I  2  2  5  2  2  3 ( a )  4 ( d )  ( a ) type of centre ( b ) population of centre ( c ) standardized adoptive times ( d ) order of adoption  Figure 3 . 2 . Adoptive times f o r centres in a closed four level central place system; K = 2 , { p j l < m < 4} = {1000, 4000, 16000, 64000}; Source: Table 3.32. 00  ro  83  standardized to p a r t i c u l a r t h r e s h o l d s .  In order to compare "actual times" -  say, f o r i n s t a n c e , the times that d i f f e r e n t processes take to expire - i t i s necessary that t h i s f a c t o r be accounted f o r .  (i)  Closure  Since closure d r a m a t i c a l l y influences the p o t e n t i a l a t centers on (at) a system's boundary (endpoints), nant o f systemic adoptive  expressed  i t i s a prime determi-  patterns.  When the marginal  ( M - l ) s t l e v e l places adopt r e l a t i v e l y  quickly,  as they tend to do with systemic openness, they may become secondary f o r the d i f f u s i o n process. that d i f f u s i o n  poles  This i s c o n s i s t e n t , of course, with the idea  i s & -state ordered process whose e a r l i e r a t t r i b u t e s  a c o n s t r a i n i n g e f f e c t on l a t e r  have  attributes.  Openness ( e s p e c i a l l y i n the areal case) seems to have two d i s t i n c t consequences.  F i r s t l y , i t emphasizes community s i z e as a f a c t o r  promoting e a r l y adoption.  Secondly, i t creates an adoptive lag a t the  lowest l e v e l of the hierarchy: there i s a rather great discrepancy between the adoptive time f o r the f i r s t l e v e l places ( I ) nearest the Mth l e v e l 1  place and the adoptive times f o r a l l other f i r s t l e v e l places ( I , ! , * * * ) 2  3  in the system (see Tables 3.1, 3.18, 3.25). The s i g n i f i c a n c e of openness seems e s p e c i a l l y r e l a t e d to the dimensioning of the system.  In l i n e a r systems - where the nesting  factor  i s minimized - there are more exceptions to the above statements. C l o s u r e , on the other hand, contributes adoptive pattern (Tables 3.4, 3.12, 3.20).  7  to a more wave-like  This generally a p p l i e s to the  set of a l l centers i n the system but p a r t i c u l a r l y  a p p l i e s w i t h i n each  84  size c l a s s .  C l o s u r e , then, emphasizes the distance to the o r i g i n a l  of d i f f u s i o n  ( i . e . the  source  M th l e v e l center) as a prime f a c t o r i n adoptive  ordering. In a d d i t i o n , openness (as one n a t u r a l l y expects) i s conducive to a more rapid completion of the systemic process.  (ii)  Central Place Populations  From one perspective i t i s the s i z e d i s t r i b u t i o n of centers that l a r g e l y determines the s p a t i a l c h a r a c t e r i s t i c s of inter-urban sion.  Most s i g n i f i c a n t l y , when the  multipliers  diffu-  (see Chapter 2)  d e c l i n e rather r a p i d l y (as i n the r a n k - s i z e as opposed to progression component case) and the system i s open, the adoptive lag ( i n the set of f i r s t l e v e l places) tends to be diminished.  However, under the same  c o n d i t i o n s , a new lag may be introduced before the very f i r s t centers of the system adopt from the  Mth l e v e l p l a c e .  This l a t t e r tendency seems  to occur, though, only when the urban system i s complex (compare Tables 3.3 and 3 . 7 , 3.4 and 3 . 8 ) . From another perspective (and a l e s s r e l a t i v e one), process comp l e t i o n times are reduced c e t e r i s  paribus by large o v e r a l l urban popula-  tions ( i . e . high d e n s i t i e s ) : that i s , when of the sequence  k , 2  k , 3  One additional  •••  ki  i s large and the elements  do not decline too r a p i d l y .  point must be included.  It should be understood  that the r a n k - s i z e r u l e , even when r e s t r i c t e d to an exponent of one, i s c o n s i s t e n t with various d i f f e r e n t sets of community s i z e (see Appendix C ) . Hence i t would be a b i t tenuous to suggest that a s p e c i f i c  diffusionary  pattern (in space and time) might be t y p i c a l l y related to the r a n k - s i z e  85  structure.  However, i t  i s i n t e r e s t i n g to note that the adoptive  orderings  per se seem to change very l i t t l e in disparate r a n k - s i z e s e t t i n g s : closed systems e s p e c i a l l y r e t a i n orderings when l i n e a r and areal systems of the same complexity (number of h i e r a r c h i a l l e v e l s ) are being compared.  (iii)  Distance Exponent  The " f r i c t i o n of d i s t a n c e , " of course, expresses a s i g n i f i c a n t e f f e c t on the o v e r a l l rate of systemic adoption.  On the other hand, i t  apparently has a permuting i n f l u e n c e on adoptive ordering only during the e a r l y to middle stages of the process. the f i r s t , second, e t c . order places ( I , 1  The acceptance pattern amongst 2 , ' " ) nearest the o r i g i n a l 1  source may vary according to the value of the distance exponent but t h i s e a r l y permuting v i r t u a l l y disappears as the smaller and more d i s t a n t places adopt (compare Tables 3.2 and 3.4, 3.15 and 3.16). In a d d i t i o n , there seems to be a tendency f o r a l a r g e r exponent to reduce the adoptive l a g in open systems and promote r e l a t i v e l o c a t i o n (with respect to the  M th l e v e l center) as opposed to merely c i t y s i z e  as an important determinant of adoptive ordering (compare Tables 3.1 and 3.3).  In closed systems, increased f r i c t i o n of distance i s  conducive to wave-like d i f f u s i o n a r y patterns.  (iv)  definitely v  H i e r a r c h i a l Levels  The complexity of the system does not, c e t e r i s p a r i b u s , seem to be as strong a d i r e c t i v e f a c t o r as one might expect.  Taking into account  that d i f f e r e n c e s in h i e r a r c h i a l s t r u c t u r i n g are accompanied by other  86  d i f f e r e n c e s ( e . g . the absolute population of the  M th l e v e l c e n t e r , the  distance to the boundary (endpoints) of the system) - a l b e i t these are i n t i m a t e l y r e l a t e d - d i f f u s i o n appears to proceed in much the same r e l a t i v e fashion in systems which have disparate h i e r a r c h i a l development (but which otherwise share the same parameters). N a t u r a l l y , though, the existence of a greater number of s i z e c l a s s e s i n more complex systems tends to even out the percentage adoptive times ( r e l a t i v e to the t o t a l time needed f o r process e x p i r a t i o n ) of i n d i v i d u a l centers and must be considered i f one intends to r e l a t e these adoptive times to other f a c t o r s ( s p e c i f i c a l l y , to the cumulative number of adopting centers as in the l o g i s t i c argument).  (v)  Geometry  Geometric e f f e c t s have already been noted i n s o f a r as closure and the s i z e of community populations have a t o p o l o g i c a l b a s i s . D i m e n s i o n a l i t y , however, should be emphasized.  Linear systems  seem to be, c e t e r i s p a r i b u s , much more conducive to wave-like d i f f u s i o n than t h e i r areal counterparts (compare Tables 3.7 and 3.29). Secondly, v a r i e t y in the nesting f a c t o r may induce some change i n adoptive p a t t e r n s .  The p r i n c i p l e s of organization i n areal systems  do not, though, seem as c r i t i c a l as other f a c t o r s in promoting s p e c i f i c acceptance p a t t e r n s . T h i r d l y , and perhaps most s i g n i f i c a n t l y , the geometry influences the cumulative numbers of equivalent centers that may adopt at c e r t a i n times.  This again i s c r i t i c a l to remember when one wishes to hypothesize  about the forms of l o g i s t i c curves.  87  (vi)  Cumulative Adoption  Hudson's s t o c h a s t i c argument generated an S-shaped cumulative acceptance (by center) curve which varied i n shape according to the nesting f a c t o r and the number of h i e r a r c h i a l l e v e l s i n the central place system. : In the more h e u r i s t i c format here, nothing so e x p l i c i t may be stated.  N a t u r a l l y , under c e r t a i n parametric c o n s t r a i n t s , S-shaped cumula-  t i v e adoption may be c h a r a c t e r i s t i c ; t h i s , however, cannot be supported as a general hypothesis. As mentioned above, the most c r i t i c a l determinant of the nature of cumulative adoption i s simply the number of h i e r a r c h i a l l e v e l s : f o r t h i s defines the number of centers and therefore the number of points (types of centers) that may be f u n c t i o n a l l y r e l a t e d (percentage cumulative centers versus percentage cumulative time).  Quite n a t u r a l l y , a smoother  c u r v e , gradually r i s i n g with few plateaus ( i . e . l i t t l e tendency toward very high or very low s l o p e s ) , i s more c h a r a c t e r i s t i c of complex systems simply because there are more points to r e l a t e . In a d d i t i o n , closure and cumulative knowing appear to be i n t i mately t i e d .  The adoptive lag of open systems tends to produce an S-shape  i n cumulative acceptance curves: t h i s e f f e c t seems e s p e c i a l l y prevalent i n those systems which are concomitantly areal and complex as well (Tables 3.1, 3.3).  On the other hand, some closed systems - again when areal and  complex - appear to be characterized by r e l a t i v e l y l i n e a r cumulative adoptive curves (Tables 3.6, 3.19). F i n a l l y , on a s l i g h t l y d i f f e r e n t note, i t should be emphasized that cumulative acceptance r e f e r s only to the number of equivalent centers  88  and t h a t , t h e r e f o r e , the population  adopting may vary considerably in  accordance with the s i z e d i s t r i b u t i o n of the places in the system.  (vii)  A B r i e f Synthesis  H i e r a r c h i a l d i f f u s i o n and the notion of a d i f f u s i o n wave (contagious growth) were phenomena more than s l i g h t l y at odds u n t i l Hudson's contribution.  It i s now p o s s i b l e to i s o l a t e some f a c t o r s that seem to  promote e i t h e r of these p a t t e r n s : ^ (i)  hierarchial  diffusion:  (a)  systemic  openness;  (b)  r e l a t i v e l y slow d e c l i n e i n t h e v a l u e s o f the elements o f t h e s e t {k I I < m < M}; m low v a l u e o f t h e f r i c t i o n o f d i s t a n c e coeff ici ent; 1  (c) (d) (ii)  a r e a l d i m e n s i o n a l i t y o f t h e system;  wave-like  diffusion:  (a)  systemic  (b)  r e l a t i v e l y rapid decline i n the values of the elements o f t h e s e t {k I I < m < M}; m ' high v a l u e s o f t h e f r i c t i o n o f d i s t a n c e coefficient;  (c) (d)  closure;  l i n e a r d i m e n s i o n a l i t y o f t h e system.  In the previous i l l u s t r a t i o n s , Table 3.1 perhaps best exemplifies ( i ) above while Table 3.8 (the areal case) and Table 3.30 (the l i n e a r case) best represent  (ii). The argument lends support to some of the hypotheses which Hudson  and Pederson suggested - p a r t i c u l a r l y to the a s s e r t i o n s that disparate  89  d i f f u s i o n patterns may be emobidied i n the same s p a t i a l structure and that cumulative adoption by central places may be S-shaped in nature - but, of course, with the many stated r e s e r v a t i o n s .  3.5  A d d i t i o n a l Comments on the A l t e r n a t i v e Model The basic d e f i c i e n c y of the newly o u t l i n e d model (and of the  seminal models, f o r t h a t matter) revolves about i t s ( t h e i r )  testability.  Since a l l these arguments are based on a s t a t i c spatial-economic theory which i s i t s e l f i d e a l i s e d and somewhat unamenable to t e s t i n g ,  isolating  domains of the real world which would accurately r e f l e c t the modelling postulates becomes an extremely d i f f i c u l t t a s k .  Now some of the s p e c i f i c  methodological issues are attended t o . F i r s t of a l l , the present model, l i k e Pederson's, i s d e t e r m i n i s t i c . In contrast to the i d e a l i s e d s t a t i c s e t t i n g , centers of the same type (here, too, might a r i s e a c l a s s i f i c a t i o n problem) i n the real world would not be expected to adopt a s p e c i f i e d item at e x a c t l y the same time.  Of  course, the computed i d e a l i s e d times could be construed as the means of adoptive p r o b a b i l i t y d i s t r i b u t i o n s f o r d i f f e r e n t  types of central places  but t h i s would n e c e s s a r i l y r a i s e c e r t a i n other problems of a more s t a t i s tical nature.^  Even Hudson's s t o c h a s t i c argument, when phrased i n the  terms of equivalent centers (see Appendix B ) , does not r e a l l y circumvent this  issue. 12 Secondly, the model i t s e l f i s s t a t i c .  Real world c e n t r a l  place a t t r i b u t e s would l i k e l y be changing over time and t h i s might have a s i g n i f i c a n t e f f e c t on the properties of an ongoing s p a t i a l process l i k e diffusion.  Therefore, i f a model i s to become a v i a b l e p r e d i c t o r i t must  90  accommodate parametric change in some fashion (whether that change be exogenous or dependent on the adoptive scheme i t s e l f ) .  The a l t e r n a t i v e  model which has been discussed can s u c c e s s f u l l y represent such change as i s shown in the upcoming chapter, by assuming that the system i s passing through successive stages of e q u i l i b r i u m - and t h i s c o n s t i t u t e s one advantage of making the threshold of adoption e x p l i c i t in the argument. T h i r d l y (and t h i s a p p l i e s to the other models as w e l l ) , i t  is  necessary to s p e c i f y the domain of items to which the proposed model i s applicable.  The postulates have been stated i n such g e n e r a l i t y that the  model should be appropriate f o r representing the adoption of most items such as o r g a n i z a t i o n s , consumer goods, d i s e a s e s , e t c . - which researchers have used i n the past as i n d i c a t o r s of a d i f f u s i o n a r y process.  In any  c a s e , the model would be most useful for t y p i f y i n g those processes where adoption, i n the r e s e a r c h e r ' s o p i n i o n , could be conceptualized according to a t e l l i n g - h e a r i n g dichotomy.  I t might be added that adoption of c e r t a i n  items might depend on p a r t i c u l a r modes of the e n t i r e  transportation-communi-  c a t i o n matrix and t h i s should be considered i n any empirical  qualification  of the proposed model. On the other hand, i t must once again be emphasized that a l l these models deal with primary ( f i r s t ) adoption.  For a more general s t a t e -  ment on inter-urban d i f f u s i o n - one taking into account intra-urban  telling  and inter-urban feedbacks f o r secondary adoption - an approach s i m i l a r to that of C a s e t t i (1969) would have to be integrated i n t o the present argument. F i n a l l y , t h i s a l t e r n a t i v e model assumes that a l l places in the system tend to contribute to the d i f f u s i o n a r y p a t t e r n .  By applying the  threshold c o n s t r a i n t s of the c e n t r a l place format, i t would be possible to  91  s a t i s f y the notion that c e r t a i n small places could not adopt higher order economic items.  Besides, propositions which were based on the idea of  process completion (as i n the previous s e c t i o n of t h i s c h a p t e r ) , might be given special a t t e n t i o n s i n c e , in the real w o r l d , there are numerous s i t u a t i o n s where a process may terminate prematurely because of a s u b s t i t u t e (economic i t e m ) , remedy (disease item), or some other  3.6  factor.  Concluding Remarks In t h i s chapter a t t e n t i o n has l a r g e l y been devoted to combining  s p a t i a l process and s p a t i a l s t r u c t u r e - f>r the c e n t r a l place context - i n a coherent and systematic f a s h i o n .  As much emphasis was placed on ( i ) de-  l i m i t i n g the postulates needed f o r t h i s operation and ( i i )  q u a l i f y i n g the  sparse l i t e r a t u r e which deals with the t o p i c , the d i s c u s s i o n has a decided "methodological" r i n g to  it.  Some s p e c i f i c consequences of a proposed a l t e r n a t i v e model (which i s capable of dealing with much more parametric v a r i a t i o n than e x i s t i n g models) were then g i v e n : to the degree that these r e s u l t s were n e i t h e r discordant with the scanty empirical work a v a i l a b l e nor c o n f l i c t i n g with i n t u i t i v e thought, they were construed as o f f e r i n g support of t h i s a l t e r n a t i v e model as being representative of central place d i f f u s i o n . In s h o r t , the model suggests that i t should not be s u r p r i s i n g to discover t h a t : (i)  i d e n t i c a l items may d i f f u s e q u i t e d i f f e r e n t l y in d i s p a r a t e c e n t r a l p l a c e s e t t i n g s ; and  (ii)  d i s p a r a t e items may d i f f u s e q u i t e d i f f e r e n t l y in i d e n t i c a l (or r e a s o n a b l y s i m i l a r ) c e n t r a l place s e t t i n g s .  92  The author has defended, as w e l l , an h e u r i s t i c approach to the argument at hand.  It was f e l t that s t r i c t f o r m a l i s a t i o n would be s e l f -  defeating at the present time and that a n a l y t i c a l r i g o r should be i n t r o duced only when the confirmation status of the modelling procedure has been s u b s t a n t i a l l y enhanced.  In the meantime, the author would welcome  r e p e t i t i o n s of the procedure (with d i f f e r e n t parameters) in order to f u r t h e r q u a l i f y the propositions discussed above.  FOOTNOTES TO CHAPTER 3  " See Bowers (1937), McVoy (1940), Crain (1966), Hagerstrand (1952), Hudson (1969), Pyle (1969), and Pederson (1970).  "•Central place theory i s more than a b i t tenuous where i t concerns the issue of consumer i n t e r a c t i o n amongst d i f f e r e n t communities. The comp l e x i t y o f the topic has precluded, to d a t e , the a r t i c u l a t i o n of any " l a w - l i k e " statements which would r e s t f i r m l y on the tenets of economic theory - a host of i n t e r r e l a t e d f a c t o r s , such as the state of t r a n s p o r t a t i o n , technology , the frequency of multipurpose household t r i p s , the incomes of those households, e t c . would have to be l i n k e d in a coherent and cons i s t e n t manner. F o r t u n a t e l y , g r a v i t y - p o t e n t i a l theory, when modified to the s p e c i f i c domination scheme of the central place s t r u c t u r e , does o f f e r a s u i t a b l e surrogate methdology. Equation (3.2) expresses the idea that i n t e r a c t i o n between centers of the same s i z e - given that they are elements of d i s t i n c t systems or of the same system at d i f f e r e n t points i n time may vary considerably f o r a number of unspecified reasons: f o r i n s t a n c e , d i f f e r e n c e s i n transportation technology, i n the p r o c l i v i t y of customers to t r a v e l , e t c . In a d d i t i o n , i t should be emphasized that the p o t e n t i a l formula, while being most appropriate f o r a s i n g l e mode of t r a n s p o r t a t i o n (communication), may be extended and made a p p l i c a b l e to multimodal i n t e r a c t i o n (by combining the various i n d i v i d u a l p o t e n t i a l functions) as w e l l .  ^Closure i s now interpreted somewhat d i f f e r e n t l y than i t was i n the previous (second) chapter. Here closure refers to whether or not d i s t i n c t systems share boundary points (where potential i s expressed) w h i l e e a r l i e r that term alluded to the existence of a well-ordered f r e quency d i s t r i b u t i o n (which, i n t u r n , would be geometrically determined) of centers i n the c e n t r a l place s t r u c t u r e . Berry (1964, 1967), on the other hand, would perhaps have an e n t i r e l y d i f f e r e n t and more functional approach to the use of c l o s u r e . Unfortunately, no other expression seems so adaptable to the d e p i c t i o n of such systemic p r o p e r t i e s .  93  94  "'This author i s of the opinion that the r e l a x a t i o n of postulates a v a r i a t i o n of Harvey's (1969) "as i f " methodology i f you l i k e - i s i n dispensable to a n a l y t i c advance in many spheres of the s o c i a l s c i e n c e s . He a l s o shares P a r r ' s (1970) more s p e c i f i c views on the inherent f l e x i b i l i t y of the h i e r a r c h i a l argument: e s p e c i a l l y i t s a b i l i t y to accommodate various geometries, n o n - t e r t i a r y a c t i v i t i e s , e t c . Therefore, i t should not be s u r p r i s i n g that the interconnection of independent systems i s presently being advocated. While t h i s would be d i f f i c u l t to r a t i o n a l i z e t h e o r e t i c a l l y i n s o l e l y economic terms (the s i n g l e system has enough of i t s own problems h e r e ) , i t i s a perspective with obvious conceptual and empirical merit Berry and Pred (1961:18), f o r i n s t a n c e , have mentioned C h r i s t a l l e r ' s own concern over " . . . the r e l a t i v e strength of any one system and adjacent systems. . . . " I t should be pointed out that with asymmetry in sets of systems e i t h e r i n terms of the geometry or central place populations of such systems - the s t r u c t u r a l symmetry of any one system would no longer be a s u f f i c i e n t c o n d i t i o n f o r that same system to be c h a r a c t e r i z e d by symmetric d i f f u s i o n a r y patterns.  "'Openness i s maximized i n a t o p o l o g i c a l sense when a system i s completely enveloped by other systems (and t h i s i s the only case considered i n t h i s t h e s i s ) . However, openness must a l s o depend on the other parameters ( i . e . the populations of c e n t e r s , the f r i c t i o n of d i s t a n c e , e t c . ) of those adjacent systems. Besides, t h i s conception of openness i s a l s o useful i f the researcher wishes to consider an adoptive process beginning at a smaller center ( i . e . a center of order m < M) i n any c e n t r a l place system ( i t might be emphasized that Hudson's argument cannot accommodate t h i s p o s s i b i l i t y e i t h e r ) . Of course, an asymmetric adoptive pattern would be chara c t e r i s t i c here as w e l l .  -' Pederson has d e f i n i t e l y dealt with a closed system (due, in large measure, to his e r r i n g i n t e r p r e t a t i o n of the C h r i s t a l l e r i a n geometry).  As implied e a r l i e r , t h i s simply means that potential i s an impetus to d i f f u s i o n between only those centers which are f u n c t i o n a l l y related.  The assumption that the threshold i s equal f o r a l l centers i s , admittedly, of an a p r i o r i nature. The form of the general argument would s t i l l be a p p l i c a b l e , however, as long as adoptive thresholds remained a function of c i t y s i z e . This threshold was made e x p l i c i t in Pederson's case but was dealt with only i m p l i c i t l y by Hudson.  95  There are two points of concern here. F i r s t l y , the tables which were omitted d i d not c o n f l i c t with any of the tables which are presently i n c l u d e d : f o r i n s t a n c e , adoptive times f o r C h r i s t a l l e r ' s data (see Beckmann and McPherson (1970:31) were s i m i l a r to those given in other tables of t h i s chapter - hence, they were not i n c l u d e d . Secondly, some very s l i g h t discrepancies might e x i s t i n the percentage (adoptive) t o t a l times because, in a few cases, these times were computed with an excessive number of s i g n i f i c a n t f i g u r e s (see the t e l l i n g process in Appendix C f o r a case i n p o i n t ) . J  Statements (b) in ( i ) and ( i i ) deserve some a d d i t i o n a l comment. To begin w i t h , i t i s simply being asserted t h a t , c e t e r i s p a r i b u s , o v e r a l l large central place populations would tend to induce h i e r a r c h i a l d i f f u s i o n while o v e r a l l small central place populations would tend to give r i s e to more wave-like patterns. D i r e c t i n g a t t e n t i o n to the rate of decline i n the s e r v i c e m u l t i p l i e r s i s but a preferable way - in the a n a l y t i c a l sense - of s t a t i n g such an hypothesis. To take an example, the strong h i e r a r c h i a l e f f e c t e x h i b i t e d f o r the open case i n Table 3.1 and the closed case in Table 3.2 (where r = 2000 and {k } = {.3333, .1667, .1250, .0937, .0703}) i s s i g n i f i c a n t l y diminished foY the comparable (open and closed) cases in Tables 3.3 and 3 . 4 , r e s p e c t i v e l y (where n = 2000 but {k = .3333, .1185, .0779, .0462, .0313}). In a d d i t i o n t h i s " d i s t r i b u t i o n a l " approach i s superior to s t a t i n g that the d i f f u s i o n pattern depends on the sum of the m u l t i p l i e r s alone. Considerable v a r i a t i o n in the rate of d e c l i n e of the m u l t i p l i e r s - and, hence, considerable v a r i a t i o n in d i f f u s i o n a r y patterns - i s e n t i r e l y consonant with an o v e r a l l f i x e d sum f o r those m u l t i p l i e r s . In f a c t , t h i s poses the i n t e r e s t i n g question of determining the d i s t r i b u t i o n of s e r v i c e m u l t i p l i e r s which would - given c e r t a i n other parameters of the system (the nesting f a c t o r ( s ) , the f r i c t i o n of d i s t a n c e , e t c . ) as well as the sum of those m u l t i p l i e r s - minimize the time of l a s t adoption in a central place system. lu  x  m  P a r t i c u l a r problems would a r i s e , f o r i n s t a n c e , over the s p e c i f i c a t i o n of appropriate time i n t e r v a l s ( f o r these would determine whether or not centers of the same type adopted at " e x a c t l y " the same time) and the actual p o s t u l a t i o n of a relevant adoptive p r o b a b i l i t y d i s t r i b u t i o n (normal perhaps?) for each central place type.  ""The property of s t a s i s induces a t h e o r e t i c a l problem of some s i g n i f i c a n c e . For an economic innovation to occur - except f o r , perhaps, " f a s h i o n " changes i n new automobiles, c l o t h e s , a p p l i a n c e s , e t c . - there would l i k e l y be a concomitant disturbance or s h i f t in the system's e q u i l i brium p o s i t i o n . While t h i s represents an inconsistency of s o r t s i n the proposed model, i t i s a shortcoming which cannot - in t h i s a u t h o r ' s opinion -  96  be resolved with the t h e o r e t i c a l tools which are presently at hand (see footnote 2 above). In any case the author does give some a t t e n t i o n , i n the f o l l o w i n g chapter, to d i s c e r n i n g changes which might occur i n d i f fusionary patterns as the whole system s h i f t s i t s e q u i l i b r i u m p o s i t i o n (which i s r e f l e c t e d in the changing s i z e d i s t r i b u t i o n of urban communities). A l s o , the author f e e l s that t h i s inconsistency i s more of a t h e o r e t i c a l rather than pragmatic issue and the model-tester should not view i t as being an important b a r r i e r to his empirical q u a l i f i c a t i o n procedures.  Chapter 4  PARAMETRIC  INFLUENCES  ON S T R U C T U R E  CENTRAL  4.1  PLACE  AND PROCESS  IN T H E  SYSTEM  Introduction In the second chapter of t h i s t h e s i s i t was stated that central  place theory, as t r a d i t i o n a l l y  r e c e i v e d , i s merely a static  formulation:  that i s , a d e p i c t i o n ( a l b e i t somewhat i d e a l i z e d ) at one point i n time of a c e r t a i n domain of the real world.  Perhaps, however, the reader has  already been convinced that there i s considerable merit in r e l a x i n g t h i s narrow viewpoint.  In f a c t , the author contends that the property of s t a s i s  per se represents only a prima f a c i e - and not f a r - r e a c h i n g - c o n s t r a i n t to the advancement of sound a n a l y t i c a l work in the subject area. For i n s t a n c e , i t has been twice demonstrated j u s t how the s p a t i a l economic p r i n c i p l e s inherent to the h i e r a r c h i a l model of c i t y s i z e provide a useful framework f o r d i s c u s s i n g systems over a period of time.  attribute  changes  w i t h i n central place  More e x p l i c i t l y , the author was able to  c l a r i f y (by making the relevant c e n t r a l place geometries i n v a r i a n t over time) the manner in which:  97  98  (i)  (ii)  aggregate c h a r a c t e r i s t i c s of central places (numbers o f f u n c t i o n s , employees, e t c . ) and t h e i r r e l a t e d systems ( t h e u r b a n / r u r a l p o p u l a t i o n r a t i o ) couId be s h i f t e d d u r i n g a p o i n t s p e c i f i c s t r u c t u r a l t r a n s f o r m a t i o n ; and p a r t i c u l a r c h a r a c t e r i s t i c s of central places (has a c e n t e r adopted a new a c t i v i t y ? ) and t h e i r r e l a t e d systems (how many c e n t e r s have a c c e p t e d t h i s new a c t i v i t y ? ) c o u l d be s h i f t e d during a p o i n t - s p e c i f i c d i f f u s i o n a r y process.  U s u a l l y , however, we tend to perceive the transformation the process i n ( i i )  in ( i )  and  as being somehow interwoven as economic a c t i v i t y  i s extended throughout a r e g i o n .  But t h i s o v e r l o o k s , of course, a very  i n t e r e s t i n g and s i g n i f i c a n t problem i n i t s own r i g h t : that being howinnovat i o n s , adoptions, and s t r u c t u r a l  changes are continuously  blended in w e l l -  defined spatial-economic systems as growth proceeds. Naturally i t  remains a much e a s i e r task j u s t speculating l o o s e l y  about the form of a modelling scheme which might s a t i s f a c t o r i l y deal with t h i s complex problem rather than a c t u a l l y d e v i s i n g a set of  relevant,  l o g i c a l l y connected, and e m p i r i c a l l y t e s t a b l e statements.^  Mind y o u ,  the author has no present i n t e n t i o n s of putting together j u s t such a set of statements - t h i s would simply require too much novel a n a l y t i c a l and empirical work; on the other hand, he wishes to o f f e r some a d d i t i o n a l i n s i g h t s i n t o how change and transformation may be perceived (and discussed) w i t h i n the a b s t r a c t c e n t r a l place context. from a corpus of p a r t i a l  It  ultimately  i s hoped that  i n s i g h t s a more general t h e o r e t i c a l model -  including  feedbacks, m u l t i p l i e r s , and the l i k e - may e v e n t u a l l y a r i s e and that t h i s will  lead to a s u b s t a n t i a l l y improved understanding of the s p a t i a l conse-  quences of economic growth in the real w o r l d .  99  To begin w i t h , i t i s quite possible to focus on d i f f e r e n t parameters (population, technology, e t c . ) in order to s p e c i f y how t h e i r i n d i v i d u a l v a r i a t i o n may be r e l a t e d to s t r u c t u r a l place systems over time.  By u t i l i z i n g the e q u i l i b r i u m conditions of the  h i e r a r c h i a l model, the instantaneous initially  changes in diverse c e n t r a l  impact of such parameters may be  e s t a b l i s h e d ; f o l l o w i n g t h i s i t becomes possible to move on to  the case of the long run where d i s c r e t e system-wide changes (due, f o r i n s t a n c e , to the migration of productive f a c t o r s ) may be incorporated i n t o the central place format. The former technique i s most t y p i c a l of the comparative s t a t i c s approach in which a system i s taken to be in a state of e q u i l i b r i u m and it  i s then "shocked" and taken to proceed to a subsequent state of  equilibrium.  3  To the author's knowledge Nourse (1968) represents the sole attempt to introduce parametric impact with e x p l i c i t system-wide "before 4 and a f t e r " c o n d i t i o n s .  Unfortunately his argument was based on the  obsolete Beckmann (1958) model - a model, i n c i d e n t a l l y , which remains i n f l e x i b l e even when properly reformulated - so that a l l of Nourse's comments must be greeted with some r e s e r v a t i o n .  H o p e f u l l y , too, when the  reader has compared the upcoming d i s c u s s i o n v i s - & - v i s Nourse's c o n t r i bution he may be able to expel some of Nourse's pessimism about the a p p l i c a b i l i t y and usefulness of the h i e r a r c h i a l perspective f o r dealing with regional economic growth. Secondly, the author reviews the macro-diffusionary of the previous chapter in a new l i g h t .  argument  By v i s u a l i z i n g independent  parametric change as an important force reshaping s p a t i a l adoptive processes  TOO  i t becomes f e a s i b l e to hypothesize regarding the p a r t i c u l a r impacts that changes in central place populations, in l e v e l s of per c a p i t a income o r , perhaps, i n transportation technology might have upon the o v e r a l l acceptance pattern of a given item.  The author suggests that t h i s new per-  spective seems most appropriate when looking at passive  items ( e . g . a new  a r c h i t e c t u r a l form, a new household appliance) which would not be expected to engender s i g n i f i c a n t feedbacks i n the central place s t r u c t u r e as other much more active  items ( e . g . a new mass t r a n s i t scheme, a new community  h o s p i t a l ) most c e r t a i n l y would.  5  One o v e r a l l concluding remark i s deemed necessary.  The tenor  of argument i s somewhat uneven as i t v a r i e s from the quite rigorous s t a t e ments of the comparative s t a t i c s approach to the more d e s c r i p t i v e a n a l y s i s used in the d i s c u s s i o n of long run s t r u c t u r a l change and, f i n a l l y , to the h e u r i s t i c treatment of central place d i f f u s i o n .  As in the t h i r d chapter,  the author sees no p a r t i c u l a r and immediate advantage i n strengthening the a n a l y t i c a l tone i n c e r t a i n parts of his d i s c u s s i o n .  Once a g a i n ,  c e r t a i n hypothetical examples are given i n order to i l l u s t r a t e the e f f e c t s of temporal changes i n the relevant central place s e t t i n g s .  4.2  Population To begin with consider the case of population growth w i t h i n a  c e n t r a l place system.  Scrutiny of the argument i n the second chapter should  make i t apparent that one can i s o l a t e the instantaneous e f f e c t s of change a change which i s s a i d to be exogenously induced - i n the employment (popul a t i o n ) c h a r a c t e r i s t i c s of the basic sector of a central place upon the o v e r a l l employment (population) c h a r a c t e r i s t i c s of that same central p l a c e .  101  Note, however, that our e x i s t i n g framework seems e n t i r e l y a p p l i c a b l e f o r d i s c u s s i n g long run population changes as w e l l .  Theory  would lead one to suspect that c e t e r i s paribus population growth over the long run could refashion systemic structure and process in but a few general  ways; the precise impact, though, would appear to depend upon: (i)  (ii)  (iii)  t h e a c t u a l magnitude o f such growth ( p u t d i f f e r e n t l y , i f p o p u l a t i o n were t o expand t h r o u g h o u t the system a t a c o n s t a n t r a t e , would t h e long run o r t h e very long run be the r e l e v a n t time s c a l e ? ) ; t h e t h r e s h p l d o f emergence f o r new c e n t e r s t h a t i s , one must c o n s i d e r " . . . t h e f o r c e s o f i n e r t i a which may p r e s e r v e t h e number o f h i e r a r c h i a l l e v e l s and even t h e o v e r a l l number o f c e n t e r s . . ." ( P a r r and Den i k e ( 1 9 7 0 : 5 7 4 ) ) ; and t h e degree o f m o b i l i t y i n p r o d u c t i v e f a c t o r s l a b o r , o f c o u r s e , i s o f paramount i n t e r e s t here.  The reader should note that population decline may be conceptua l i z e d in a symmetrical but opposite manner. Before proceeding to the a n a l y s i s per se a few statements of a d e f i n i t e methodological c h a r a c t e r , d i s c l o s i n g the author's personal views about the especial advantages of the ( s t r i c t ) systems approach, might be of some i n t e r e s t to the reader.  The author p a r t i c u l a r l y favors the systemic  model since i t guarantees a measure of consistency i n the a n a l y s i s as concern varies over d i f f e r e n t temporal and s p a t i a l s c a l e s .  More e x p l i c i t l y ,  the reader should recognize that change may be appropriately examined w i t h i n : (i)  a s p e c i f i c c e n t r a l p l a c e - which i s an individual i t s e l f as w e l l a s an e l e m e n t o f t h e much w i d e r system - a t a p a r t i c u l a r i n s t a n t i n t i m e ; o r  (i i)  a s p e c i f i c c e n t r a l p l a c e system - i t s e l f an i nd i v i duaI - o v e r a p a r t i c u l a r i n t e r v a l o f t i m e .  102  I t i s perhaps a truism to remind the reader that the long run changes alluded to i n ( i i )  above would be made apparent i n the same c e n t r a l places  which would e x h i b i t the instantaneous changes of  (i).  In the f i r s t of these cases an exogenous-endogenous dichotomy i s r e a l i z e d w i t h i n the central place i t s e l f  (since the change in external  employment (population) n e c e s s i t a t e s a d i r e c t change i n the s e r v i c i n g or basic sector which, in t u r n , brings about successive rounds of change in the non-basic s e c t o r ) .  In the second case, however, t h i s dichotomy -  if,  i n f a c t , i t e x i s t s - must be somehow r e a l i z e d at the "boundary" of the e n t i r e system. In order to c l a r i f y t h i s l a s t point the reader should observe that t h i s d i s t i n c t i o n would be v a l i d , f o r i n s t a n c e , i f the e x i s t i n g c e n t r a l place populations were to remain i n n a t e l y stable while migration p e r s i s t e d in the long run.  Then i t would be e n t i r e l y c o r r e c t to a s s e r t that systemic  population growth was brought about by an exogenous change.  In c o n t r a s t ,  the e f f e c t would be i d e n t i c a l but the d i s t i n c t i o n could not be s t r i c t l y upheld i f a l l growth were i n t e r n a l  (caused by an increase in the b i r t h  rate perhaps). The author simply f e e l s that the exogenous-endogenous dichotomy i s a very useful one f o r ensuring p r e c i s i o n or methodological r i g o r in an argument but he wishes to emphasize that the meaning of that d i s t i n c t i o n must be considered when the temporal and/or s p a t i a l scales become enlarged. Nevertheless, t h i s a b i l i t y to l i n k successive scales (see Harvey (1969:452) i s a major reason why the systemic approach has both conceptual and i n s t r u mental merit when one i s undertaking the complex task of modelling some portion of the real world.  103  In c l o s i n g t h i s foreword i t should be noted that since populat i o n change - both i n the sense of growth and r e d i s t r i b u t i o n - i s a symptom of other forms of parametric change, t h i s i n i t i a l  section i s considerably  more d e t a i l e d than the subsequent two sections which focus upon the impact of changes in per c a p i t a income and d i f f e r e n t  types of technology on  systemic e q u i l i b r i u m .  (i)  Structure  Examination of the discussion in Section 2.3 should make i t apparent t h a t an exogenous s h i f t i n employment (population) f o r a p a r t i c u l a r c e n t r a l place would be r e a l i z e d throughout the rural areas and the various smaller communities served by that central p l a c e .  On the other hand,  t h i s s h i f t i s u l t i m a t e l y resolvable to a change in the rural density alone, since a l l community populations are " b u i l t up" from t h i s rural base. The upcoming d i s c u s s i o n i s i n i t i a l l y concerned with the former (and more general) case wherein a s h i f t in any of the components B B e  m 2 ' * * * ' mm e  °^  a n  m  ^  e-|,  l e v e l center has an impact upon the t o t a l number  of employees (persons) r e s i d i n g there.  Unfortunately, though, the recur-  sive format of the h i e r a r c h i a l model makes the exercise of d e r i v i n g these p a r t i c u l a r impacts more than a b i t tedious.  For t h i s reason the author  includes an a l t e r n a t i v e method - one which gains in conceptual and computational  s i m p l i c i t y what i t loses in a n a l y t i c a l g e n e r a l i t y - wherein a  s h i f t in the rural density of the system may be r e l a t e d to a change i n the employment (population) of any central p l a c e .  The advantages of t h i s second  approach become much c l e a r e r l a t e r in the chapter when i t proves useful to B B B e l i m i n a t e the sectors e ^ , 2 » * * * » ^ argument. e  e  m  r o m  m m  t n e  104  To begin w i t h , the "matrix" approach advocated in the second chapter seems e s p e c i a l l y amenable f o r i n d i c a t i n g i n the components  B e . mj  (1 < j < m) -  the impact of  of the basic sector  l e v e l place upon the t o t a l employment (population)  B e m  shifts  of an  m th  of that same p l a c e .  R e c a l l i n g (2.21) or ( 2 . 3 2 ) , the reader sees t h a t : . + ml  B e  e . = e .. + mi mi  + m2 A m  +  B e  B e  mm  (4.1)  where:  A  - 1 -  m  m I k i=l  (4.2)  1  This e f f e c t i v e l y divides the t o t a l employment  e^  engaged in  i th l e v e l  a c t i v i t i e s i n t o i t s basic and non-basic p o r t i o n s . Assuming that ( i )  technology remains constant ( i . e . that there  i s no v a r i a t i o n i n any of the service m u l t i p l i e r s ) B e  and that ( i i )  all  T  mi ' mi e  ^  -  1  - ^ m  differentiate  a r e  in a s u f f i c i e n t l y  borhood of t h e i r e q u i l i b r i u m p o s i t i o n s as given i n ( 4 . 1 ) , k. + A _j m m  3e . mi T  k. A m 1  for  small neighthen:  i=j (4.3)  for  i/j  3e . T  The reader should note that  l e v e l c e n t e r , the direct  — i n  (4.3) represents, f o r any  m th  impact of a " s m a l l " exogenous change i n employment  105  B e .  T However, i t mi must be stressed (perhaps the reader might review the second chapter) that in the sector  any sector B ^ mi e  I  1  5  e , 1  upon the t o t a l employment sector  mj  < J  (2 < j < m) }  s o  t  n  a  t  e  may be stated as a function of e  the total  impact of a s h i f t in  and  0  j th level  a c t i v i t i e s becomes: m Y  k 3e 3e  3e  B  m a  a=j 3e . mj A. m  mi  3e  for j < i  mi  (4.4)  3e "mj l  B 3e k T — ^ i . . B* a=j 3e mj A m m  K  l  for j > i  3e' 3e' mi and mi stems from the r e c u r s i v e B* 3e^ 3e "mj "mj properties of domination which c h a r a c t e r i z e the central place argument. The discrepancy between the terms  The former term indicates the impact (exerted upon the body of a l l employees i n an  m th l e v e l place engaged in  of a s h i f t i n employment i n any of the  j  i th l e v e l  (1 < j < m)  activities)  individual  basic  sectors delineated by the central place topology (each of these i s i n d i cated in equation (2.17) above).  An a l t e r n a t i v e perspective i s to think  of these s h i f t s as occurring in any one of the  j  which the  m th level center serves i n the capacity  j th,***,  or  m th l e v e l p l a c e .  On the other hand, the term  complementary areas of a  1st,  2nd,"*,  T  3e'. — ^  i n d i c a t e s the impact upon  mj of a s h i f t in all  j th  (1 < j < m) l e v e l basic sectors - not j u s t the  j e^ mi  106  s i n g l e sector which i s provided with the with the  m th bundle by the  j th bundle - that are provided  m th level p l a c e .  N a t u r a l l y , most of these  basic sectors (complementary areas) are provided with lower order goods and s e r v i c e s by the various turn m-dominated by the  i th  (1 < i < m) l e v e l places which are i n  m th level p l a c e .  Now a number of a t t r i b u t e s of these impacts may be i n f e r r e d from the two equations.  If,  for i n s t a n c e , the s e r v i c e m u l t i p l i e r s d e c l i n e i n a  well-ordered fashion ( i . e . as i n the usual case where  ki > k  > • • • > k ), m  2  then (4.3) suggests t h a t : 9e ma 9e "mj  T 9e mb 9e "mj  1 < a < b <m a,b  f  j  (4.5)  and: 9e' ma 9e ma  9e mb  9e 'ma 9e  9e mb B* 9e "mj  9e  ae'  B* 9e "ma  B* 9e mb  9e mb  1 < a < b < m  (4.6)  while (4.4) suggests t h a t :  l  'mj  l < a < b < j < m  (4.7)  and:  mi  with the p a r t i c u l a r consequence t h a t :  mi  1 < a < b < m  (4.8)  107 9e 9e . ma ^ mb T  T  3e , ma  >  1 < a < b < m  9e . mb  m  (4.9)  Equations (4.7) and (4.8) are perhaps the most s i g n i f i c a n t and hence deserve some i n t e r p r e t a t i o n  at the present time.  The former states  t h a t , given the set of i d e n t i c a l basic sectors i n which the exogenous s h i f t might occur and the condition that in the sector sector  e^  e^  a < b < j , the impact would be greatest  of a l l f i r s t l e v e l employment, second greatest i n the  of a l l second l e v e l employment and would p r o g r e s s i v e l y diminish  f o r the subsequent  j-2  sectors of higher order a c t i v i t y .  The l a t t e r  equation complements t h i s in that i t s t a t e s , given the endogenous sector in which the various exogenous s h i f t s might be expressed, the greatest impact would be induced by a change i n f i r s t l e v e l a c t i v i t i e s , the second greatest by a change in second l e v e l a c t i v i t i e s , and that t h i s e f f e c t would i n c r e a s i n g l y decline f o r the  M-2  higher order a c t i v i t i e s .  In a d d i t i o n , a f t e r r e c a l l i n g (2.22) where t o t a l employment was shown to be:  e  m" ml e  +  It follows (assuming now that  e  e^  m 2  < - >  +  4  is d i f f e r e n t i a t e  small neighborhood of i t s e q u i l i b r i u m p o s i t i o n ) de m TBde . mj  k!  T  =  k , " , 7T /T m m 2  +  m  +  +  k. + A , j m , ^ A — m +  in a s u f f i c i e n t l y  that:  "  k , m A m -  ,0  108  where  ^ - i s the export base m u l t i p l i e r f o r an m i s nearly a truism to point out t h a t : de m de 'mj  m th l e v e l p l a c e .  It  de  1  j < m< n < M  de 'nj  (4.12)  L  On the other hand: T T de m 3e . m _ y mi B* ^ B* de . i=l 3e . mj mj  (4.13)  B  so t h a t : de' m de 'mj c  de' de  j < m< n <M  (4.14)  L  nj  As implied above, however, the body of employment  e^  can be  resolved i n t o an expression t h a t only involves the r u r a l employment density eo  ( i . e . the number of r u r a l employees in each basic u n i t a r e a ) , the  set of nesting c o e f f i c i e n t s (K^  | 1 < i < m-l},  multipliers  This argument was f i r s t presented by  {k^  | 1 < i < m}.  and the set of s e r v i c e  Beckmann and McPherson (1970:27) in terms of the r u r a l population density ri  although, i t should be added, those two authors apparently were not  i n t e r e s t e d in d i s c e r n i n g the r e l a t i v e e f f e c t s of s h i f t s in parameters - more s p e c i f i c a l l y , s h i f t s in multipliers  k.  r  x  different  i t s e l f and i n each of the  (1 < i < m) - upon the population  p  m  of an  m th l e v e l  central place. F i r s t of a l l the reader might wish to return to the notational format o u t l i n e d i n Section 2.2 of the second chapter.  There the reader  109  should see that the t o t a l employment ( i . e . the employment of the  E  i n an  m  m th l e v e l market area  m th l e v e l place as well as that of the  complementary area which i t m-dominates) i s :  E  ™ i  = e m  +  (4  -  15)  .m R e c a l l i n g next (2.22) makes i t apparent that f o r  e  After defining  D  for  m m" A  m> 2  e  m-l  Yl  =  m > 2:  i  <4-16>  as:  D = e' m m  and u t i l i z i n g (4.15) and ( 4 . 1 6 ) , i t follows  D  m  (4.17)  'm-l  that:  • ^  (4.18)  Now by observing t h a t :  mm = K m-l  i t may be demonstrated that:  'mm-1 k m-l  V l  - 1  'm-l  (4.19)  no  where, from (2.2) and (2.4)  ej  ejj_  m-l  K. A.  Ei  But t h i s means that for  (4.21)  m > 2:  a  = eo_  E  m  Therefore, a f t e r r e d e f i n i n g  A  e m  3  n  i  i  A  i 1  > i=l  (4.22)  +  as:  e' = e m  i t follows from (4.18) and (4.19)  m  n D.i  +  .  (4.23)  = 2  that: m-l  n K.  k m  k i + Mn. e H Ai A A X  + k KiK 3  +  0  2  A A 2  i=l A , A m-l m m  2  3  1  (4.24)  which proves to be an extremely useful statement i n the upcoming discussion For present purposes, however, i t  i s s u f f i c i e n t to note t h a t : m-l  de' m  n K.  k M  1=1  A m-l  1  (4.25)  A m  where the a s t e r i s k i s a simple reminder that t h i s change i n r u r a l employment density i s system-wide. The author now wishes to g e n e r a l i z e some of the above statements f o r the case of population (as opposed to employment a l o n e ) . w i t h , r e c a l l (2.24) where i t was stated that the t o t a l  To begin  population  p^  in  of an  m th l e v e l place which i s supported by  i th  (1 < i < m)  level  employees i s :  "II""  Now assuming d i f f e r e n t i a b i l i t y  +  d  i  of a l l  )  e  mi  < - 6) 4  pj.  2  i t should become apparent  that: 3p . .  3e .  T  -8 eJ " . mj 1  T  (1  < - > 4  3e . mj  1  l  27  and that: 3p . 3e . = (1 + d.) * \i B* 3e . 3e . mj mj T  T  m i  m i  u  ;  B  3e . —pp- and 3e . mj T  where  D  tively.  3e . — ^ 3e . mj T  have been derived above in (4.3) and (4.4) respec-  B  Of course, statements s i m i l a r to ( 4 . 5 ) , ( 4 . 6 ) , ( 4 . 7 ) , (4.8) and  (4.9) - only now f o r the e f f e c t of an exogenous change in employment on the various population sectors - may be formulated by c r e a t i n g appropriate ordering c o n d i t i o n s . F i n a l l y , the reader should r e c a l l (2.24) where the argument demonstrated that:  Pm Pml Pm2 "* PL =  +  +  <- )  +  Then i t f o l l o w s , assuming the d i f f e r e n t i a b i l i t y  4 29  of  p , that: m  112  dp  8e  m  m  T  l - H ' * ^ - f  (4-30)  dp 7TF =  (4.31)  and that: m I (1 *  d  3e^. i ) ^  which i n d i c a t e t h a t :  d  Pm m  d  Pn  —g— < —g— d e . d e . mj nj  for  j < m< n < M  (4.32)  for  j < m< n < M  (4.33)  and t h a t : dp dp —jpr < — ^ de . de . mj. nj  Unfortunately, however, (4.25) does not lend i t s e l f to a simple extension f o r the population case. The wary reader has perhaps noted that the s t a t i c format i s useful i n other ways for analysing the impacts of small exogenous changes i n s p e c i f i e d parameters upon other v a r i a b l e s of a more endogenous chara c t e r - e s t a b l i s h i n g some of these other p o s s i b i l i t i e s , in f a c t , comprises a good part of the remainder of t h i s chapter.  But now i n t e r e s t turns to  examining changes i n central place populations over much greater periods of time.  The f o l l o w i n g discussion may be construed as both a c r i t i q u e  and a r e v i s i o n of the argument found in Nourse (1968:209-212). Consider, to begin w i t h , the elementary case where a long run increment of population - say, for s i m p l i c i t y , that a l l rural  (basic)  113  areas and c e n t r a l places grow by one percent - occurs i n a central place system but there i s no tendency f o r any new centers to emerge.  In other  words, due to the r e l a t i v e l y small increase in p o p u l a t i o n , the system would show no i n c l i n a t i o n to e i t h e r e x h i b i t areal expansion or f u r t h e r h i e r a r c h i a l d i v i s i o n (which, of course, would be a r e a l l y i n t e r n a l  to the  system). Quite o b v i o u s l y , then, consumer purchasing power must increase w i t h i n the market area of each central place in the system.  This in turn  i n d i c a t e s t h a t there would be a tendency f o r a s h i f t to occur - assuming, o f course, f a c t o r m o b i l i t y - i n the system's hierarchial  marginal  goods  and services (to use a phrase coined by Berry and Garrison (1958) which r e f e r s to the upper endpoints of a l l bundles in the system except the l e v e l one) due to a type of import s u b s t i t u t i o n .  Put d i f f e r e n t l y ,  M th  a surplus  of purchasing power would be created i n the market areas of a l l centers on the  m th  (1 < m < M)  level by t h i s population increase and that  surplus might well be s u f f i c i e n t - here i t  i s taken that a one percent  increase is s u f f i c i e n t - to induce a c t i v i t i e s to migrate away from centers on the  (m+l)st, ( m + 2 ) n d , " « ,  the next lower l e v e l s . by using the t r a d i t i o n a l  and  M th l e v e l s i n t o those centers on  This slippage of a c t i v i t i e s may be demonstrated cost and demand methodology of spaceless economics  or by employing the threshold conception of C h r i s t a l l e r i a n theory: f o r a l u c i d synthesis the reader should r e f e r to Parr and Denike (1970). Unfortunately, outside of s t a t i n g that k^  ki  increases and that  decreases in the long run, at t h i s time i t i s only possible to speculate  about the nature of change in the other m u l t i p l i e r s of the set {k |m=l,2,'*',M} m  c h a r a c t e r i s t i c of the general h i e r a r c h i a l model.  114  Theoretical improvements, focussed on the composition of each m u l t i p l i e r , are sorely needed before even q u a l i t a t i v e changes can be s p e c i f i e d f o r those remaining  M-2  multipliers.  One perspective that appears useful for dealing with such r e a l l o c a t i o n i s to i s o l a t e the changes, between the i n i t i a l  and subsequent  states of e q u i l i b r i u m , in the numbers of persons engaged in r u r a l , o r d e r , second o r d e r , * " , and the e n t i r e system.  M th order a c t i v i t i e s r e s p e c t i v e l y  first throughout  The r e s u l t a n t changes i n the central place populations  follow accordingly.  These activity  populations and r e l a t e d  coefficients  (which, s i g n i f i c a n t l y , are i d e n t i c a l to the service m u l t i p l i e r s ) expressing those populations as r a t i o s of the t o t a l systemic population may be computed as shown in Appendix E. Consider the f o l l o w i n g hypothetical case designed to the above remarks.  Suppose that a four l e v e l  to with  and  r-i = 2000  {k  m  [t ]> 0  = {.3333,  K=3  illustrate  system e x i s t s at time  .1667, .1250, .0937}.  the population of t h i s system expands by one percent up to a time but there i s no r e d i s t r i b u t i o n occurring i n the time i n t e r v a l and  t i , then  ri  would increase to  2020  where  {k  [ti]}  m  If ti  between = {k  m  to  [t ]}. 0  E v e n t u a l l y , however, the slippage of goods and services would take place as argued above so that the system might be characterized by a new set {k  m  [t ]} 2  of m u l t i p l i e r s at the s t i l l  l a t e r time  t .  From the second  2  chapter i t should be apparent that a property of t h i s  growth-real!ocation  scheme would be that  l k m=l  m  Ct.] -  I k m=l  m  [ ] t l  -  I k m=l  m  [t,]  (4.34)  115  If  i t i s assumed that only  then  {k  m  [t ]} 2  = {.3500,  ki  and  ki+  .1667, .1250, .0770}  state of the h i e r a r c h i a l system at time p a r t i c u l a r change, e s p e c i a l l y as i t (i)  (ii)  change during the  t . 2  redistribution  would represent a  possible  The i m p l i c a t i o n s of t h i s  affects:  t h e numbers o f persons r e s i d i n g of d i f f e r e n t s i z e s ; 9 and t h e numbers o f persons engaged activities;  in centers  in specific  are i l l u s t r a t e d in Tables 4.1 and 4.2 r e s p e c t i v e l y .  The reader should  note that in order to f a c i l i t a t e computations the t r a d i t i o n a l (with no account given of d i f f e r e n t i a l  notation  family s i z e ) of the l i t e r a t u r e  is  utilized. I t i s i n t e r e s t i n g to observe t h a t , at l e a s t under these s p e c i a l r e s t r i c t i o n s , r e d i s t r i b u t i o n appears to be a function of c i t y s i z e :  the  percentage c o e f f i c i e n t s f o r the f i r s t , second, t h i r d , and fourth order places are (looking at column 6 i n Table 4.1 and column 8 in Table 4.2 as well as neglecting rounding e r r o r s ) : 0 . 7 3 , 0.81 (0.60 + 0 . 2 2 ) , (0.77 + 0.24 + 0 . 2 9 ) , and -2.85 (-0.43 - 0.46 - 0.29 - 1.67) The intermediate  1.31  respectively.  t o t a l s 1.67, 0.00, 0.00, and -1.67 in column 8  of Table 4.2 represent the changes i n urban a c t i v i t y r a t i o s and c l e a r l y r e f l e c t , in percentage form, the hypothesized changes in the o r i g i n a l  set  . , . . ,. 10 of s e r v i c e m u l t i p l i e r s . £  The author must c a u t i o n , however, that these stated properties of incremental growth and r e d i s t r i b u t i o n must in no way be taken as gene r a l i z a t i o n s : they simply r e s u l t from the specific  assumptive framework.  On the other hand, a technique has been created which allows comparisons  116 Table  4.1  A Proposal of a One Percent Growth and R e d i s t r i b u t i o n Level Central Place System; {k  .1250, .  m  0937}; {k [t ]} = m  Scheme in a K=3 Four  .1667, {.3500, .1667, .1250, .0770} =  [to]}  (k  m  [t.J}  = {. 3333,  2  (1)  (2)  (3)  (4)  (5)  (6)  Time  Population Sector  Population Per Unit  No. of Units  Sectoral Population  % Total Population  FT  2000  27  54000  28.13  PT  1000  18  18000  9.37  pT  4000  6  24000  12.50  pT  16000  2  32000  16.67  pT  64000  1  64000  33.33  192000  100.00  to  2020  27  54540  28.13  1010  18  18180  9.37  PT  4040  6  24240  12.50  PT  16160  2  32320  16.67  64640  1  64640  33.33  193920  100.00  rT pT ti  pV  t  2  rT  2020  27  54540  28.13  PT  1088  18  19584  10.10  PT  4304  6  25824  13.31  pT  17430  2  34860  17.98  pV  59112  1  59112  30.48  193920  100.00  Table 4.2 An Analysis by A c t i v i t y Sectors of the Proposed One Percent Population Growth and R e d i s t r i b u t i o n Scheme; (k  m  Activity Sector 1  P  P  P  0  = {k (2)  (1)  r  [t ]}  1  Population Sector  = {.3333, .1667, .1250, .0937}; {k (3)  (4)  Time  Time  to  ti  [t ]} 2  (5)  (6)  % Total to ,  = {.3500, .1667, .1250, .0770}  tl  (7)  - -  (8)* '  Time t  % Total t  % Total t % Total t i  2  2  2  54000  54540  28.13  54540  28.13  0.00  pT  18000  18180  9.37  19584  10.10  0.73  pT  12000  12120  6.25  13284  6.85  0.60  pT  12000  12120  6.25  13616  7.02  0.77  pi  22000 64000  22220 64640  11.46 33.33  21397 67881  11.03 35.00  -0.43 1.67  pi  12000  12120  6.25  12540  6.47  0.22  pi  8000  8080  4.17  8556  4.41  0.24  pi  12000 32000  12120 32320  6.25 16.67  11228 32324  5.79 16.67  -0.46 0.00  pi  12000  12120  6.25  12688  6.54  0.29  pi  12000 24000  12120 24240  6.25 12.50  11554 24242  5.96 12.50  -0.29 0.00  pi  18000  18180  9.37  14933  7.70  -1.67  192000  193920  100.00  193920  100.00  0.00  3  TOTALS  x  "FT  2  P*  [t ]}  .  118  of r e a l l o c a t i o n s to be made amongst diverse central place systems with d i f f e r e n t m u l t i p l i e r s , growth increments, t o p o l o g i e s , and the l i k e . This p e r s p e c t i v e , r e s t i n g on the twin suppositions that (i)  systemic population would expand incrementally and then s t a b i l i z e and  that ( i i )  no new centers would emerge because t h i s expansion was so s l i g h t ,  must be considerably modified i f population growth i s taken to p e r s i s t i n the (very) long run.  Theory suggests that the extension of  interstitial  purchasing power would, at some time, induce the emergence of new f i r s t order c e n t e r s .  Concomitantly, centers previously on the  m th,  (m+l)st,  ( m + 2 ) n d , * " , and ( M - l ) s t l e v e l s would be expected to acquire (m+l)st, (m+2)nd, ( m + 3 ) r d , ' " , and  M th order a c t i v i t i e s r e s p e c t i v e l y as w e l l .  Unfortunately, e x i s t i n g theory provides no r a t i o n a l e in suggesting the nature of a threshold  f o r such emergence - i t appears, once a g a i n ,  that considerable a t t e n t i o n must be d i r e c t e d in the future to the  attributes  of the i n d i v i d u a l commodity bundles before even a tenuous s o l u t i o n to the problem of emergence may be forwarded. G e n e r a l l y , however, i t seems reasonable to a n t i c i p a t e  that  extensive population growth would, c e t e r i s p a r i b u s , induce f u r t h e r systemic d i v i s i o n according to the i n i t i a l of the central place s t r u c t u r e . system, a two-fold increase  organizational  (geometrical)  To take an example, in a  M level  K=3  in population over a lengthy period of time  would lead to the a r t i c u l a t i o n of three  identical  the area o r i g i n a l l y housing the s i n g l e  M l e v e l system.  on the  principles  M th level would serve ( i . e .  M l e v e l systems w i t h i n The i n i t i a l  center  M-dominate) one of these "new"  systems and t h i s system, in t u r n , would be enveloped by s i x o n e - t h i r d portions of two "new" i d e n t i c a l  (equivalent)  systems.  These l a t t e r  two  119  equivalent systems would each be  M-dominated by an equivalent center  which was previously of the ( M - l ) s t order (in the o r i g i n a l system) before the long run population growth commenced.  This systemic r e s t r u c t u r i n g  would take place i n a l l s i z e classes so that i t might be s a i d , in summary, that there occurs a one bundle import s u b s t i t u t i o n in a l l e x i s t i n g centers except the o r i g i n a l  M th l e v e l p l a c e .  This i n t e r p r e t a t i o n d i f f e r s s i g n i f i c a n t l y from that suggested by Nourse: he would argue that the o r i g i n a l  M th order place simply  becomes an (M+l)st order place i n a more complex s i n g l e system which i s confined to the o r i g i n a l a r e a . ^  While t h i s appears perhaps more consonant  with the real world experiences of economic  growth in a r e g i o n , i t  i s not -  in t h i s author's opinion - a v a l i d c e t e r i s paribus argument (and t h i s what Nourse d e s i r e s ) f o r the impact of population  growth alone.  is  For such  h i e r a r c h i a l extension to become e v i d e n t , i t would be necessary to have new economic a c t i v i t i e s ( s p e c i a l i z e d goods and s e r v i c e s which could only be supported by the o v e r a l l s h i f t i n aggregate demand) accompany the growth in population. On the other hand, even by "assuming away" the dilemma of an emergence threshold - suppose, f o r i n s t a n c e , that an  x - f o l d increase 0  in systemic population is s u f f i c i e n t f o r a l l "new" centers to appear still  it  does not remain a simple task to speculate about the nature of  systemic r e s t r u c t u r i n g due to an x - f o l d nesting f a c t o r )  (xo < x < K - l , where  K  i s the  increase i n the long run.  However, i f i t  i s f u r t h e r supposed t h a t , a f t e r such emergence,  growth and r e d i s t r i b u t i o n occur so that ( i ) i s retained and ( i i )  the o r i g i n a l set of m u l t i p l i e r s  the population sectors a l l expand at the same r a t e ,  then i t may be demonstrated t h a t :  12  120  (i)  i n each o f t h e K "new" systems, t h e p o p u l a t i o n o f each u n i t i n t h e s e c t o r s r i , p i , p 2 , * " ,p^ | x+1 is - T r - times t h e p o p u l a t i o n o f t h e same u n i t K  b e f o r e t h e p o p u l a t i o n i n c r e a s e and c o n c o m i t a n t h i e r a r c h i a l d i v i s i o n occur (note, of course, t h a t these same u n i t s do n o t occupy t h e same l o c a t i o n s as b e f o r e ) ; (ii)  (iii)  t h e K - l e q u i v a l e n t c e n t e r s which a c q u i r e a c t i v i t i e s p r e v i o u s l y found i n o n l y t h e M t h l e v e l p l a c e have p o p u l a t i o n s consonant w i t h t h e p r o p e r t i e s o f t h e "new" e q u i v a l e n t systems which they p a r t i a l l y M-dominate; and the o r i g i n a l M t h level place r e t a i n s i t s populat i o n but transforms i t s a c t i v i t y s e c t o r s in a manner whereby a c o n t i n u a l l y d i m i n i s h i n g " s u r p l u s " p o p u l a t i o n - t h a t i s , t h e amount o v e r and above the p o p u l a t i o n s o f t h e o t h e r K - l emerging M t h l e v e l p l a c e s - p r o v i d e s t h e r e m a i n i n g goods and s e r v i c e s t o t h e K "new" s y s t e m s .  What i s important to observe here i s that the commodity set i s being cont i n u a l l y r e d e f i n e d , although always with a t o t a l of the time when  x+1 = K and that during t h i s  k M +  where  *  p  M  and  *  P  M  l  = "  ' ^ m  M+l  redefinition:  'PM  'Jr  are the populations of the o r i g i n a l  and system r e s p e c t i v e l y .  Note also that the m u l t i p l i e r  x -* K - l , which simply i n d i c a t e s that with a  bundles, u n t i l  (4.35)  M th l e v e l place -> 0  as  ( K - l ) - f o l d increase in systemic  p o p u l a t i o n , K "new" and i d e n t i c a l systems e x i s t as was s t i p u l a t e d above. It should be apparent t h a t , with the r e d e f i n i t i o n of the commodity set over time, t h i s argument has a s i m i l a r i t y to the proposal  121  o u t l i n e d by Nourse; however, by r e t a i n i n g a s t r i c t e r conception of c e n t r a l place p r i n c i p l e s (as well as having at hand a more powerful model), the present author f e e l s he i s giving a more accurate d e p i c t i o n of the i m p l i cations of long run c e t e r i s paribus population growth w i t h i n a central place system. Quite o b v i o u s l y , the above formulation  i s an i d e a l i s a t i o n but  i t does represent an elementary a n a l y t i c a l model of systemic transformation.  In a d d i t i o n , i t could be refined (to accommodate s h i f t i n g  v a r i a b l e t o p o l o g i e s , e t c . ) but the present format seems e n t i r e l y  multipliers, sufficient  f o r the time being. Population decline may be examined i n a symmetric f a s h i o n . Depending on the postulated nature of such d e c l i n e , varying amounts of activity  loss would be expected f o r the system as a whole and the threshold  f o r a c t i v i t i e s would c e r t a i n l y migrate up through the hierarchy ( i . e . an m th (1 < m < M) l e v e l place would tend to lose a c t i v i t i e s to places on the (m+l)st l e v e l and perhaps even higher l e v e l s i f the population decline persisted).  (i i )  Process  Gravity - potential  theory suggests t h a t , under c e r t a i n circum-  s t a n c e s , population increases throughout a central place system would tend to accelerate a d i f f u s i o n a r y process which had already commenced.  For  many items ( e . g . most consumer goods) the e f f e c t s of t h i s growth might be considered n e g l i g i b l e as termination of primary adoption would be expected to occur somewhat q u i c k l y ; nevertheless, f o r some items (frequently  those  122  i n v o l v i n g more c a p i t a l r i s k : new productive techniques,  infrastructure  improvements, e t c . ) d i f f u s i o n would l i k e l y proceed in a moderate fashion and concomitant systemic growth would doubtless have an influence on the nature of s p a t i a l adoption. To begin w i t h , suppose a four l e v e l the previous section where  K=3  system e x i s t s as in  {p > = {1000, 4000, 16000, 64000}.  Suppose,  m  f o r convenience, that ( i ) a l l central places grow at the same r a t e , that (ii)  t h i s rate may be standardized according to the threshold time of f i r s t  adoption (see the previous c h a p t e r ) , and that ( i i i )  effective  potential  i s enhanced only a f t e r an increment has been added to the t e l l i n g and hearing c e n t e r s .  These assumptions are simply useful f o r conceptualizing  continuous growth in discrete  terms and they could be modified according  to the wishes of the researcher. Consider the fourth l e v e l center. threshold time i s  t  If,  f o r i n s t a n c e , the adoptive  and growth i s one percent per u n i t of  t,  then the  p o t e n t i a l l y e f f e c t i v e population of t h a t center at each of the times: 0, t, 2 t , 3 t , " ' , nt (64000)(1.01 ) . n  i s , respectively:  64000, 64640, 65286, 65939,  With growth conceptualized as such, the modelling scheme  of the previous chapter becomes amenable to s p e c i f y i n g divergences that would be expected between d i f f u s i o n which occurs i n a s t a t i c m i l i e u and d i f f u s i o n which occurs i n a more dynamic m i l i e u . illustrate  Tables 4.3 and 4.4  such differences for two growth rates i n open and closed systems. It should be apparent to the reader that population  differentially  growth  contracts adoptive times depending upon the nature of  systemic c l o s u r e : e a r l y adoptive times in the open case but l a t e r times i n the closed case seem e s p e c i a l l y i n f l u e n c e d .  Interestingly  adoptive enough,  123  Table 4.3 The Impact of Population Growth on Spatial Adoption in an Open K=3 Four Level Central Place System; {p > m  = {1000, 4000,  16000, 64000}, b = 2  Adopting Center  No Growth  1% Growth  2% Growth  4  0.00  0.00  0.00  3  1.00  1.00  1.00  2  3.00  2.94  2.87  5.07  4.88  4.70  9.02  8.37  7.89  9.53  8.81  8.37  1 l  1  3  1  2  124  Table 4.4 The Impact of Population Growth on S p a t i a l Adoption in a Closed K=3 Four Level Central Place System; {p > m  = {1000, 4000,  16000, 64000}, b = 2  Adopting Center  No Growth  1% Growth  2% Growth  4  0.00  0.00  0.00  3  1.00  1.00  1.00  2  1.30  1.29  1.28  1  1  1.91  1.89  1.87  1  2  3.53  3.46  3.36  1  3  10.00  9.21  8.57  125  i n both instances the adoptive orderings are retained from the s t a t i c argument of the previous chapter. It should be a truism that ( i ) c e n t r a l place system, or ( i i ) finally, (iii)  s i m i l a r growth in a d i f f e r e n t  d i s s i m i l a r growth in the same system, o r ,  d i s s i m i l a r growth i n a d i f f e r e n t system each represents a  v a r i e t y of circumstances i n which adoptive divergences could be remarkably unlike those indicated in Tables 4.3 and 4 . 4 .  But t h i s i s a whole new  Pandora's box - one, unfortunately, which must be opened i f proper empirical q u a l i f i c a t i o n of the h e a r i n g - t e l l i n g model i s to be undertaken - and i t was decided by the author not to extend the d i s c u s s i o n at the present time.  4.3  Technology The impact of a technological change - upon e i t h e r the  attributes  of central place structure or process - would be i n t i m a t e l y t i e d to the very nature of that change.  Perhaps two r e l a t i v e l y simple examples, e x h i b i t -  ing how long run s h i f t s in the general state of technology may have cont r a d i c t o r y e f f e c t s on the structure of the space-economy, can help elucidate this  point.  F i r s t l y , a s u b s t a n t i a l c e t e r i s paribus improvement i n a g r i c u l t u r a l p r o d u c t i v i t y would be expected to induce a large number of rural households - and, i n d i r e c t l y , many households in the lower order central places as well - to migrate i n t o the very l a r g e s t central places in the system. On the other hand, the growth of the more intermediate sized places might be d i s p r o p o r t i o n a t e l y enhanced by improvements i n communications technology which would allow very s p e c i a l i z e d s e r v i c e s (for i n s t a n c e , s c i e n t i f i c  126  research or higher education) to " f i l t e r down" from the very l a r g e s t centers ( t h i s i s a type of urban d e c e n t r a l i z a t i o n ) . This section opens by considering the e f f e c t s of instantaneous changes in the service m u l t i p l i e r s upon the e q u i l i b r i u m states of the 14 endogenous employment and population s e c t o r s .  Specific attention  given to the issue of breaking down these m u l t i p l i e r s  into their  is  individual  components so that i t becomes possible to d i s t i n g u i s h between the impact of: (i)  (ii)  a s h i f t in the aggregate m u l t i p l i e r i t s e l f (which might be induced by t h e a d d i t i o n o f a new. a c t i v i t y t o an e x i s t i n g bundle ( p o s i t i v e s h i f t ) o r by an improvement i n t h e p r o d u c t i v e e f f i c i e n c y of the given a c t i v i t i e s i n that same bundle ( n e g a t i v e s h i f t ) ) ; and a s h i f t i n a p a r t i c u l a r component ( i . e . an individual a c t i v i t y ) of that m u l t i p l i e r ( n e g a t i ve sh i f t ) .  This d i s t i n c t i o n i s useful because i t  leads to the e x p l i c i t a n a l y s i s -  a l b e i t in a somewhat naive fashion - of supply and demand i n the central place argument. Subsequently the longer run p o s s i b i l i t i e s of s t r u c t u r a l are considered.  variation  The above cases are mentioned again in t h i s new context  and a d d i t i o n a l concern i s d i r e c t e d to increases i n a g r i c u l t u r a l and transportation  production  improvements - both of which would be expected to  accompany technological progress in a space-economy. F i n a l l y , the discussion b r i e f l y turns to analyse the s i g n i f i c a n c e of these induced s t r u c t u r a l  changes in d i s t u r b i n g or a l t e r i n g  schemes in the central place s e t t i n g .  adoptive  127  (i)  Structure  The impact of a s h i f t in any element of the set is f i r s t resolved.  {k.  | 1 < j < m}  This s h i f t should be construed as being aggregate in  nature, since i t could be induced by a s h i f t i n any of the s p e c i f i c a c t i v i t i e s which comprise the  j th bundle.  There are two approaches to t h i s problem.  The f i r s t and more  encompassing perspective i s to consider "shocking", the employment sectors e  mi  ( I f i f m )  i n (4.1) by any s e r v i c e m u l t i p l i e r  k..  The formulations  which describe t h i s e f f e c t are r e l a t i v e l y complex but they do, conveniently enough, c o l l a p s e somewhat when the o v e r a l l e f f e c t on  e^  i s taken up.  The a l t e r n a t i v e approach i s to u t i l i z e (4.24) immediately although now, of course, only the o v e r a l l e f f e c t on arguments are presented in the usual  e^  may be a s c e r t a i n e d .  Since the  m th l e v e l format, the reader may  wish to r e f e r to Appendix F f o r a comparison of the two methods i n the context of a second l e v e l central p l a c e . Turning now to (4.1) and assuming d i f f e r e n t i a b i l i t y the m u l t i p l i e r s B T e •, e .  k.  (1 5 j 5 m)  of each of  and each of the employment sectors  (1 5 i < m), i t follows t h a t :  128  (A ) * V  ,  9 e B  -  m i  3k.. J  +A  f  u m i K  l  ^  . B m de i V ma . 3k.  m k- I b-l  +  B  k m b  1  -V  a=j  R  e  * V  3e  mi  3k. J  (A ) m>  [  m 3e^  f  +A 3k. m .1 011  for 1  D  m  m  n  A  (4. for . B m oe^ y  A fk m i  +  m a  I.  K  ak.J  m y B . i mb D  k K  i  e e  for j > i  which means, r e c a l l i n g (4.10) and assuming d i f f e r e n t i a b i l i t y  of  e , that: m T  B m 3e m 'ma I e° + A I m 3k. m L. i =- l mi  T de m dk. J  m  L  ( m J' A  +  T e  ni  y -ma^ . 3k. J  a=J  A m This i n d i c a t e s , f o r i n s t a n c e , that: de m dk. J T  for 1 < j < m  (4.37)  j < m< n < M  (4.35  15  de n dk. J  T  <  In a d d i t i o n , combining (4.26) and (4.29) with the assumption that each c e n t r a l place population  p  m  is d i f f e r e n t i a t e ,  suggests t h a t :  129  dp  3e . T  m  m  HkT-  I  J  ( 1  (4.39)  0.wr  +  d  '=1  j  so that: dp  dp  dk.  dk.  J  j < m < n .< M  J  (4.40)  On the other hand, d i r e c t d i f f e r e n t i a t i o n of (4.24) makes the impact of a s h i f t in  ki  upon  e : m T  r  de  T  dk,  e  1 0  + k K (A +A ) 2  (Ax)  2  1  (A )  1  2  x  2  (A )  +  k K K (A +A,) 3  1  2  (A )  2  2  2  (A )  2  2  +  2  3  m-l k  m  X  K  i l  A  I = I  ^  ( m-l) A  m-l'  ~  +  (V  2  A  "m  (4.41) !  The same operation may be performed f o r the remaining  m-l  multipliers  i n order to reach the general r e s u l t t h a t : T de 'm i dki = e i { (AJ* 0  T de 'm dk. = e -4 [ 0  n  KC  c-l  e  0  1  V (A.)  ^ 2  c=l  K  ° J  d  c=l v  m  k (A + A ) i-1 J—i_i !_ K 2 / . x 2 , ( A . , , ) " ( A ^ c=l  1  N  (A ) •  n  Ji dk  i=2 ( A . ^ )  2  +  j-l C  f o r 2 < j < m-l  m-l 6  . ? MVi  Kc  nr  (4.42)  130  This i n d i c a t e s , for i n s t a n c e , that:  a-1  n  K  c=iJ i=a+l  (A^)  (A.)'  c=l'  (A )  C  de ^ de m> m dk, < dk. a b T  2  b  T  for 2 < a < b < m  As i n the previous s e c t i o n , d i f f e r e n t i a t i o n  (4.43)  of (4.24) does not  provide a statement analagous to (4.39) except, of course, for the  trivial  case when d = constant = d^ (1 < i < m). Suppose now that each of the m u l t i p l i e r s  k^  i s resolvable i n t o  a set of smaller m u l t i p l i e r s : these l a t t e r c o e f f i c i e n t s would represent the i n d i v i d u a l goods and services which make up the  j th bundle.  In par-  t i c u l a r , consider the l i n e a r case where:  ki = k n +  ki2  = k i +  k 2  k2  2  2  + ••• + k +  k. = k., + k . + . . J Jl j2 0  k m  + k  + k  Izi 2z  2  jz.  = k + k + ' « « + k ml m2 mzm 1  0  (4.44)  If the reader wishes to think of these m u l t i p l i e r s as being ordered so as to r e f l e c t increasing threshold requirements ( i . e . the threshold requirement  131  for function function  f\  f . , with associated m u l t i p l i e r Ja  b  if  a < b ) , then the set  {kj  z  k. , i s l e s s than that f o r ja | 1 < j < m}  sent the h i e r a r c h i a l marginal goods i n the system.  It  would repre-  should immediately  f o l l o w , because of the l i n e a r r e l a t i o n s h i p s hypothesized in ( 4 . 4 4 ) , t h a t : mi mi W~ WT jn j 3e  1 < 1, j < m , 1 < n < z.  3e  =  (4-45)  and that:  d e  I  d  jn  if differentiability  of  k^  n  for  d  e  k  I  1 <j <m  j  1 < n < z  1 < j < m and  1 < n < z^.  i s assumed.  (4.44) becomes e s p e c i a l l y u s e f u l , however, when a d i s t i n c t i o n can be drawn between the demand per employee (household) f o r each i n d i v i d u a l function and the various outputs per employee (household) which c h a r a c t e r i z e production of the d i f f e r e n t  i n d i v i d u a l goods and s e r v i c e s .  i e n t , then, to define the m u l t i p l i e r  k j  where:  (i)  x  i n n  k^  = ^ S  n  It  as:  (4.47)  jn  i s t h e demand p e r employee (household) o f  > n  function  f ^ ; implicit to this  argument i s t h a t  x.  n  particular  i s identical  for a l l  employees ( h o u s e h o l d s ) i n t h e system; and (ii)  i s conven-  s^. i s t h e o u t p u t p e r employee o f t h o s e n  employees who a r e engaged i n p r o v i d i n g  f  .  132  The i m p l i c a t i o n of ( 4 . 4 7 ) , a f t e r assuming t h a t  x^  n  d i f f e r e n t i a b l e and that demand i s held constant ( i . e  d k  in  '".in  d s  jn  (s. )  .  and dx^  are both n  =0),  i s then  - ,jn k  2  s  n  (4.  jn  so that in the aggregate case (see ( 4 . 3 7 ) ) , f o r instance: de d e dk. _m _ m jn ds . dk- d s . jn jn jn T  T  . B K• ( m e JH_ l + y < s. A m . k jn m a=j JIV T T  e  m  (4.  L  n  This l a s t statement i n d i c a t e s , f o r an  m th l e v e l c e n t e r , the extent of  the rate of decline in t o t a l employment  e\ which would be p r e c i p i t a t e d m by a " s m a l l " c e t e r i s paribus improvement in the productive e f f i c i e n c y of function  f^.  This formulation has only partial  m e r i t , however, since  i t does not account f o r any s u b s t i t u t i o n s in demand amongst the various goods - more s o p h i s t i c a t e d approaches in the future w i l l doubtless eliminate t h i s problem though. The author f e e l s that the impact of r u r a l p r o d u c t i v i t y changes could be u s e f u l l y analysed w i t h i n a s i m i l a r framework.  In t h i s case  it  would simply become necessary to d i s t i n g u i s h i n d i v i d u a l m u l t i p l i e r s - one f o r each of the various a g r i c u l t u r a l a c t i v i t i e s in the system - which, together, would comprise the rural s e r v i c e m u l t i p l i e r  k  0  (see the second  chapter for some comments here). In c l o s i n g the comparative s t a t i c s portion of t h i s section should be pointed out that a " s m a l l " change in the e x i s t i n g  it  transportation  133  technology might be construed much as an income subsidy to the consumer and,  t h e r e f o r e , a n a l y s i s of that case i s included in the upcoming section  which i s concerned w i t h s h i f t s i n per c a p i t a income. It  i s somewhat speculative to consider these technological  changes i n a c e t e r i s  paribus fashion over the long run.  The author  r e f r a i n s from making extremely s p e c i f i c statements, in a cause and e f f e c t c o n t e x t , about the i n t e r r e l a t i o n s h i p s of technological changes ( i n c l u d i n g t e c h n i c a l , marketing, and organizational innovations) and v a r i a t i o n s  in  the central place structure because: (i)  (ii)  most i m p o r t a n t i n n o v a t i o n s tend t o c l u s t e r t o g e t h e r i n space and t i m e ( i . e . they a r e r a r e l y independent phenomena); and t h e s e i n n o v a t i o n s tend t o be so enmeshed i n the whole p r o c e s s o f r e g i o n a l economic growth t h a t i t i s d i f f i c u l t t o d i s c e r n what m a t e r i a l improvements a r e a t t r i b u t a b l e t o a t e c h n o l o g i c a l change per s e .  However, i t may not be too hazardous to comment in a very general fashion on how systemic a t t r i b u t e s might vary with c e r t a i n technological changes. H o p e f u l l y , the b r i e f upcoming d i s c u s s i o n seems a b i t more p l a u s i b l e than the argument given by Nourse (1968:215-218). Urban p r o d u c t i v i t y may be enhanced by new marketing schemes, c a p i t a l - i n t e n s i v e techniques, agglomeration economies, and the l i k e .  In  any c a s e , as Parr and Denike (1970) have i l l u s t r a t e d to some l e n g t h , the cumulative e f f e c t of such changes in technology i s a reinforcement of f u n c t i o n a l c e n t r a l i z a t i o n i n the i n t e r m e d i a t e - t o - l a r g e i n central place systems.  s i z e d communities  In other words, the a b i l i t y of large firms  employ economies of s c a l e , new a d v e r t i s i n g schemes, e t c . l e a d s , in the  to  134  most p a r t , to a t r a n s f e r of goods and s e r v i c e s from lower to higher l e v e l s of the h i e r a r c h y . Rural p r o d u c t i v i t y , as pointed out near the beginning of t h i s very s e c t i o n , may be enhanced by increased farm mechanization, new f e r t i l i z e r s , institutional  reform, and the l i k e .  C e t e r i s p a r i b u s , the improve-  ment of a g r i c u l t u r a l output per u n i t of labour input would be expected to induce a stream of r u r a l - t o - u r b a n m i g r a t i o n . coefficient  R° = A^  As a consequence, the  (see Appendix E ) , representing the r a t i o of rural  to  t o t a l employment (population), would tend to d e c l i n e in the long run. A s i g n i f i c a n t transportation improvement - say, perhaps a new mode of t r a v e l or an upgrading, of some e x i s t i n g mode(s) - which i s systemwide in extent would allow benefits to accrue to households s i n c e : (i)  (ii)  a g r i c u l t u r a l p r o d u c t s c o u l d be moved i n a more e f f i c i e n t manner t o c e n t r a l p l a c e s (Beckmann (1968:77-83) g i v e s a b r i e f b u t i n t e r e s t i n g a c c o u n t o f such f l o w s ) ; and d e l i v e r e d p r i c e s o f f i n i s h e d goods ( i n c l u d i n g p r o c e s s e d a g r i c u l t u r a l commodities) would be reduced as we I I J ^  It f o l l o w s , then, that a transportation  improvement would be  conducive to functional c e n t r a l i z a t i o n : that i s , a strengthening of the r e l a t i v e importance of the l a r g e r centers i n the system.  F i r s t of a l l ,  the enhancement of customer m o b i l i t y would permit households to trade  off  some of t h e i r shorter and more frequent t r i p s (to acquire convenience goods) f o r more multi-purpose t r i p s ( e . g . consider the symptomatic d e c l i n e of r u r a l s e r v i c e centers on the Canadian P r a i r i e s  and i n the American Midwest).  Secondly, these same customers would enjoy an increase i n real  income  (what Long (1971) has termed the transport income e f f e c t ; see Appendix G)  135  which would favor the purchasing of income-elastic goods and s e r v i c e s . Since many low order commodities are of the i n c o m e - i n e l a s t i c type - f o r i n s t a n c e , f o o d , c l o t h i n g , and personal services - i t seems reasonable to generalize t h a t t h i s transport income e f f e c t would stimulate  additional  growth of the l a r g e r places i n a central place system. In g e n e r a l , then, technological improvements are c l e a r l y conducive to a progressive increase i n the urban/rural population r a t i o of the spaceeconomy.  More s i g n i f i c a n t l y , though, they tend to promote growth i n only  the intermediate-sized and l a r g e r places i n the system.  In terms of the  h i e r a r c h i a l model, the higher order s e r v i c e m u l t i p l i e r s  {•••,  k.,»»*,k..} 3  would tend to increase at the expense of the lower order {ki,  # , ,  , k . ,}  **  multipliers  ( r e c a l l that the s e r v i c e m u l t i p l i e r s represent the  activity  r a t i o s f o r the various goods and s e r v i c e s ) . (ii)  Process  Systemic adoption could seemingly be modified i n d i v e r s e ways by technological change.  To begin w i t h , a c e t e r i s paribus  transportation  improvement could have a d e b i l i t a t i n g e f f e c t on the " f r i c t i o n ( i . e . reduce the distance exponent i n the g r a v i t y - p o t e n t i a l t h a t the comments i n ( 3 . 4 : i i i )  of distance"  formula) so  above might be worth reviewing.  On the  other hand, t h i s improvement might be construed as being a surplus of income f o r central place residents so that the population terms in the potential  function would have to be weighted accordingly (see Isard (1960:  507) f o r comments on t h i s weighting procedure).  Unfortunately, at the  present time, theory does not permit any estimations of the  relative  s i g n i f i c a n c e of these two e f f e c t s ; h o p e f u l l y , though, the considerable t a l e n t s being d i r e c t e d to the understanding of the demand f o r travel  will  136  soon o f f e r a s o l u t i o n to t h i s problem (at l e a s t in the somewhat s t r i c t c e n t r a l place c o n t e x t ) . In a d d i t i o n , there are a few other points - perhaps of a more speculative nature again - which should be touched upon.  The s t r u c t u r a l  changes in the space-economy (mentioned above) would be expected, c e t e r i s p a r i b u s , to stimulate i n t e r a c t i o n amongst the l a r g e r centers of the system a t the expense of diminished i n t e r a c t i o n between those l a r g e r places and the various low order centers which they dominate.  The consequences of  t h i s would be t h a t : (i)  (ii)  4.4  a d o p t i o n ( i n a r e l a t i v e sense) would tend t o be t e m p o r a l l y r e s t r i c t e d t o even a g r e a t e r e x t e n t in t h e d i s t a n t low l e v e l p l a c e s i n t h e system; and a d o p t i o n might not even o c c u r (due t o a l o s s o f p o p u l a t i o n ) i n many o f t h e low l e v e l p l a c e s i n the s y s t e m .  Per Capita Income Before examining the s i n g u l a r impact of a change in per c a p i t a  income on the employment (population) and adoptive a t t r i b u t e s of central p l a c e s , a comment on the nature of personal income i s r e q u i r e d . perspective of the central place argument has t r a d i t i o n a l l y  allowed  t h e o r i s t s to avoid making e x p l i c i t statements about the s p a t i a l of income i n the relevant systems:  The p a r t i a l  distribution  t h i s being a p r a c t i c e , i n c i d e n t a l l y ,  which r e f l e c t s the f a c t that l i t t l e a t t e n t i o n has been devoted to the balance of trade (payments) in e i t h e r the C h r i s t a l l e r i a n or Loschian 1g formulations. Parr (1970:228) has pointed out that an i m p l i c i t postulate of the h i e r a r c h i a l model i s :  137  . . . t h a t p a t t e r n s o f consumption w i l l be i d e n t i c a l between t h e r u r a l a r e a s and t h e d i f f e r e n t l e v e l s o f center.  In f a c t , Parr might have generalized t h i s statement even f u r t h e r . It can be argued that the relevant postulate i s that a l l households i n the system would earn the same' "apparent" real income - in other words, that each household would consume an i d e n t i c a l amount of each 19 commodity bundle.  T h i s , of course, i s the same assumption which was  u t i l i z e d i n the previous section (on technology) of t h i s For present purposes i t  chapter.  i s not necessary to r e l a t e real income  and money income f o r a l l central place l o c a t i o n s - though that might prove a useful exercise - but i n Appendix 6 the author demonstrates those cond i t i o n s which are necessary and s u f f i c i e n t for real income to remain constant  in the neighborhood of any given  location.  The assumption of constant real income i s appropriate simply because e x p l i c i t supply and demand conditions (which would ensure the s p a t i a l e q u i l i b r i u m of a l l households) for the central place scheme have not as yet been s p e c i f i e d - in t h i s absence, having real income constant i s at l e a s t s u f f i c i e n t for r e t a i n i n g the s t r i c t h i e r a r c h i a l which c h a r a c t e r i z e that body of  (i)  properties  theory.  Structure  To begin w i t h , r e c a l l (4.47) and postulate t h a t , with " s m a l l " changes in the m u l t i p l i e r change i n output  s.  k.  (i.e.  and household demand ds  i n  = 0).  x . , there i s no  Then i t f o l l o w s t h a t :  138  d k  in  1  k  in  jn  jn  x  jn  1  -  J  -  m  1 < n < x. J  Given any  m th  (1 < m < M)  l e v e l place s t i p u a t e next that the  per c a p i t a income of a l l households in that place i s Y. x.  i s taken to be a continuously d i f f e r e n t i a t e dk.  then the r e l a t i v e e f f e c t  (real)  Now i f demand  function of income Y,  of a change i n the m u l t i p l i e r  k.  n  due to  a s h i f t in every household's income Y, becomes: dk. jn dY  dk. jn dx.  =  dx. ,m dY  k. dx. jn jn x. dY jn  =  (4.51)  In p a r t i c u l a r note that t h i s e n t a i l s t h a t : Y dk e  where  e.  jn'  multiplier  and  v Y  k^  k . ,Y jn' e  n  jn'  Y dx.  j n  k. dY jn  =  x . dY jn  n =  e  ( ' 4  Y  and household demand  Xj  n  with respect to income  Y.  More s i g n i f i c a n t l y , though, i t should be pointed out that the de -jjy- of a s h i f t in income upon the t o t a l employment e^ of an T  impact  x . ,Y jn'  a r e , r e s p e c t i v e l y , the e l a s t i c i t i e s of the  v  x  =  m th l e v e l place i s (from (4.37) and ( 4 . 4 6 ) ) :  5 2  )  139  m dY  (4  This asserts (again, in a p a r t i a l sense o n l y , because there i s no s u b s t i t u t i o n amongst productive inputs) j u s t how great the rate of increase (decline) in the employment body  would be, given an income-induced  change in demand f o r the income-elastic ( i n c o m e - i n e l a s t i c ) good or function  f^.  One property of t h i s i n c r e a s e (decline) to note i s i t s depen-  dence, in an inverse manner, on the production c o e f f i c i e n t  s.  n  (where  Consider now the case of a long run increase i n per c a p i t a income.  This increase must r e f l e c t an outward s h i f t in the space-economy's  p r o d u c t i o n - p o s s i b i l i t y curve - a s h i f t which may be brought about by a v a r i e t y of f a c t o r s , a c t i n g alone or in combination with one another. Improvements i n technology and o r g a n i z a t i o n , better s k i l l s in labour, a p r o c l i v i t y on the part of firms to s u b s t i t u t e c a p i t a l f o r other i n p u t s , population growth, and the emergence of external economies would be numbered amongst the most important of such f a c t o r s .  In other words, a  long run expansion in personal income cannot be postulated to be some s o r t of independent phenomenon - as i t i s viewed in Nourse (1968:212-214), f o r instance - but must be seen as a symptom of economic growth. In the context of the h i e r a r c h i a l model there i s one s a l i e n t point to consider (and perhaps the reader r e c a l l s  t h i s from the previous  140  section).  As a per c a p i t a income increase took p l a c e , s h i f t s would  n e c e s s a r i l y follow in the elements of the set  {k  m  | 1 < m < M} due to  differences in the r e l a t i v e (income) e l a s t i c i t i e s of demand f o r the i n d i v i d u a l goods and services (which comprise the system's bundles). That i s , as a consequence of s h i f t s i n demand alone, changes would be induced in the numbers of employees (persons) who were involved i n the p r o v i s i o n of f i r s t order, second order, e t c . a c t i v i t i e s throughout e n t i r e central  place system.  the elements of the set  {k^}  the  It might prove u s e f u l , then, to construe as being appropriate demand ratios  (for  the  20 various bundles) which c o u l d , of course, be redefined over time.  There-  f o r e , the impact of a long run change i n personal income could be conc e p t u a l i z e d much l i k e a long run change in population ( t h i s case was 21 analysed e a r l i e r and i l l u s t r a t e d in Tables 4.1 and 4 . 2 ) . Besides, while commodity bundles would be comprised of a mixture of i n c o m e - i n e l a s t i c ( i n f e r i o r ,  necessary) and income-elastic (normal,  luxury) goods and s e r v i c e s , i t could be expected t h a t these bundles would be weighted i n a p r o g r e s s i v e l y income-elastic fashion as t h e i r orders 22 increased. The r e s u l t i n g i m p l i c a t i o n should be apparent.  Regional growth  c o n t r i b u t i n g to income increases would tend to be l o c a l i z e d in the higher l e v e l places - whose primary purpose i t i s to o f f e r the more s p e c i a l i z e d , c a p i t a l i n t e n s i v e , and income-elastic goods and services - so that i n creased functional c e n t r a l i z a t i o n would occur in the c i t y system. This i n t e r p r e t a t i o n Nourse.  d i f f e r s considerably from that suggested by  While he recognized the importance of considering i n c o m e - e l a s t i c i t y ,  he was e i t h e r r e l u c t a n t or unable to integrate i t s consequences into his  141  argument.  In any case, his hypothesis that a system would e x h i b i t  hier-  a r c h i a l truncation due to c e t e r i s paribus [ s i c ] income changes (over the long run) i s more than a b i t d i f f i c u l t to accept ( e s p e c i a l l y when i t  is  recognized, as argued above, that such increases can only be i n d i c a t i v e  .  of long run changes in the performance of the space-economy).  (ii)  Process  The influence of a per c a p i t a income increase upon the  attributes  of a d i f f u s i o n a r y process may be conjectured to be s i m i l a r to the case given f o r a population i n c r e a s e .  By weighting (see Isard (1960:507))  the  populations i n the p o t e n t i a l function according to such income i n c r e a s e s , i t becomes p o s s i b l e to represent - in an aggregate sense - such notions as consumers enhancing t h e i r demand for travel ductive innovators perceiving p r o f i t a b i l i t y  (or communication), pro-  i n a d i f f e r e n t fashion (due  to market increases i n purchasing power), and the l i k e . More s p e c i f i c a l l y , Tables 4.3 and 4.4 might j u s t as well represent standardized one percent and two percent growths i n personal income (population held constant) as increases i n p o p u l a t i o n . Of course, there i s again the concomitant problem of d e l i n e a t i n g how such income s h i f t s might a f f e c t the value of the distance exponent in the p o t e n t i a l function but an answer here, unfortunately, must await the synthesis of theory r e l a t e d to the demand f o r travel to central place theory per se (see footnote 2 of the t h i r d c h a p t e r ) .  142  4.5  Concluding Remarks In t h i s chapter the s t a t i c modelling argument o u t l i n e d e a r l i e r  i n the t h e s i s was r e l a t e d to parametric change.  Instantaneous  shifts,  as well as long run transformations, i n central place a t t r i b u t e s  (employ-  ment, p o p u l a t i o n , e t c . ) were seen to r e s u l t from system-wide changes in p o p u l a t i o n , technology, and per c a p i t a income. Some new t h e o r e t i c a l f i n d i n g s were engendered by introducing a type of comparative s t a t i c s a n a l y s i s into the d i s c o u r s e ; unfortunately, on the long run side the argument tended more to r e a l i z e the paucity of s o l i d t h e o r e t i c a l ground (upon which the h i e r a r c h i a l model r e s t s ) rather than make substantive a n a l y t i c a l advances. R e l a t i v e l y l i t t l e a t t e n t i o n was given to the e f f e c t s of such changes on the d i f f u s i o n a r y patterns of central place systems - research i n t h i s area might, in f a c t , be stymied u n t i l f u r t h e r a r t i c u l a t i o n of the central place and g r a v i t y - p o t e n t i a l  theories can be fashioned.  Perhaps, however, one point d i d c r y s t a l l i z e from the various perspectives.  It seems that changes in central place s t r u c t u r e (as  r e f l e c t e d i n the s i z e d i s t r i b u t i o n of a system's communities) are generally dependent upon the balance between population growth (or d e c l i n e ) and economic growth (or d e c l i n e ) .  If population increases were to  outstrip  p r o d u c t i v i t y in a dramatic way, f o r i n s t a n c e , then a long run trend towards systemic d i v i s i o n ( i n t o r e l a t i v e l y independent subsystems) would be expected to occur.  On the other hand, i f innovations (or, perhaps, regional com-  parative advantage) proved p e r s i s t e n t l y successful in enhancing product i v i t y , then systemic properties  would be r e i n f o r c e d but i n such a way  that favored the population growth (decline) of the intermediate-sized and l a r g e r (smaller) places in the system.  FOOTNOTES TO CHAPTER 4  'This statement applies in p a r t i c u l a r to those observers l i k e Maruyama (1963), Pred (1965, 1966), Lampard (1968), and Thompson (1968) who have advocated a cybernetics framework for d i s c u s s i n g growth and transformation i n an inter-urban context. In a l l honesty, though, i t should be stressed that those arguments are not t o t a l l y confined to the domain which central place theory purports to cover.  ''In the case of an instantaneous s h i f t the researcher would seem to be on r e l a t i v e l y safe grounds for s t i p u l a t i n g whether or not a s p e c i f i c systemic change e x h i b i t e d ceteris paribus c h a r a c t e r i s t i c s . When the argument moves on to the long run case, however, i t becomes apparent that the e f f e c t s of a l l parameters cannot be analysed holding "other things the same." Misconceptions of t h i s s o r t have occurred in the economic l i t e r a t u r e i n the past (see Friedman (1949) and h i s treatment of the Marshallian demand curve) so i t might prove a useful exercise to remain aware of the nature and l i m i t a t i o n s of the c e t e r i s paribus p e r s p e c t i v e .  The e q u i l i b r i u m states for the h i e r a r c h i a l model are given i n equations (2.18) and (2.33). In comparative s t a t i c s a n a l y s i s , a t t e n t i o n i s being devoted to the direction of any change from such an equilibrium state. With the paucity of a n a l y t i c a l statements in central place theory which concern e q u i l i b r i u m per se and the genuine d i f f i c u l t i e s which p e r s i s t in v i s u a l i z i n g e q u i l i b r i u m in real world systems, perhaps i t i s encouraging to r e c a l l Isard ( 1 9 5 6 : i x - x ) : J  Despite the consequent failure to attain equilibrium in the secular sense, there is s t i l l value in equilibrium analysis. It is thought pertinent and worthwhile by some who conceive of the socio-economic system as a body tending toward a moving equilibrium and by others who find in equilibrium analysis categories of reference with which the extent of disequilibrium can be measured.  143  144  Most important, equilibrium analysis is valuable it enables one to grasp better the laws of change the workings of a system.  because and  "'There are two points of i n t e r e s t here. F i r s t l y , the best demonstration of how systemic e q u i l i b r i u m may be a l t e r e d ( i n a s p e c i f i c central place) i s found in Parr and Denike (1970) but those authors were not so concerned with the system-wide i m p l i c a t i o n s of such a s h i f t . Secondly, long run change was examined by Nourse but i t remains d i f f i c u l t to see how such changes could r e a l i s t i c a l l y be fashioned by the individual v a r i a t i o n of d i f f e r e n t parameters (see footnote 3 above).  ^An "innovation c l u s t e r , " as suggested by Berry and Horton (1970:87), would be an extreme example of an a c t i v e item; such an item might well induce s u f f i c i e n t systemic change so as to r e d i r e c t i t s own expected adoptive pattern. The temporal r e s o l u t i o n scale used to examine the adoption of such an item would, of course, be much greater than the appropriate scale used f o r the adoption of a l e s s s i g n i f i c a n t item.  Perhaps a r e l a t i v e l y simple example would have considerable i l l u s t r a t i v e m e r i t . The reader might wish to r e f e r to Appendix D where equation (4.4) i s outlined for a second l e v e l p l a c e .  For  The text includes only the simple example where 1 < a < b < j < m. the case of 1 < j < a < b < m, note the f o l l o w i n g : B 3e 'mc k. - k I B* B c=j 3e . ^ 3e , mj > mb A_ < „_B* 3e: m mj m  L  D  3e  3e  L  ma B* 3e 'mj  ^ 3e , ma > mb . B* < _ B* 3e . 3e . mj mj  Each m u l t i p l i e r  k.  vJ  T  (1 < j < m)  T  i s delineated into smaller  component m u l t i p l i e r s f o r the purposes of comparative s t a t i c s a n a l y s i s in Section 4.3 of t h i s chapter (where i n t e r e s t focuses on the impact of techn o l o g i c a l change on the central place s t r u c t u r e ) . In the case of the long  145  run, however, e f f o r t s must f i r s t be taken to s p e c i f y the threshold r e q u i r e ments of the i n d i v i d u a l functions (goods and s e r v i c e s ) before d i s c r e t e s h i f t s in the aggregate m u l t i p l i e r s may be a s c e r t a i n e d .  The  population sectors  ri  and  {p  m  | 1 < m < 4}  s p e c i f y the  total populations r e s i d i n g in r u r a l areas, f i r s t l e v e l p l a c e s , second l e v e l p l a c e s , t h i r d l e v e l p l a c e s , and the fourth l e v e l place r e s p e c t i v e l y .  N o t e that .0000, -.0167}. 10  {k  m  [t ] 2  - k [ti] m  1  | 1 < m < 4} = {.0167, .0000, -  It i s only f a i r to note that Nourse was dealing with the e a r l y Beckmann (1958) model which, even when properly reformulated, does not e l i c i t a set of r e l a t e d s e r v i c e m u l t i p l i e r s .  ""Although t h i s argument may appear to c o n f l i c t with the i n c r e mental perspective (used f o r the example of one percent population growth; see Tables 4.1 and 4.2) upon cursory examination, t h i s i s not r e a l l y the case. Here i t i s presently assumed that the m u l t i p l i e r s remain constant throughout systemic growth and r e d i s t r i b u t i o n because, f o r each emerging c e n t r a l p l a c e , the population served i s proportionately less as w e l l . In the previous incremental argument, however, r e d i s t r i b u t i o n led to increases or decreases in the m u l t i p l i e r s due to r e l a t i v e shifts - from the "ground up" - of the populations served at each l e v e l . 13  The cases of "no growth" are those already shown i n Tables 3.10 and 3.12 r e s p e c t i v e l y .  ^Beckmann and Schramm (1972) a l s o d e a l t with the e f f e c t s of changes i n the service m u l t i p l i e r s upon systemic properties although there impacts were not r e l a t e d to the central place populations per se; r a t h e r , i n t e r e s t was given to the e f f e c t s of technical change upon ( i ) the c i t y / market area population r a t i o s and ( i i ) the population d i f f e r e n c e s between centers on adjacent h i e r a r c h i a l l e v e l s .  The i n e q u a l i t i e s expressed in ( 4 . 3 8 ) , ( 4 . 4 0 ) , and (4.43) depend upon a p o s i t i v e incremental increase i n the s e r v i c e m u l t i p l i e r kj (i.e. dk. > 0) - a condition which would r e s u l t from a technical innovation  146  adding a new function or a c t i v i t y to the j th level bundle (see footnote 16 below). These i n e q u a l i t i e s would n e c e s s a r i l y be reversed i f the techn i c a l innovation were considered to occur amongst the e x i s t i n g stock of j th l e v e l a c t i v i t i e s ( i . e . representing c a p i t a l deepening or the subs t i t u t i o n of c a p i t a l for labor in the production of a s p e c i f i c good or s e r v i c e in the j th bundle). This l a t t e r case i s taken up i n more d e t a i l i n equations (4.44) through (4.49) where the appropriate s e r v i c e m u l t i p l i e r s are resolved i n t o t h e i r more elementary ( m u l t i p l i e r ) u n i t s .  The function function {f^}  f  or function set  (1 < n < z ) {f  i s a member of the composite  | 1 < m < M}  which was introduced i n  the second chapter.  I t would be e s p e c i a l l y useful to have, in the words of Henderson (1972:437), " . . . a set of prices and p o s s i b l y a set of demand and supply conditions that would y i e l d balanced trade. . ." amongst the r u r a l and urban u n i t s .  See von Boventer (1963) and Henderson (1972).  ' T h i s point deserves some added c o n s i d e r a t i o n . F i r s t l y , the assumption of constant real income may be s u i t a b l y relaxed without destroying the h i e r a r c h i a l properties of the model. It could be p o s t u l a t e d , f o r i n s t a n c e , that there are z q u a l i t a t i v e categories of labor ( u n s k i l l e d , p r o f e s s i o n a l , e t c . ) , each i n a state of e q u i l i b r i u m , and that the employment (population) body engaged in providing a c e r t a i n bundle may be weighted according to these z c a t e g o r i e s . This would simply create a spaceeconomy with z d i f f e r e n t l e v e l s of real income. Secondly, the usual u t i l i t y and constant money income argument of t r a d i t i o n a l economics may be extended from the intra-urban scale to the inter-urban scale by a multicenter argument f i r s t suggested by Papageorgiou (1971) and then f u r t h e r embellished by Papageorgiou and C a s e t t i (1971) and Papageorgiou (1973) (see Appendix G). But here a s i g n i f i c a n t t r a d e - o f f must be considered: i s i t worth f o r f e i t i n g geometrical elegance (and i t s very obvious conceptual advantages) f o r a somewhat t i g h t e r a n a l y t i c a l package? At issue here i s the reason why real income i s most s u i t a b l e v i s - c l - v i s the l a t t i c e approach ( i . e . where places are equally spaced from t h e i r surrounding neighbors). In that c l a s s i c a l approach i t i s assumed that intra-urban a c t i v i t i e s are dimensionless ( i . e . there i s a p e r f e c t s c a l e dichotomy between the two r e s o l u t i o n l e v e l s mentioned above); hence, the consumption of r e s i d e n t i a l land (and i t s related component in the household's u t i l i t y function) must be precluded f o r the urban case. N a t u r a l l y , t h i s makes i t expedient to suppose that a l l households throughout the system would consume equal amounts of the "other" goods and s e r v i c e s . J  147  In order to accommodate constant money income i n t o the s t r i c t h i e r a r c h i a l format, a lengthy l i s t of quite u n r e a l i s t i c assumptions - as i n von Boventer (1963) - would be required ( e l s e households would migrate to the cheaper goods sources - the l a r g e r c e n t e r s ) . The s p a t i a l d i s t r i bution of money income, given constant real income, may be e s t a b l i s h e d i n the neighborhood of a s p e c i f i c l o c a t i o n as shown i n Appendix G. 20  Tinbergen (1967) has used demand r a t i o s to o u t l i n e the prop e r t i e s of a s i m i l a r , but not i d e n t i c a l , inter-urban system. 21  S i m i l a r i t i e s at l e a s t e x i s t f o r the case of an incremental long run change.  " S e e Parr (1970:235) for some relevant comments. The problem i s s i m i l a r (and i n t r i c a t e l y related) to the dilemma which faced Thompson (1965:146) when he s t a t e d , while commenting on community economic s t a b i l i t y , that: . . . deduction fails us here. For every incomeelastic import (e.g., automobiles and tourism) one can cite, one can come up with an income-inelastic import (e.g., food, cigarettes, and fuel). And for every income-elastic local good or service (e.g., entertainment, restaurant dining, and home repairs), one can cite a likely income-inelastic one (e.g., rent, u t i l i t i e s , and local transportation).  Chapter 5  A MODIFICATION OF THE STRICT HIERARCHIAL FORMAT  5.1  Introduction In the previous chapter a t t e n t i o n was given to the temporal  v a r i a t i o n of h i e r a r c h i a l a t t r i b u t e s it  in a central place system.  However.,  i s quite possible to extend the m u l t i p l i e r argument i n a nontemporal  sense as w e l l . To the author's knowledge, Parr (1970:228) has been the only observer to speculate on the a d d i t i o n of localized  a c t i v i t i e s of a non-  c e n t r a l place type ( e . g . s p e c i a l i z e d manufacturing, resource e x p l o i t a t i o n , e t c . ) to the s t r i c t h i e r a r c h i a l f o r m a t J  However, t h i s addition was not  given e x p l i c i t c o n s i d e r a t i o n : that i s , the t o t a l population r e l a t e d to such l o c a l i z e d increments was not a l l o c a t e d amongst the d i f f e r e n t centers of the c i t y system.  sized  The remainder of t h i s chapter deals with  the i m p l i c a t i o n s - i n regards to both systemic structure and process of such an a l l o c a t i o n scheme.  5.2  Structure The case here concerns the i n t e g r a t i o n of but one new a c t i v i t y  i n t o the h i e r a r c h i a l format, although numerous such a c t i v i t i e s may be  148  149  concurrently integrated by u t i l i z i n g the same procedure.  The real concern,  however, i s over the s p a t i a l a l l o c a t i o n of the accompanying s e r v i c e s e c t o r , given the s i z e and l o c a t i o n of the employment (population)  body engaged  in this localized a c t i v i t y . To keep the case r e l a t i v e l y simple consider an central place system and an increment  M level  K=3  p of nonnodal a c t i v i t y . Suppose, a, n i n a d d i t i o n , that the rural population i s s u f f i c i e n t l y productive to provide 2 f o r p ^ and the population which services i t . Then, i r r e s p e c t i v e of M  a  where  p  a , IM M  i s located i n the system, an a d d i t i o n a l increment  p  a, n M  such  that: M Pa,M p  a,M  p ^  k  M I k m=l  1 -  i s needed to service  I  [  (and i t s e l f ) .  a  m  ]  m  The t o t a l population  pj  M  to be a l l o c a t e d i s simply:  I,M  P  =  p  a,M  +  p  a,M  a,M M " I m=l P  1  Now i f e n t i r e population  p  g  pj  i s located at the  M  M  (5.2) k  m m  M th level c e n t e r , then the  i s a l l o c a t e d to that center and t h i s  allocation  may be added to the figure generated by the h i e r a r c h i a l model (see Chapter 2).  Suppose t h i s a l l o c a t i o n may be expressed n o t a t i o n a l l y a s :  150  a,M  M a,M  (5.3)  M a,M;M  p  p  whose meaning should become c l e a r as the argument proceeds. If,  however,  p  Q  M  is located at a (one-third) component of  a ( M - l ) s t l e v e l equivalent center, then: a,M  M-l a,M;M-l P  (5.4)  M-l  1 "  I k m  m=l  i s a l l o c a t e d to that equivalent center and:  M-l a,M;M  M-l a,M;M-l  P  P  M mil  i s a l l o c a t e d to the  (5.5)  m  M th l e v e l center where:  a,M ~ M-l a,M " M-"Ta,M;M-l  M-l a,M;M p  (5.6)  These two a l l o c a t i o n s may be added to the i n d i v i d u a l central place popula3  tions generated by the h i e r a r c h i a l model as w e l l . . To take the argument one step more, i f  p  a  ^  i s located at one  of the s i x (M-2)nd l e v e l p l a c e s , then: a,M  M-2 a,M;M-2 P  1 -  (5.7)  M-2  T k  L. m=l  r  151  i s a l l o c a t e d to that (M-2)nd l e v e l p l a c e , and: 'M-1 M-1  M-2 a,M;M-l P  m=l  (5.8)  M-2 a,M;M-2 P  m  i s a l l o c a t e d to each of two surrounding ( M - l ) s t l e v e l components, and: 4  J 1-2 a,M;M K  1  k. M-1 M " I m=1  M-2 a,M;M-2 p  k  m  L m  M M  V  1 -  i s a l l o c a t e d to the  P  . m= I L  M-2 a,M;M-2 P  +  2  (5.9)  M-2 a,M;M-l P  m  M th level place where:  T T T a,M = M-2 a,M = M-2 a,M;M-2 P  P  +  +  2  C j 1 lM-2 a,M;M-lJ P  (5.10)  M-2 a,M;M P  Once again these a l l o c a t i o n s may be appended to the relevant central place populations generated by the h i e r a r c h i a l model. The argument becomes progressively more d i f f i c u l t to express a b s t r a c t l y as  p,a , n i s located i n lower and lower l e v e l s of the hierarchy M  but the l o g i c underlying the above i s simply continued. s a l i e n t difference does a r i s e :  However, one  d i f f e r e n t i a t i o n amongst the various  types  152  (see Appendix B) of central places in the same s i z e c l a s s becomes a possibility.  The reader should r e f e r to Appendix H where two numerical  examples may be found. In the f o l l o w i n g t a b l e , t h i s a l l o c a t i v e scheme i s f o r a l o c a l i z e d increment of 1000 persons ( t h a t i s , a t o t a l change of 4742) i n a f i v e l e v e l K=3 system where 64000, 256000} p o s s i b l e typical  illustrated population  {p > = {1000, 4000, 16000,  according to the h i e r a r c h i a l model.  m  Seven of the twelve  l o c a t i o n s for t h i s nonnodal a c t i v i t y are accounted for -  a number which should be e n t i r e l y s u f f i c i e n t to demonstrate  variability  i n the a l l o c a t i o n of the concomitant s e r v i c e s e c t o r , given the s i z e and l o c a t i o n of the l o c a l i z e d a c t i v i t y . Two a t t r i b u t e s deserve c l o s e r a t t e n t i o n .  F i r s t of a l l , the  a l l o c a t i o n i s r a d i a l l y confined according to the topology of the system. If a r a y , o r i g i n a t i n g at the f i f t h l e v e l c e n t e r , i s swept through the area influenced by the a l l o c a t i o n , then ( i )  the minimum influence i s zero radians  ( l o c a l i z e d a c t i v i t y at a fourth l e v e l component) and ( i i )  the maximum  influence i s IT radians ( l o c a l i z e d a c t i v i t y at a f i r s t l e v e l center nearest the f i f t h l e v e l c e n t e r ) .  Secondly, a l l o c a t i o n i s dependent upon the type  of center possessing the l o c a l i z e d a c t i v i t y and not j u s t that c e n t e r ' s size.  This i s made most evident when the f i f t h l e v e l place i s again con-  s i d e r e d : as s i z e c l a s s i s held constant, but the center of i n t e r e s t  is  moved p r o g r e s s i v e l y more d i s t a n t from the f i f t h l e v e l place ( i . e . type of center v a r i e s ) , then the service population a l l o c a t e d to the f i f t h l e v e l place declines in l i k e f a s h i o n .  This follows from the p r i n c i p l e of m-  domination in the central place argument.  153  Table 5.1 V a r i a t i o n in Central Place Populations as a Consequence of a L o c a l i z e d Activity p  a  , = 1000; K=3, M=5, { k }  = {.3333, .1667,  .1250, .0937, .0704}, n  = 2000  Populations of Centers Type of Center  Equiv. No.  No L o c a l . Activity  Location of L o c a l . A c t i v i t y 5 4 3  5  1  256000  260742  257187  257483  4  2'  64000  64000  67555(1 )* 64000(1)  64592(1) 64000(1)  3  6  16000  16000  16000  18667(1) 16000(1)  2  1  6  4000  4000  4000  4000  2  2  6  4000  4000  4000  4000  2  3  6  4000  4000  4000  4000  1  1  6  1000  1000  1000  1000  1  2  6  1000  1000  1000  1000  3  12  1000  1000  1000  1000  1*  12  1000  1000  1000  1000  6  1000  1000  1000  1000  12  1000  1000  1000  1000  I  l I  5  6  CONTINUED  154  Table 5.1 (Continued) Populations of Centers Type of Center  Equiv. No.  2  1  Location of L o c a l i z e d A c t i v i t y l 1  1  3  5  1  4  2  64100(1) 64000(1)  65062(1) 64000(1)  64008  64509(1) 64000  3  6  16222(2) 16000(4)  16222(2) 16000(4)  16037(1) 16018(2) 16000(3)  16760(1 ) 16037(2) 16000(3)  258198  257236  258820  257602  2  1  6  6000(1) 4000(5)  4000  4167(2) 4000(4)  4167(1) 4000(5)  2  2  6  4000  6000(1) 4000(5)  4000  4167(1) 4000(5)  2  3  6  4000  4000  4000  4000  6  1000  1000  2500(1) 1000(5)  1000  6  1000  1000  1000  1000  3  12  1000  1000  1000  2500(1) 1000(11)  1*  12  1000  1000  1000  1000  6  1000  1000  1000  1000  12  1000  1000  1000  1000  l  1  1 I  l  5  l  6  2  Parenthesized f i g u r e s are the numbers of equivalent centers of the same type but with d i f f e r e n t populations.  155  5.3  Process The asymmetric structure of the modified central place system  would c e r t a i n l y be expected to induce asymmetry in an adoptive process and so i t does.  The nature of t h i s process asymmetry, given the h i e r a r c h i a l  parameters of the underlying system, would be dependent upon the s i z e and absolute l o c a t i o n of the l o c a l i z e d a c t i v i t y  increment.  This represents another advantage of the d i f f u s i o n a r y perspective given i n the t h i r d chapter: a n a l y s i s i s permitted despite the nonexistence of s t r i c t h i e r a r c h i a l p r o p e r t i e s .  However, with t h i s added v a r i a b i l i t y ,  i t becomes a somewhat tedious chore to generate adoptive times - theref o r e , only one example (see Table 5.2) i s given.  The a l l o c a t i o n f o r  this  example may be found i n case ( i ) of Appendix H and i s i l l u s t r a t e d i n the f i f t h column of Table 5 . 1 . By r e f e r r i n g to Table 5 . 2 , i t may be ascertained that s t r u c t u r a l asymmetry has a rather s i g n i f i c a n t e f f e c t on adoptive times.  In f a c t ,  adoptive orderings may be so a l t e r e d that some members of one type adopt b e f o r e , and other members a f t e r , a l l the members of another type.  The  chain of (dominating) r e l a t i o n s with the fourth l e v e l component ( i . e . onet h i r d equivalent center) i s the stimulus of t h i s  alteration.  In c l o s i n g , i t must be stressed that i t would be hazardous to speculate on the i n t e r r e l a t i o n s between h i e r a r c h i a l parameters and the properties of the l o c a l i z e d a c t i v i t y in generating asymmetry - an approach s i m i l a r to that of the t h i r d chapter would have to be u t i l i z e d .  Likewise,  accommodating several l o c a l i z e d a c t i v i t i e s - d i f f i c u l t enough f o r the s t r u c t u r a l argument - would be an extremely d i f f i c u l t  task.  156  Table 5.2 A Comparison of Adoptive Times (Standardized) Between ( i ) Centers in a Closed K=3 Five Level System and ( i i ) One L o c a l i z e d A c t i v i t y )  Centers in a Modified (Due to  Version of that System; b=2, {k } m  .1667, .1250, .0937, .0704}; n  = {.3333,  = 2000, p , = 1000 a, o and Located at 4th Level Component  Type of Center  Adoptive Time in S t r i c t H i e r a r c h i a l System  a  Adoptive Time i n Modified System  5  0.00  0.00  3  1.00  1.00  1.30  1.30  1  1.76  1.76  4  1.91  1.67 1.91  3.53  3.44 (1) 3.53 (5)  5.47  5.48  7.80  7.75 (2) 7.81 (4)  3  10.01  9.58 (2) 10.02 (4)  1"  10.96  10.76 (2) 10.98 (4)  I  12.42  12.28 (2) 12.45 (4)  12.99  11 .97 (2) 13.02 (4)  2 I  2 l I  2  2  3  2  l  1  5  6  (1/3)* (5/3)  Parenthesized figures r e f e r to the numbers of equivalent centers adopting at that time.  157  5.4  Concluding Remarks This argument was s o l e l y intended to support P a r r ' s (1970) con-  tention that a l o c a l i z e d a c t i v i t y could be embraced i n the s t r i c t format.  hierarchial  Extensive asymmetry in structure and process were stressed as  concomitant features of t h i s m o d i f i c a t i o n .  158  FOOTNOTES TO CHAPTER 5  'There are two key supposition here: ( i ) that the population body engaged i n t h i s new a c t i v i t y locates a t an e x i s t i n g c e n t r a l place s i t e and the systemic topology i s retained (perhaps a new technology j u s t becomes a v a i l a b l e f o r e x p l o i t i n g a nonubiquitous r e s o u r c e ) , and ( i i ) that the a d d i t i o n a l populations do not promote the slippage (import s u b s t i t u t i o n ) of goods and s e r v i c e s as discussed i n the previous chapter.  Rural population increases may be considered by using the argument i n Chapter 2 but t h i s only complicates the issue a t hand.  Care must be taken with the ( M - l ) s t centers in terms of a l l o c a t i o n ; when d i s c u s s i n g s t r u c t u r e t h i s i s not c r i t i c a l but i t must be remembered that a component (and not a "whole" equivalent center) receives the a l l o c a t i o n when concern turns to d i f f u s i o n .  Chapter 6  SUMMARY AND CONCLUSIONS  6.1  Introduction In t h i s t h e s i s the d i s c u s s i o n was i n i t i a l l y concerned with estab-  l i s h i n g various a t t r i b u t e s of a general h i e r a r c h i a l model which r e f l e c t s the tenets of C h r i s t a l l e r i a n (central place) thought.  faithfully From there  i n t e r e s t passed to recognizing j u s t how t h i s s t r u c t u r a l argument could provide a useful framework f o r modelling inter-urban d i f f u s i o n a r y processes.  (adoptive)  F i n a l l y , the discourse closed with the author considering the  e f f e c t s of d i f f e r e n t parametric s h i f t s - both continuous and d i s c r e t e i n nature - upon the properties of the s p e c i f i e d models. In other words the essence of the d i s c u s s i o n has been ( i ) the a r t i c u l a t i o n of e x i s t i n g statements and ( i i ) the r e s o l u t i o n of new s t a t e ments about inter-urban structure of c e n t r a l place p r i n c i p l e s ) .  and process  (at l e a s t within the context  While t h i s exercise has been attempted - i n  a more i m p l i c i t fashion i t should be added - i n the past, the author f e e l s that h i s present c o n t r i b u t i o n i s s i g n i f i c a n t l y more general important of a l l , considerably more consistent have been.  and, most  than these past endeavors  The author, i n f a c t , i s hopeful that h i s present work w i l l  159  160  contribute to the a r t i c u l a t i o n , at some future date, of a general s t r u c ture-process theory of inter-urban growth and development. Now the chain of argument has a purposeful design in that the reader f i r s t became acquainted with the conventional s t a t i c cases (models) and then proceeded to see how ( i )  temporal (instantaneous and long run)  change, with s p a t i a l r e g u l a r i t i e s in the space-economy held constant, and (ii)  s p a t i a l change, as r e f l e c t e d i n s t r u c t u r a l asymmetry with time held  constant, could both be embraced w i t h i n these o r i g i n a l models.  As such  i t was necessary f o r the author to c o n t i n u a l l y jump back and f o r t h between the polar concepts of structure (form-function) to e l i c i t various p a r t i c u l a r a s s e r t i o n s .  and process when he wished  It seems most a p p r o p r i a t e , then,  that i n t h i s f i n a l summary chapter the more s i g n i f i c a n t of these a s s e r t i o n s should be grouped together under the two general thematic headings. In a d d i t i o n , t h i s chapter contains a few b r i e f points in passing which confirm the author's strong advocation of a systems methodology f o r t h i s appropriate topic a r e a , that i s - and then closes i t s e l f by suggesting a number of paths along which future research might  6.2  follow.  Structure The second chapter demonstrated that the most general of the  e x i s t i n g h i e r a r c h i a l models of c i t y s i z e could be f u r t h e r extended by incorporating a d i s t i n c t i o n between employees and dependents f o r a l l households i n a central place s e t t i n g .  the  This led to the formulation of a  r e l a t i v e l y simple statement f o r the b a s i c / n o n - b a s i c employment r a t i o of an  m th (1 < m < M) l e v e l place as well as a s l i g h t l y more complicated  expression for the b a s i c / n o n - b a s i c population  r a t i o of that same p l a c e .  161  Both these expressions were seen to be independent of the topology of the central place system which was being considered and the employment r a t i o was also shown to decline for p r o g r e s s i v e l y l a r g e r centers  (this  would hold t r u e , as w e l l , f o r the population r a t i o i f there was l i t t l e or no v a r i a t i o n i n family  size).  I t was then demonstrated ( e x p l i c i t l y  f o r the case of a system  w i t h a constant nesting f a c t o r and no v a r i a t i o n i n family s i z e ) that the urban/rural  population balance of any central place system (or subsystem  f o r that matter) would equal the inverse of the b a s i c / n o n - b a s i c population r a t i o of the l a r g e s t center i n the system (subsystem). It was a l s o shown that the " c l a s s i c a l " argument, i n v o l v i n g recurs i v e formulas to describe the employment and population in the market areas o f centers of d i v e r s e s i z e ( i . e . centers on the various l e v e l s of the h i e r a r c h y ) , could be interpreted i n a somewhat d i f f e r e n t manner so that  it  bore a generic resemblance to the input-output scheme of spaceless economics. The e s s e n t i a l d i f f e r e n c e between the two cases was seen to l i e i n the categories of a n a l y s i s : in the former, linkages are delimited between the various hierarchial  sectors of central places w h i l e , i n the l a t t e r ,  linkages are expressed between the various industrial  sectors of the economy.  A f t e r e s t a b l i s h i n g the e q u i l i b r i u m conditions f o r a c e n t r a l place system, instantaneous s h i f t s in the employment and population c h a r a c t e r i s t i c s of the d i f f e r e n t communities could be determined by a comparative s t a t i c s methodology.  By c l o s e l y adhering to the systemic guidelines of the  h i e r a r c h i a l model, exogenous s h i f t s could f i r s t be expressed for employment ( p o p u l a t i o n ) ,  technology ( o u t p u t ) , and income (consumption) and,  162  then, the impacts of such s h i f t s upon those e q u i l i b r i u m conditions could be discerned in t u r n . A considerable number of statements - which led to the inference of several q u a l i t a t i v e hypotheses - were included in the d i s c u s s i o n so as to demonstrate the h i e r a r c h i a l d i f f e r e n t i a t i o n impacts.  of these instantaneous  Perhaps the most s i g n i f i c a n t of these hypotheses were that  the greater the s i z e of the central p l a c e , ( i )  the greater would be the  impact of a s h i f t i n any given basic employment (population) (ii)  sector and  the greater would be the p o s i t i v e (negative) impact of a technical  innovation bringing f o r t h a new a c t i v i t y  ( c a p i t a l deepening i n any e x i s t -  ing a c t i v i t y ) . The long run i m p l i c a t i o n s of such changes were taken up as well.  However, t h i s part of the text noticeably suffered i n terms of r i g o r  because such a paucity of analytical  e f f o r t has been devoted (to date,  that i s ) to a r t i c u l a t i n g issues l i k e ( i )  the threshold requirements, the  advantages found i n s p e c i a l i z a t i o n and agglomeration, e t c . on the part of firms with ( i i )  the incomes gained, the frequencies of multipurpose  t r i p s , e t c . on the part of households in the central place context.  As  a consequence, i t i s presently impossible to s t i p u l a t e j u s t how the s e r v i c e m u l t i p l i e r s would discretely t i o n of the s h i f t ( s )  change i n the long run as a r e f l e c -  i n the relevant parameter(s).  Out of these arguments, however, emerged a few i n t e r e s t i n g points which had been h i t h e r t o overlooked i n the l i t e r a t u r e .  It was shown,  f o r i n s t a n c e , that the s e r v i c e m u l t i p l i e r s also represented the actual proportions of a system's t o t a l population (with family s i z e again held constant) which'were devoted to f i r s t l e v e l , second l e v e l , t h i r d l e v e l ,  163  etc. activities respectively.  In a d d i t i o n , i t became apparent that  these same m u l t i p l i e r s also represented demand r a t i o s f o r the  different  bundles of goods and s e r v i c e s provided throughout the system. F i n a l l y , the s t r u c t u r a l argument was modified in two s p e c i f i c ways.  In the second chapter, the s t a g e - l i k e properties of the h i e r a r c h i a l  model were incorporated i n t o an elementary evolutionary scheme which retained the various c h a r a c t e r i s t i c s of the conventional s t a t i c case. The f i f t h chapter, on the other hand, introduced s t r u c t u r a l asymmetry due to a noncentral place type of a c t i v i t y i n t o the pure C h r i s t a l l e r i a n model.  (manufacturing, mining, e t c . )  -  In both these cases rather complicated  procedures were o u t l i n e d i n order to a l l o c a t e the extra  population  (this,  of course, would be d i s t r i b u t e d throughout the e x i s t i n g systems) which would be required to s e r v i c e or support these new increments.  6.3  Process The t h i r d chapter was e n t i r e l y devoted to r e s o l v i n g some methodo-  l o g i c a l issues in the l i t e r a t u r e  regarding central place d i f f u s i o n and,  then, a r t i c u l a t i n g a model which was (hopefully)  superior to the e x i s t i n g  formulations. In that chapter the author f i r s t c l a r i f i e d , i n a somewhat f a s t i d i o u s manner, his reasons for u t i l i z i n g a strict when modelling inter-urban d i f f u s i o n .  systems methodology  Its a p p l i c a t i o n led him to recognize  that the boundary conditions of any central place system could d r a m a t i c a l l y a f f e c t the c h a r a c t e r i s t i c s of any processes occuring w i t h i n that system. I t was decided, as a consequence, that i t would be a useful exercise to  164  d i s t i n g u i s h between ideal  closed and open central place systems ( a l l  real world systems are at l e a s t p a r t i a l l y open) in the d i s c u s s i o n . The h i e r a r c h i a l model was used in conjunction with g r a v i t y potential  theory in order to generate appropriate  (Hagerstrand-1ike)  d i f f u s i o n a r y patterns for central place systems with d i f f e r e n t (nesting f a c t o r s , s i z e d i s t r i b u t i o n of communities, e t c . ) .  properties  I t was demon-  s t r a t e d that h i e r a r c h i a l and wave-like adoptive patterns seemed to be concepts i n that ( i )  •polar  systemic openness ( c l o s u r e ) , ( i i )  a relatively  slow (rapid) decline in the values of the s e r v i c e m u l t i p l i e r s ,  (iii)  a  r e l a t i v e l y low (high) value of the f r i c t i o n of distance c o e f f i c i e n t  in  the p o t e n t i a l formula, and ( i v ) areal ( l i n e a r ) dimensionality appeared to induce the former ( l a t t e r ) type of d i f f u s i o n a r y  pattern.  This argument was subsequently modified so that changes in such patterns - from the ideal s t a t i c case, that i s - could be discerned f o r (i) (ii)  independent parametric s h i f t s (in p o p u l a t i o n , income, e t c . ) and the a d d i t i o n of noncentral place type a c t i v i t i e s .  The second case  was perhaps the more i n t e r e s t i n g in that r a d i a l discrepancies emerged in the adoptive scheme, r e f l e c t i n g the f a c t that centers i n t i m a t e l y t i e d to a l o c a t i o n of incremental a c t i v i t y would adopt sooner than the other centers of the same type throughout the system.  6.4  D i r e c t i v e s for Future Research It  i s only r i g h t t h a t , before t h i s t h e s i s c l o s e s , a few comments  should be advanced which might s p e c i f y appropriate (in the author's opinion anyways) areas of concern f o r future r e s e a r c h .  165  F i r s t of a l l , there are numerous a n a l y t i c a l d e f i c i e n c i e s on the s t r u c t u r a l  side of the argument alone.  As has been repeatedly pointed  out i n the t e x t , the s e r v i c e m u l t i p l i e r s of the h i e r a r c h i a l model require a more p r e c i s e i n t e r p r e t a t i o n conditions.  Only then w i l l  i n terms of e x p l i c i t supply and demand  i t become possible to i n d i c a t e , i n a s u i t a b l y  rigorous manner, the e q u i l i b r i u m s t a t e ( s ) through which a central place system would be expected to pass over the long run. Another useful exercise - and t h i s i n p a r t i c u l a r would c e r t a i n l y enhance the a b i l i t y of the s t r u c t u r a l model to represent c e r t a i n domains of the real world - would be to introduce dimensions of purposeful and/or random v a r i a t i o n into the d e t e r m i n i s t i c and i d e a l l y fashioned h i e r a r c h i a l model.  This m o d i f i c a t i o n might be i n s t i t u t e d e i t h e r in terms of the  areal s i z e (depicting random v a r i a t i o n of community l o c a t i o n s from the p e r f e c t l a t t i c e ) or the employment (population) density d i f f e r e n c e s in the p r o d u c t i v i t y of a g r i c u l t u r a l  (reflecting  land) of the basic rural  areas or i t might be introduced into the s e r v i c e m u l t i p l i e r s themselves (so as to i n d i c a t e random differences i n labor or management s k i l l s o r , perhaps, even s p a t i a l v a r i a t i o n i n the d i s t r i b u t i o n of amenities f o r an otherwise f e a t u r e l e s s and bland w o r l d ) . Another useful change would be to introduce d i s t i n c t i o n s between productive and consumptive linkages amongst central places.  In that way,  the expenditures leaking out of communities to higher l e v e l places could be considered in both those sectors of the space-economy ( i . e . f o r both i n d i v i d u a l firms and consumers in an m th level place purchasing goods and s e r v i c e s from (m+l)st, (m+2)nd, e t c . l e v e l p l a c e s ) .  This would a l s o  help to r e s o l v e ' t h e balance of payments problem in the t r a d i t i o n a l  argument  166  and would provide a framework f o r considering even other topics - such as the c y c l i c a l s t a b i l i t y of the i n d i v i d u a l  urban economies or the  inter-  r e l a t i o n s of investment and long run change throughout the system - under the h i e r a r c h i a l  format.  Besides (and t h i s i s a topic of special i n t e r e s t to the author), there are promising avenues along which to integrate the rent and populat i o n d i s t r i b u t i o n models of intra-urban economics with the model of community s i z e .  inter-urban  The d i s t i n c t i v e h i e r a r c h i a l t r a i t s of the  o v e r a l l system could then be discerned - in a t h e o r e t i c a l  sense at l e a s t -  w i t h i n the i n d i v i d u a l centers of that system. On the s t r i c t l y empirical s i d e , the author would welcome t e s t i n g of his proposed d i f f u s i o n a r y model.  With appropriate care taken in  s e l e c t i n g a s u i t a b l e study a r e a , a s e r i e s of d i f f e r e n t  items which have  been adopted, e t c . some of the hypotheses of the t h i r d chapter could perhaps be r e a l i s t i c a l l y q u a l i f i e d . In addition i t would be useful to r e l a t e the temporal changes in the service m u l t i p l i e r s  of such a study area to changes in other  parameters (such as average household income, travel times to nearest neighbor communities, and the l i k e ) so as to suggest - in the absence of an e x i s t i n g rigorous a n a l y t i c a l argument - how long run changes in such m u l t i p l i e r s might be s p e c i f i c a l l y r e l a t e d to various i n d i c a t o r s of regional economic growth (or d e c l i n e ) . T h i r d l y , i t would be i n t e r e s t i n g to test the a l l o c a t i v e scheme given in the f i f t h chapter for noncentral place type a c t i v i t i e s .  This  could be done, a g a i n , by s e l e c t i o n of a s u i t a b l e area with only a few such a c t i v i t i e s located at d i f f e r e n t  sites.  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Human Behaviour and the Principle of Least an Introduction to Human Ecology. Cambridge: Addison Press.  Effort:  Wesley  APPENDIX A  STAGE MATURATION OF A FOUR LEVEL K=3 CENTRAL PLACE SYSTEM  Assumptions:  k r  x  0  = .05, k [to]  P® [ t j  p?  x  2  = .20, k  3  = .15, ku = .10  = 750  ti:  A l l o c a t i o n in A t  = .30, k  x  [  A  t  l  ]  =  = .3 (750) = 225  ^_12251  Pio  [Ati]  = 17  Pii  [Ati]  = 104  =  1 2 1  (numbers of dominated units in parentheses)  r i = 17 (1)  ri [ti  + A t i ] = 750 + 17 = 767  Pi  + A t i ] = 225 + 104 = 329  [ti  174  t :  Pz [ t ]  2  = .2 (3(329) + 3(767)} = 658  2  S  p  [  A  t  2  ]  =  ^ 6 5 8 )  =  P20 [ A t ]  = 73  pli [At ]  = 439  2  2  8 0 4  pi2 [ A t ] = 292 2  A l l o c a t i o n in A t : 2  Pa [ t  t : 3  r i = 24  (3)  pin  (2)  = 10  pi  1 2  = 419  r i [t  2  + A t ] = 767 + 24 = 791  Pi [ t  2  + A t ] = 329 + 10 = 339  (1)  2  2  + A t ] = 329 + 419 + 658 + 292 = 1698  2  2  p? [ t ] 3  = .15 (3(1698) + 6(339) + 9(791)} = 2137  p [At ] = S  3  3  7 ( 2 1  „  -3  3 7 )  = 4986  [At ]  P30  3  = 356  P?i [ A t ] .= 2137 3  pL  [At ]  = 1425  P?  [At ]  = 1068  3  3  3  Allocation in A t : 3  r  = 40  x  (9)  P 3 i i = 17  p  2  P  U:  3  [t  3  pf  (6)  1 2  = 24  (2)  p?  P 3 2 2  = 62  (2)  p  1 3  S i23  r i [t  3  + A t ] = 791 + 40 = 831  Pi [ t  3  + A t ] = 339 + 17 = 356  [t  = 1987  (1)  = 1 301  (1 )  3  3  + A t ] = 1698 + 24 + 62 = 1784  3  3  + A t ] = 1698 + 1987 + 1301 + 2137 + 1068 = 8191 3  p? [ U ] = .1  (3(8191 ) + 6(1784) + 18(356) + 27(831)} = 6412  p  s  [ A U ]  =  M w z i  =  25648  177  p?o [ A U ] = 1 6 0 3  p5i [ A U ] =  9618  p?  [AU]  p,3  [Ati,] = 4 8 0 9  p5-  [AU]  2  =6412  = 3206  Allocation in A t , :  ri  = 59  p5n = 25  (27)  (18)  p5i2 = 36  (6)  p5i3 = 51  (2)  p?m =  8850  (1 )  p5  (6)  P5  = 1 30  (2)  pL- =  5606  (1 )  pl  = 401  (2)  22  = 91  23  33  p  2  p  3  p„  [U  + At,]  [U  ri  [ U + A U ] = 831 + 59 = 8 9 0  Pi  [ U + A U ] = 356 + 25 = 381  P  5  3l|  = 4007  [ U + A U ] = 1784 + 36 + 91 = 1911 + AU]  = 8191 + 51 + 1 3 0 + 401 = 8 7 7 3  = 8 1 9 1 + 8 8 5 0 + 5606 + 4 0 0 7 + 6 4 1 2 + 3206 = 3 6 2 7 2  (1 )  Table A . l Populations of Rural Areas and Central Places in a K= 3 Four Level Central Place System Which Matures in a Stage-Like Fashion*  Time 750 (27)** 20250  to  t i + Ati t t  2  3  + At  2  + At 3  t, = At,  Pi  P2  P3  P*  -  -  -  -  Total  20250  -  -  -  767 (27) 20709  329 (27) 8883  791 (27) 21357  339 (18) 6102  1698 (9) 15282  831 (27) 22437  356 (18) 6408  1784 (6) 10704  8191 (3) 24573  890 (27) 24030  381 (18) 6858  1911 (6) 11466  8773 (2) 17546  29592  -  42741  64122 36272 (1) 36272  96172  Figures are subject to rounding e r r o r s . Figures in ;the parentheses r e f e r to the equivalent number of r u r a l areas, f i r s t l e v e l p l a c e s , e t c .  CD  \  179  Note that while incremental growth in r u r a l population remains a constant f r a c t i o n (0.05) of incremental growth i n t o t a l (rural plus urban), the r u r a l / t o t a l  population  r a t i o d e c l i n e s in the sequence 1.00,  0.70, 0.50, 0 . 3 5 , 0.25 and the urban/rural r a t i o increases in the sequence 0.00, 0 . 4 3 , 1.00, 1.86, 3.00.  APPENDIX B  THE  NOTION  OF  EQUIVALENT  CENTERS  TELLING  IN  HUDSON'S  STOCHASTIC  PROCESS  In order to i l l u s t r a t e Hudson's t e l l i n g process i t to delineate the actual t e l l i n g c h a i n s .  While this reduces the elegance  of the o r i g i n a l mathematical argument, a much truer of a d i f f u s i o n a r y process i s a t t a i n e d .  i s useful  spatial  perspective  In s h o r t , i t i s possible to  demonstrate p r o b a b i l i s t i c symmetry i n s p a t i a l adoption: a property that should be expected i n the i d e a l i s t i c central place s e t t i n g . Consider, for i n s t a n c e , a f i v e l e v e l sequence: 1, 2, 6, 18, 54 class.  K=3  system with the  representing the number of centers by s i z e  Table B.l c l a s s i f i e s these centers i n t o types according to  their  distances from the s i n g l e f i f t h l e v e l placeJ In Hudson's p r o p o s a l , only t h i s f i f t h l e v e l place would know (have adopted) at time  t . x  In f a c t , i t would know with complete c e r -  t a i n t y ; i . e . with a p r o b a b i l i t y of one.  However, s i x surrounding  fifth  l e v e l p l a c e s , each the geometric center of an i d e n t i c a l system, would a l s o know at the same time.  At time  180  t , 2  then, the f i f t h level place  Table B.l Types of Equivalent Centers i n a K=3 Five Level Central Place System  Order  Type  Equivalent Number  5  5  1  0  4  4  2  3/3  3  3  6  3  2  Distance from 5th Order. Center  18 2  1  6  /3  2  2  6  2^3  2  3  6  1  54 l  1  6  l  l  2  6  2  I  3  12  1"  12  l I  5  6  6 12  n  4  182  of i n t e r e s t i s surrounded by s i x fourth l e v e l places - each knowing with p r o b a b i l i t y 1 - and s i x t h i r d , second, and f i r s t l e v e l places each knowing with p r o b a b i l i t y 0.333.  In equivalent terms, two places  of each l e v e l have adopted with complete c e r t a i n t y . The t e l l i n g process for t h i s p a r t i c u l a r system may be represented by the statement:  t  : g/hj  i  which means t h a t a t time  t.  each dominated by  e  p r o b a b i l i t y by  (0 < f.  i n each type and  c  f^ g  e(b)(c)  = f.  (B.l)  (i > 1 ) , centers of type  center(s) of type g , increase t h e i r adoptive < 1).  The increment of p r o b a b i l i s t i c adoption  place at the previous time  t.  i s represented by  i s simply the r e c i p r o c a l of the nesting f a c t o r  process a t e a c h time  h. (j = 1 , 2 , • • • ) » '  t.  (1 < i < 5) i n the f i v e l e v e l  K. K=3  b  The t e l l i n g system i s then:  183  ti t  2  5/4  5/3,2M  3(1) [.333) = 1 .333 (1) [.333)  1  4/3 4/2 ,2 ,l 3/2\2 ,2 3/l ,l ,l\l 2V1M 2  3  2  2  2(1)1 .333) ( I X .333) 2( .333)1 .333) .333)1 .333) 21 .333)< .333) .333)( .333)  6  3  3  5  2  3/2 ,2 ,2 3/l ,l ,l\l 2V1M 271 2 /l ,l\l 2 /l\l 2 /l x  2  2  3  3  5  2  3  2  3  3  6  6  3  5  271M 271 2 /l ,l\l 2 /l\l 2 /l 3  6  6  3  21 .667)( .333) z: .444 .667)( .333) = .222 21 .222)( .333) = .148 .222)( .333) = .074 .556) ( .333) = .185 .556)( .333) = .185 .370 2( .556)( .333) 2( .444)( .333) .444)( .333) .444) ( .333) 2( .444)( .333) .444)( .333)  2  3  5  which s p e c i f i e s the f o l l o w i n g incremental  1,  4 :  0,  3  0,  0,  0, 1, 0, .333, .667,  o,  0  0,  0  0,  0  2  1  0, .333, .222, .444,  0  2  2  0,  0, .556, .444,  0  2  3  0,  0, .556, .444,  0  l  1  0, .333, .222, .148, .296  l  2  0,  0, .333, .370, .296  I  3  0,  o, .222, .481, .296  1-  0,  0, .111, .592, .296  1  5  0,  o, .111, .592, .296  6  0,  0, .333, .370, .296  l  .296 .148 .148 = .296 = .148 = -  states of p r o b a b i l i s t i c adoption  f o r a l l members o f each type: 5  =  .667 .333 .222 .222 .222 .111  —  3  2  = = = = =  184  Therefore, the sequential  states  p r o b a b i l i s t i c adoption, representing  sums of incremental s t a t e s , a r e :  5 :  1,  1,  1,  1,  1  4 :  0,  1,  1,  1,  1  3 :  0, .333,  1,  1,  1  2 :  0,  0, .556,  1,  1  2 :  o, .333, .556,  1 ,  2 :  o,  1,  1  l :  o, .333, .556, .704,  1  l : z  o,  0, .333, .704,  1  l : 3  o,  0, .222, .704,  1  1":  o,  0, .111, .704,  1  l : 5  o,  0, .111, .704,  1  l :  o,  0, .333, .704,  1  1  2  3  1  6  0, .556,  1  These two l i s t s are useful f o r i d e n t i f y i n g the d i f f e r e n t adoptive patterns of centers that are of the same s i z e but which have d i f f e r e n t locations.  relative  D i f f u s i o n i s symmetric in the sense that a l l centers of the  same type have i d e n t i c a l adoptive p a t t e r n s . The incremental l i s t i s the counterpart to Hudson's f i r s t - h e a r i n g matrix [A].  Note, f o r i n s t a n c e , that at time  t , 3  the  equivalent number of f i r s t l e v e l places hearing a message (adopting) f o r the f i r s t time i s : a  3 5  = .222(6) + .333(6) + .222(12) + .111(12) + .111(6) + .333(12) = 12  185  which i s an element of the matrix:  [A] =  0  0  0  0  0  2  2  2  2  0  0  4  8 12  0  0  0  8 24  0  0  0  0 16  for a l l f i r s t - h e a r i n g centers. The above argument i s s o l e l y intended to be a s p a t i a l q u a l i f i c a t i o n of Hudson's p r o p o s a l .  I t i s t h i s author's contention that i t  is  simply more accurate to state that centers adopt with c e r t a i n p r o b a b i l i t i e s rather than to state that equivalent enters adopt with c e r t a i n t y . other hand, the same empirical t e s t i s used to v e r i f y e i t h e r  On the  interpretation.  FOOTNOTES TO APPENDIX B  ^On a hexagonal l a t t i c e the distance from one point to each surrounding point can be c a l c u l a t e d by applying the formula: d  i d  = (i  2  + ij + j ) * 2  1 < i < j  where i and j are integers representing distances on a s i x - f o l d a x i s with unit t r a n s l a t i o n periods and a p e r i o d i c r o t a t i o n angle of T T / 3 . See Dacey (1965). 2 The inconsistency of Hudson's argument may be found a t time t . In a d d i t i o n minor rounding errors e x i s t because the p r o b a b i l i s t i c values are expressed i n decimal form. 2  186  APPENDIX C  THE ALTERNATIVE DIFFUSIONARY MODEL: PROPERTIES OF THE TABLES AND AN OUTLINE OF THE TELLING PROCESS  In t h i s appendix a t t e n t i o n i s f i r s t given to the t h i r t y - t w o tables of generated data i n Chapter 3.  In order to complement the  table headings, the f o l l o w i n g l i s t provides a d d i t i o n a l information about the c e n t r a l place populations in open (closed) systems: Tables 3 . 1 , 3 . 2 , 3 . 3 , 3 . 4 ; Populations:  1  1000, 4000 (2000), 16000, 64000 (21333), 256000 2  Tables 3 . 5 , 3.6, 3.7, 3.8; r a n k - s i z e r e l a t e d Populations:  1000,  2946 (1473), 8385, 21800  (7267), 54500 Tables 3 . 9 , 3.10, 3.11, 3.12; Populations:  1000 (500), 4000, 16000 (5333), 64000  187  188 Tables 3.13, 3.14, 3.15, 3.16; r a n k - s i z e related Populations:  1000 (500), 2846, 7400 (2467), 18500  Tables 3.17, 3.18, 3.19, 3.20; Populations:  1000, 4000 (2000), 16000 (5333), 64000  Table 3 . 2 1 , 3.22, 3.23, 3.24; r a n k - s i z e r e l a t e d Populations:  1000, 3062 (1531), 9800 (3267) 24500  Tables 3.25, 3.26; r a n k - s i z e r e l a t e d Populations:  1000 (500), 3857, 13500 (6750), 40500  Tables 3.27, 3.28; Populations:  1000, 4000, 16000, 64000 (32000), 256000  Tables 3.29, 3.30: Populations:  r a n k - s i z e related 1000, 1923, 3571, 6250 (3125), 12500  Tables 3.31, 3.32; Populations:  1000, 4000, 16000 (8000), 64000  The t e l l i n g process postulated i n Chapter 3 i s now f o r a four level system (open and closed) with a  K=3  illustrated  geometry.  The  equivalent central place populations of that system d i f f e r by a m u l t i p l e  189  of four and the distance exponent of the p o t e n t i a l f u n c t i o n i s two.  The  computed adoptive times are those given i n Tables 3.10 and 3.12 (with a s l i g h t d i f f e r e n c e due to v a r i a t i o n i n s i g n i f i c a n t  figures).  The centers may be c l a s s i f i e d according to types as they were i n Appendix B: i t should not be necessary to restate the appropriate distances from the  M th level p l a c e .  The. populations, then, according  to types a r e : ^  type  open system  closed system  population  4  64000  3  16000  2  4000  1  1  1000  1  2  1000  1  3  1000  4  64000  3  5333  2  4000  1  1  1000  1  2  1000  l  3  500  Since the t e l l i n g process i s now continuous i n nature, i t s s i t y (with regard to a s p e c i f i c hearing or adopting place) jumps in a step-wise fashion as new t e l l i n g centers (which have themselves j u s t  inten-  190  adopted) j o i n i n according to the p r i n c i p l e s of central place domination. Adoptive times may be standardized to the time of f i r s t adoption (for a p a r t i c u l a r process i n a p a r t i c u l a r system) outside the  M th l e v e l center  (source): f o r these r e l a t i v e times, the s c a l i n g f a c t o r  G of equation  (3.2) may be eliminated and populations may be expressed i n simpler terms ( i . e . not i n thousands). continuously at a type (tellers)  An increment of e f f e c t i v e p o t e n t i a l , expressed h  center (hearer) by  a f t e r some time  t^,  (  p  and  p^, p^ d ^  a  9/h:  g  center(s)  are populations of type  p  h  )  3__JL_ d  where  type  may be represented by the statement:  e  V  e  (Cl)  gh  g  and  h  centers r e s p e c t i v e l y  i s the distance separating them. The threshold f o r adoption i s simply determined by the greatest  p o t e n t i a l expressed outside the  (I)  M th l e v e l place in any one system.  The T e l l i n g Process in an Open System 4/3: 4/2:  3 (64 x 16) 3 2  64 x 4  341.333 = threshold 85.333  4/1 :  64 x 1 l  64  4/1 :  64 x 1 l  16  4/1 :  2 (64 x 1)  x  2  3  2  2  (/7)  2  18.286  191  (1)  (ii)  remaining resistance 341.333  3 adoptive time = 1 2 time 1  :  3/2  .  incremental  341.333 256  85.333 2 (16 x 4) = 42.667 2 (/J) 256 time = 85.333 + 42.667 = 2  adoptive time = 3 (iii)  l time 3 1  2/1  1  incremental  341.333 149.333  3 (64) = 192 2(4X1) g 1 =  time =  149.333 64 + 8  2.074  adoptive time = 5.074  (iv)  l time 1 3  4/1  3  16 x 1 = 16 1 2  time 3 2/1  341.333  18.286  3  incremental  2 (18.286 + 16) = 68.572  254.475  2 (4 x 1) 8 1 254.475 time = 6.018 34.286 + 8  adoptive time = 9.018 (v)  I  2  time 1 3/1  2  time 3 incremental  341.333  16 16 x 1 = 16 l 2  2 (16 + 16) = 64 O C1  OQ O  time = 3 2 + 3  adoptive time = 9.533  261.333  =  6.533  192  (II)  The T e l l i n g Process in a Closed System 4/3 : 4/2  6 4  3  ^  3  3  = 37.926  3  (64.x 4)  :  4/11  X  = 85.333 = threshold  64^x2  :  =  4/1 :  6  4  = 16  2  4/1 : 3  6 4  Q(/7) x  5  = 4.571  2  .  (i) (ii)  remaining resistance 85.333  2 adoptive time = 1 i time 1  85.333 21.333  1  2/1 >  :  64 • l_(4_xjl  :  incremental time =  a  8  21 333 Q = .296 64 + 8  adoptive time = 1.296 (iii)  3 time 1  :  37.926  85.333 47.407  2/3  •  (4 x 5.333) _  14.222  (^3) 47 407 incremental time = 37 926'+ 14 222  =  "  9 0 9  adoptive time = 1.909 (iv)  l time 1  85.333  2  :  16 2  2  :  1  2  =  8  time 1.909:  .909 (.16 + 8) = 21.816  3:1*  5 3^3_xj. 1  =  5 > 3 3  3  47 51 7  incremental time = 24 + 5  333  =  1 , 6 2 0  adoptive time = 3.529  l time 1:  4.571  2/l :  4 - ^ 5  3  3  time 1.909: 3/1 : 3  = 2  .909 (4.571 + 2) = 5.973 5  '  3 3 3  x 1  °74  5  = 2.667 789  incremental time = 5 5 7 1 + 2 667 adoptive time = 10.005  =  8  -  0 9 6  FOOTNOTES TO APPENDIX C  'This i s the basic progression component model f i r s t suggested by Beckmann (1958) and then revised by Beckmann (1968) and Parr (1969). In t h i s case central place populations d i f f e r by a m u l t i p l e that i s constant f o r a l l l e v e l s : i n t h i s example i t i s f o u r . The bracketed f i g u r e s represent the effective populations of places on the boundary or at the endpoints of a closed system.  ^The r a n k - s i z e rule i s indicated i n several tables of Chapter 3 by the statement p (R ) = p . the population of a place midway on the -  m  M  m th h i e r a r c h i a l l e v e l times i t s rank (see Parr (1969) f o r a good d i s cussion) equals the population of the M th l e v e l p l a c e . Even i n t h i s non-exponential case, the s i z e d i s t r i b u t i o n v a r i e s d r a m a t i c a l l y with other f a c t o r s (geometry, s i z e of smallest p l a c e s , e t c . ) .  Note that both systems contain the same central place populations but that the populations effective f o r p o t e n t i a l (and, hence, d i f f u s i o n ) are greater f o r boundary points i n the second and fourth s i z e c l a s s e s . o f the open system.  194  APPENDIX D  THE IMPACT OF EXOGENOUS EMPLOYMENT SHIFTS IN A SECOND LEVEL CENTRAL PLACE  The purpose of t h i s appendix i s simply to e l u c i d a t e (4.4) f o r the second l e v e l case.  In a d d i t i o n , statements are given to i l l u s t r a t e  (4.13) and ( 4 . 3 1 ) , each of which depends on ( 4 . 4 ) . R e c a l l i n g (2.4) and (2.7) i t should be apparent t h a t :  e  2 i  = kie  0  where, given that:  Ai = 1 - k A  x  = 1 - ki - k  2  2  (D.2)  i t follows t h a t :  3ejj  2  _ k  2  ( K - 1) t  195  (D.3)  196  The a s t e r i s k i s a reminder that the second l e v e l p l a c e , while i t  provides  the second bundle to these  provide  - 1  K  x  f i r s t l e v e l p l a c e s , does not  the same places with the f i r s t bundle as w e l l . Now by using (2.12) and ( 2 . 1 3 ) , i t should be c l e a r t h a t :  3eli 3e  ,  1 +  . k  (K:-!)  2  I  n  2 i  =  (1-ki)(1-k ) + kik (Ki-1) Ai A 2  2  2  3eL 3e  k (K -l) Ai 2  k^ A,  1  1  2 1  k (l-k )K Ai A 2  1  +  k (K -1) Ai 2  A  1  2  3eli  3eL  3e  3eL  2 2  Ml) A 2  kid-kj Ai A 2  3el B* 3ef 2  9el  2  3eL "  ,  k (l) 2  >  A  2  _ (1-ki)' A i A,  so t h a t :  (D.  197  del de i  A,  Ai A  2  2  del A  de  dp • B* de 2  del B  *  de  2  2 2  (D.5)  2  2 2  2 2  (1 + d j  2 i  dp  de  J_ A  B* 3de  (1 + d ) 2  3e i  2l  (1 + d i )  3eli 3e  2  + (1 + d ) 2  2 2  3el « B* 3e 2  2 2  (D.6)  APPENDIX E  THE DIVISION OF CENTRAL PLACE POPULATIONS ACCORDING TO ACTIVITY SECTORS  The f o l l o w i n g argument - as i t precludes v a r i a b i l i t y i n the family s i z e m u l t i p l i e r and the central place topology ( i . e . the nesting factor)  - does not have the same degree of g e n e r a l i t y as many o f the  previous assertions i n t h i s t h e s i s .  The author simply f e e l s t h a t , i n  t h i s p a r t i c u l a r case, considerable i l l u s t r a t i v e merit might be l o s t by making the notation e x c e s s i v e l y complex.  As a r e s u l t , the present argu-  ment i s phrased in terms of sectors of population - t h i s b e i n g , of course, the t r a d i t i o n a l  u n i t f o r h i e r a r c h i a l modelling - rather than sectors of  employment. Consider now an M l e v e l central place system with nesting factor  K.  The t o t a l  population  P^ of t h i s syrtem may be divided into  M+1  parts or general a c t i v i t y s e c t o r s , designated by the members of the  set  { r , p , p , " - , p , * * « , p } , where: 1  1  (i) (ii)  2  m  M  r i s the population engaged in rural a c t i v i t i e s ; and 1  (agricultural)  p i s the population engaged i n m th (1 5 m 5 M) l e v e l a c t i v i t i e s ( i . e . the sum of the populations m  198  199  i n each of the m t h , ( m + 1 ) s t , • , and M th order places engaged i n .providing the m th bundle of goods and s e r v i c e s ) . Values of these elements may be computed as: r» = K ^ r ,  p  = k {(K-l)K "  1  M  +  p  2  1  M  + (K-l) ( p ^ + r j  = k {(K-l)K -  2  (pi+ri) + (K-1)K "  M  3  2  + (p +r )} M  x  (p +r ) + ( K - 1 ) " M  2  (pa+rj  3  (p +r )  4  2  3  2  + . . . + ( K - l ) (p _-,+r ) + (p +r )} M  •  -  +  2  M  ( -l)<P .,+r ) •  +  (p r )}  K  M  2  m  M +  m  M P  =  k  M P {  + M  r  M  (E.l)  }  However, each of the a c t i v i t y  sectors  p  (1 < m < H) may be reformulated  m  so t h a t :  m  + i l l + J<2_ + J<JL_  Ai  AiA  A A  2  2  3  +  M-1V  A  (E.2)  where:  V .  1  -  J, i=l  k  i  (E.3)  200  But (E.2) may be reformulated i t s e l f by adding each of the terms one at a time:  pm  =  Now the t o t a l population  m  K  P  r  M  ,  i  ( E > 4  of the system i s (according to  (4.22) in the main t e x t ) : .M-l " so that the proportion  M  rt  R° of t o t a l population engaged in rural  activities  is:  while the proportion  R  m  of t o t a l population engaged i n  m th (1 < m < M)  level a c t i v i t i e s i s :  In other words the s e r v i c e m u l t i p l i e r s alone define the proportion  of a  central place system's t o t a l population which i s devoted t a rural production or i s engaged in providing the of goods and s e r v i c e s .  1 s t , 2 nd,  or  M th bundle  I n c i d e n t a l l y , the argument can be extended f o r a  v a r i a b l e topology or be restructured to concern employment (as opposed to population) but (E.6) and (E.7) would remain true mutatis mutandis.  x  APPENDIX F  THE IMPACT OF AN EXOGENOUS SHIFT IN A SERVICE MULTIPLIER IN A SECOND LEVEL CENTRAL PLACE  The purpose of t h i s appendix i s to i l l u s t r a t e and compare the r e s u l t s of each approach. multiplier  k  2  (4.37) and (4.42)  The-effect of a s h i f t in the  upon the t o t a l employment i n a second l e v e l place i s  considered. I f the reader r e c a l l s (2.12) and (2.13) then i t should follow that:  3 e  T  i  _  ~  8 k 2  Meg, +  e? ) +|ff-A 2  (A )  2  k  l  2  2  g  A^+eL  3eL. d k 2  "  )  (A )  +  | ^ A  2  A  1  (F.l)  2  2  so t h a t :  del d k 2  Iff (A ) 2  201  2  (F.2)  202  where: Ai = 1 - ki A  If i t  2  = 1 - ki  (F.3)  i s r e c a l l e d that:  3^22 9k 2  (Ki-1) k i e Ai  „  0 +  K l  e  °  (F.4)  then i t can be shown that:  de dk  2  :  _ Ki e  (F.5)  0  (A )' 2  But from (4.41):  el  e  0  K L . kaKi Ai AiA  (F.6)  2  which implies t h a t :  del dk 2  Ki e (A ) 2  as in (F.5) above.  ( :  (F.7)  APPENDIX G  A NOTE ON THE RELATION BETWEEN MONEY INCOME AND REAL INCOME IN THE HIERARCHIAL FORMAT  The f o l l o w i n g demonstration i s an extension of the n e o c l a s s i c a l (Hicks-Samuelson) argument on consumer behavior found i n spaceless economics.  The proposals of Long (1971) are embraced i n the "multicenter  distribution"  framework of Papageorgiou  (1971).  I t must be pointed out, however, that the argument represents a special interpretation where a g r i c u l t u r a l  of the s t r i c t h i e r a r c h i a l format: t h a t being,  goods are not immediately consumed but are sent on to  the urban centers f o r p r o c e s s i n g . only the  In other words a t t e n t i o n i s devoted to  M bundles of goods and s e r v i c e s .  i s only p a r t i a l  Needless to s a y , a n a l y s i s  i n nature.  Suppose t h a t : (i)  M composite goods and s e r v i c e s are offered accordto s t r i c t h i e r a r c h i a l  (ii)  principles;  the l o c a t i o n of any household under e q u i l i b r i u m conditions may be s p e c i f i e d by a s e t {s |m=l,2,*••,M} of distances to the nearest points of supply f o r goods of order  1,2,•,M  r e s p e c t i v e l y ; {s } may be 1 interpreted as a M-dimensional vector as w e l l ;  203  204  (iii)  the p r i c e  p  of the  m  m th bundled good may be  e x p l i c i t l y r e l a t e d to the m i l l p r i c e vector component  s , and the transport III  (per unit distance) t ; that i s :  (iv)  f,  rate  r\  the household, given i t s l o c a t i o n {s^}, a u t i l i t y function  the  maximizes  u = u ( x i , x , • • • . x ^ ) according 2  to a budget c o n s t r a i n t Y = Y({s }) in money m  income terms. Then, the usual statement of spaceless economics (see a good t e x t l i k e Samuel son (1947) or Henderson and Quandt (1958)):  d x  i  =  - ^  l i  dp  1  +  - XD .dp 2  Vl,i  - AD dp  2  ( d Y  Mi  - 1 Pl - 2 x  d  x  d p  with Lagrangian \ , determinant D, and c o f a c t o r s may be generalized t o :  M  2 - —  - M M x  d p  }  (j = 1,2,•••,M+1),  t  r  0  ,  205  dx  1  " " £  '11  M  .  M  QTM  3 p  +D Mi  9 f  li^  3 p  9 t  ds.  -Kz-  9 s  M  9Y ds.! - X] 8s i  D  +  . M + "^n d t „ + M  f  M a?: 'M  .  9 P d S  M - x  m  d f  M  +  M 3E7 M  M , 9iT M M  9 p  d t  9 p  + +  (G.3)  d S  which implies t h a t : 9x  i 9s. J  ap•  -A  D ji  3s, J  fa>  aY  D  9Y  *  Du., .  D  ap. x j • as  8Y 9s  9x.  9sT  3p  J  9s  9Y  ***  9S .  (G.4)  J  where * i n d i c a t e s that l o c a t i o n i s held constant, * * i n d i c a t e s that u t i l i t y and l o c a t i o n are held constant, and * * * i n d i c a t e s that m i l l p r i c e s , transport r a t e s , and l o c a t i o n are held constant.  T h i s , of course, i s  the same as: 9x.  ax.  9s .  9Y  i  However, i f  a  X i  *  3Y  9x.  9sT J  ap  i s set equal to z e r o , then:  ap. (G.5) *  3 s  j  206  f!ii 3Y 9s . J  3P.  9p. 9s  V  (G.6)  9x. i |9Y  9Y where  represents the s h i f t i n money income which would be required  by the household to compensate f o r a l o c a t i o n a l change away from the supply point of the  j th composite good and s t i l l  same amount of the  i th composite good.  go on consuming the  The necessary and s u f f i c i e n t  conditions f o r real income to remain constant i n the neighborhood of the 9x. household's l o c a t i o n a r e , then, determined by s e t t i n q •—• ,, 9s. = 0 for a l l J i,j = 1,2,--.,M. J  3  n  r  The above argument i s useful f o r i n d i c a t i n g the change i n money income f o r incremental l o c a t i o n a l s h i f t s .  The s p e c i f i c a t i o n of real income  at any point in the system may be e s t a b l i s h e d vis-et-vis the  M th l e v e l  center by an indexing procedure. In a d d i t i o n , i t should be pointed out t h a t :  fax,! 3 t  j  9t. - X.  fax^ 9Y  which i s i d e n t i c a l to the statement in Long.  ***  9t. J  He has c a l l e d the f i r s t term  on the r i g h t hand side the transport s u b s t i t u t i o n e f f e c t and the second term the transport income e f f e c t .  (G.7)  FOOTNOTES TO APPENDIX G  In t h i s argument i t i s assumed that a household i s in s p a t i a l e q u i l i b r i u m i f i t has the same real income as a l l other households: put d i f f e r e n t l y , i t consumes the same amounts of goods and services as other households and i t , t h e r e f o r e , has no i n c e n t i v e to change l o c a t i o n .  This i s a picky point but i s d i r e c t e d to notation e n t h u s i a s t s , {p } i s used to i n d i c a t e p r i c e s only in t h i s appendix - throughout the remainder of the t h e s i s i t r e f e r s to urban populations.  207  APPENDIX H  THE ALLOCATION OF NONNODAL ACTIVITIES  Two numerical examples of the argument in Chapter 5. are presented below.  Suppose a f i v e level  K=3  system e x i s t s with  {.3333, .1667, .1250, .0937, .0704}  and  {p }  256000}.  The a l l o c a t i o n  T  c  a  a>5 5 i y k P  L* m=l  Case  ffl  = {1000, 4000, 16000, 64000,  m  of:  p = P ,5  i s of  {k } =  =  J000 .2109  = 4 / 4 < i  n  interest.  (i): Suppose  p  r  3  =  1000  locates at a (one-third) component of a  aD 5  type 4 center;  then:  2Q°3 = 3555  i s a l l o c a t e d to that type 4 component  .0704 (3555) _ 2i g = 1187 1 1 Q  Q  . ^ . ^ _. . , i s a l l o c a t e d to the s i n g l e type 5 center 4  208  W-*>  Suppose  P 5 = 1000 locates at one type a  •^3^=  2  1  center; then  2000 i s a l l o c a t e d to that type 2 center 1  .1250(2000/3) „ , . . , . . , ^ o = 222 i s a l l o c a t e d to each of two type 3 centers 7[  n  .0937(222/3) ' - = 2813 2( .0937) (222/3) _  25  0 K  . . . . , i s a l l o c a t e d to each of two  type 4 components n component .  5 Q  i g  a  o c a t e c  | to one type 4  -  •  +  1 2 5  °^°  0 / 3 ) 9  +  ^(2000+2(222/3)}  (2000 + 2(222) + 100}  = 2198 i s a l l o c a t e d to the type 5 center  

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