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UBC Theses and Dissertations

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

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CITY SIZE DISTRIBUTIONS:  FOUNDATIONS  OF ANALYSIS by GORDON FREDRICK MULLIGAN B.Sc,  U n i v e r s i t y of B r i t i s h Columbia,  I969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the  Department of  GEOGRAPHY We accept t h i s t h e s i s as conforming to r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA February,  1972  the  In p r e s e n t i n g  this thesis in partial  an advanced d e g r e e a t the  f u l f i l m e n t o f the r e q u i r e m e n t s f o r  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 f r e e l y  a v a i l a b l e f o r r e f e r e n c e and  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e for  s c h o l a r l y p u r p o s e s may  by h i s r e p r e s e n t a t i v e s .  be  GKO£^/1  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  V/7  2-  the Head o f my  s h a l l n o t be  permission.  Department of  Date  gain  /> s4  y  Columbia  study.  copying of t h i s  thesis  Department o r  I t i s understood t h a t c o p y i n g or  of t h i s t h e s i s f o r f i n a n c i a l written  g r a n t e d by  that  publication  allowed without  my  ABSTRACT While many observers recognize the  significance  of the c i t y s i z e d i s t r i b u t i o n t o p i c , the r e s o l u t i o n of s e v e r a l apparent i n c o n s i s t e n c i e s i n the body of has not yet been achieved. sociologists,  literature  T h i s may e x p l a i n why geographers,  demographers, h i s t o r i a n s , economists,  planners e s s e n t i a l l y tend to describe i n t e r c i t y are biased toward ad hoc i n t e r p r e t a t i o n s , to making i n t u i t i v e statements i n t h e i r  and  patterns,  and are prone research.  The primary purpose of t h i s t h e s i s i s to evolve a more c o n s i s t e n t methodological v i e w p o i n t w i t h i n the community s i z e t o p i c .  E f f o r t s are made to u n i t e a n a l y t i c a l  statements r e s t i n g upon a common premise, to q u a l i f y , i n t h i s l i g h t , the approaches prevalent i n e m p i r i c a l research,  and to r e l a t e theory and empiricism by adopting  a f l e x i b l e explanatory framework.  The d i s c u s s i o n n e c e s s a r i l y  involves a c r i t i q u e of e x i s t i n g arguments and c e r t a i n extensions that, we can devise from those arguments. While there i s considerable a t t e n t i o n d i r e c t e d to presenting e m p i r i c a l methodologies, no o r i g i n a l data a n a l y s i s  is  included. Contending that the notions should be bound "together w i t h i n a systems framework, we n a t u r a l l y devote i n i t i a l emphasis to the features i  of c e n t r a l place  systems  I ii as  o u t l i n e d i n the  (I966) the  and  partial  (195^).  Losch  equilibrium theory We  C h r i s t a l l e r model, the  realistic  of the  two  place  simpler  s p e c i a l cases of  (1966).  characteristic ality may  The  Using  property  ( t h a t i s , the  economic base and  the  subtopic  interurban  illustrate  may  given  be  by  t o what degree constant  proportion-  the  c o n c e p t , we  also  provide  r e l a t i o n between  community  It is  u s e f u l i n b r i d g i n g the  felt intra-  expounded r a n k - s i z e  rule,  essentially  a consequence o f e m p i r i c a l r e s e a r c h ,  i s then  formally  attached  At  stage  h i e r a r c h i a l models.  a r g u m e n t s become i n c r e a s i n g l y r i g o r o u s certain intuitive literature.  The  notions  idea of hierarchial  e x p l a i n the  notions  c a n n o t be  i n order  sets  uni-hierarchy  c o n c l u s i o n here i s t h a t e x i s t i n g  hardly  this  t h a t seem a c c e p t e d  d e v e l o p e d t o complement the basic  1958)  scales.  widely  t o the  the  (Beckmann,  d i s t r i b u t i o n theme. be  Christal-  of empirical g e n e r a l i z a t i o n .  hierarchial v i e w s on  The  more  shown t o  most e l e m e n t a r y model  a limit  the  upon  i n t e g r a t i o n of  formulation  we  some r a t h e r n o v e l  and  paper i s an  Besides,  considered  that this  apparently  s i m p l e s t models are  a more g e n e r a l  f a c t o r ) of the  be  and  stress  s i z e models, a l l of which d i s p l a y a  larian hierarchy.  Dacey  particular  Christaller  approaches.  A major t h r u s t of the several city  of  rank-size considered  is  to q u a l i f y the  crudely  arguments.  city  phenomenon b u t totally  in  our  The  s i z e models t h a t the  incompatible.  two  * • i 11  E m p i r i c a l research methodologies are s t r e s s e d another fundamental s u b t o p i c .  We suggest c e r t a i n  avenues  along which e m p i r i c a l e f f o r t s must be strengthened either  (i)  before  rigorous i n d u c t i v e g e n e r a l i z a t i o n s or ( i i )  theory s u b s t a n t i a t i o n become more r e a l i z a b l e .  as  firm  Particular  a t t e n t i o n i s given to d e l i m i t a t i o n of the study area (and, therefore  to the scale problem), the comparison o f  frequency curves,  and the value of inferences we can  make u s i n g r a t h e r crude s t a t i s t i c a l  tools.  At t h i s stage  we introduce other skew d i s t r i b u t i o n s that are s i m i l a r to the r a n k - s i z e curve.  Furthermore,  genetically the  stochastic  models that seemingly account f o r these d i s t r i b u t i o n s are taken to complement the d e t e r m i n i s t i c theory mentioned above.  Here we support the c e n t r a l place argument as  only e x i s t i n g source o f models that e x p l i c a t e inducing s p a t i a l d i f f e r e n t i a t i o n of economic  the  those f a c t o r s activities  and, as a consequence, urban p o p u l a t i o n s . F i n a l l y , we pursue the idea of growth w i t h i n the interurban s t r u c t u r e .  At t h i s time, however, d i s c u s s i o n  i s c e r t a i n l y exploratory and so i s l i m i t e d to developing notions concerning the i n t e r r e l a t i o n s of growth v a r i a b l e s (population, income, e t c . ) broadest sense.  and h i e r a r c h i a l s t r u c t u r e  i n the  Within t h i s a n a l y t i c framework we can  suggest only the most general f a c t o r s that may be  associated  with low degrees of primacy (a q u a l i t y of interurban ture that we view as a d e v i a t i o n from a skew d i s t r i b u t i o n ) .  struc-  characteristic  This p a r t i c u l a r subtopic promises to be  iv an e x c i t i n g research theme i n i t s own r i g h t as move from e q u i l i b r i u m to dynamic m o d e l l i n g .  investigators  TABLE OF CONTENTS Page ABSTRACT  i  TABLE OF CONTENTS  v  LIST OF TABLES  viii  LIST OF FIGURES  ix  Chapter 1.  INTRODUCTION  1  2.  THE CENTRAL PLACE SYSTEM  7  A Review of C e n t r a l Place Theory Introductory Remarks  7  Case of the Single Good  8  Case of Many Goods  15  D i f f e r e n c e s Between C h r i s t a l l e r and Losch  22  Scope and Nature of the C l a s s i c a l Argument  26  Extensions  29  of the C l a s s i c a l L i t e r a t u r e  Aggregate Relations and Elemental Components  3.  7  30  H i e r a r c h i a l Structure  35  The C e n t r a l Place System Reconsidered  37  CITY SIZE MODELS AND DISTRIBUTIONS A Review of the H i e r a r c h i a l Models Terms and Notation Model It  .  The General Case v  38 38 38 41  vi Model I I J  The A g g r e g a t e A p p r o a c h  M o d e l III« Model I V i  The G e o m e t r i c M u l t i p l i e r The C o n s t a n t M u l t i p l i e r  H i e r a r c h i a l Models  4.  a n d t h e Economic B a s e  49 52 55  H i e r a r c h i a l M o d e l s and t h e R a n k - S i z e R u l e  60  H i e r a r c h i a l S e t s and t h e R a n k - S i z e R u l e  73  EMPIRICAL ANALYSIS AND INTERPRETATION  78  Background  78  The S t u d y A r e a  80  C i t y S i z e Patterns»  Skew D i s t r i b u t i o n s  and R e l a t e d C o n c e p t s  89  The R a n k - S i z e and P a r e t o D i s t r i b u t i o n s  90  Steady S t a t e D i s t r i b u t i o n s  96  Comparison  of Distributions  I n t e r p r e t a t i o n and E x p l a n a t i o n  100 103  S t o c h a s t i c Approaches  103  G e n e r a l Systems Theory  107  The A g g r e g a t e M o d e l R e c o n s i d e r e d 5.  45  CHANGING PATTERNS OF INTERURBAN STRUCTURE Growth i n a T h e o r e t i c a l C o n t e x t  113 116 117  Population  117  Income  121  Innovations  125  A Brief Synthesis  131  Graphing the Aggregate Model  132  6.  CONCLUDING REMARKS  BIBLIOGRAPHY  LIST OF TABLES Table 1. 2.  3.  Page Service M u l t i p l i e r s and B a s i c / N o n - B a s i c Ratios of Four C e n t r a l Place Systems  57  Fundamental P r o p e r t i e s of Midway C i t i e s i n Related C e n t r a l Place Systems v i a Diverse M o d e l l i n g Approaches  71  Constant Rank-Size Products Given by Independent H i e r a r c h i a l Sets v i a Model II  76  viii  LIST OF FIGURES Page  Figure 1.  P r i c e and Output Conditions f o r the I n d i v i d u a l Producer with no Competition and with Free Entry  ix  11  1  Chapter  INTRODUCTION Concern over the q u e s t i o n bution fact,  i s widespread i n the geographical i t i s a topic that  many f i e l d s .  (1939)  scientists i n  study o f the primate  a m i x t u r e o f i n t u i t i o n , weak l o g i c ,  and r a t h e r  rank-size  feature  loose  Among t h e few a c c e p t e d g e n e r a l i -  z a t i o n s are those t h a t primacy i s a s s o c i a t e d urbanization,  city  regularities.  However, c o n t e m p o r a r y e f f o r t s on t h e t o p i c o f t e n  analysis.  In  impetus on t h e e m p i r i c a l  Z i p f 's (19^-9) a c c o u n t o f r a n k - s i z e  statistical  distri-  literature.  intrigues social  The theme i s g i v e n  side through J e f f e r s o n ' s and  o f community s i z e  with  over-  c o l o n i a l i s m , and u n d e r d e v e l o p m e n t w h i l e  tendencies are associated with the interurban  !integration of economically the most s e r i o u s r e c e n t  advanced r e g i o n s .  Perhaps  e f f o r t s made t o e x p l a i n c i t y  size  d i s t r i b u t i o n s come from t h o s e f o l l o w i n g Beekmarm ( 1 9 5 8 ) who adhere t o c e n t r a l p l a c e models and f r o m o t h e r s S i m o n ( 1 9 5 5 ) who p r e f e r t h e s t o c h a s t i c argument. little  pursuing  But with  agreement on b o t h t h e o r e t i c a l and e m p i r i c a l f r o n t s ,  and  disparate  approaches f l a v o u r e d w i t h  and  redundancies, i t i s not s u r p r i s i n g that the t o p i c i s  e n v e l o p e d by an a i r o f d i s s a t i s f a c t i o n .  inconsistencies  2 From the standpoint of strengthening harmony among the diverse e f f o r t s , there i s alone s u f f i c i e n t reason t o attempt a r i g o r o u s review o f e x i s t i n g c o n t r i b u t i o n s . Besides, the tenor of present argument i n the f i e l d o f r e g i o n a l development and planning i s t h a t a much sounder knowledge o f the r e l a t i o n s h i p s among u r b a n i z a t i o n , economic growth, and c i t y s i z e arrangements i s d e c i d e d l y needed. Friedmann ( I 9 6 6 ) , f o r example, emphasizes that l i t t l e i s understood  about the substructures o f the space-economy  and the i n f l u e n c e o f s p a t i a l a c t i v i t y patterns upon r e g i o n a l growth.  H o p e f u l l y , then, t h i s study w i l l show  p r a c t i c a l b e n e f i t s as w e l l as s a t i s f y i n g personal  curiosities.  In t h i s t h e s i s , we analyse the l o g i c of e x i s t i n g t h e o r e t i c a l and e m p i r i c a l statements about community s i z e d i s t r i b u t i o n s and, when i n disagreement, present our counterarguments. the apparent  With t h i s i n mind we attempt to r e s o l v e some o f d i f f e r e n c e s between the d e t e r m i n i s t i c and  p r o b a b i l i s t i c i n t e r p r e t a t i o n s that support extent) the rank-size p r i n c i p l e . to r e l a t i n g seemingly  (to some  A l s o , a t t e n t i o n i s devoted  independent geographical  concepts  ( f o r example, economic base and d i f f u s i o n ) to the d i s c u s s i o n of c i t y s i z e s .  The purposes of the t h e s i s are c l e a r l y  twofold: (i)  To examine and attempt to r e f i n e  (in explicit  fashion) the e x i s t i n g methodology of the general problem area 1 and  3  (ii)  To  s u b t o p i c s and  o f f e r new  to extend  p r o b l e m a r e a t o the and As  i d e a s w i t h i n the  notions  t h a t bond the  general  g r o w i n g body o f g e o g r a p h i c a l  literature  theory. the c h a p t e r s  a concise this  are  d e v i s e d t o be  s k e t c h o f the e n t i r e  somewhat  study  independent,  seems a p p r o p r i a t e  at  time. We  review (I966),  are  Losch  concerned with p r e s e n t i n g a  (195^)i  and  later  students.  i n t h a t i t e l u c i d a t e s the  and  the  and  Loschian  realize  first  o f c e n t r a l p l a c e t h e o r y as d e v e l o p e d  essential  significant  the  fundamentals, both  thesis.  The  review  i d e n t i f y i n g the q u a l i t i e s  f o l l o w i n g chapter direct  employ the n o t i o n s  theory  to Special  of  hierarchial  communities.  i s the most r i g o r o u s o f  a t t e n t i o n to r e l a t i n g  h i e r a r c h i a l models i n e x p l i c i t  fashion.  the  Furthermore,  i n t e r - and  i n t r a u r b a n s c a l e s v i a the concept  base.  remainder o f the  chapter  the  various  o f the c e n t r a l p l a c e models t o  the r a n k - s i z e r e g u l a r i t y  is  Christallerian  domain o f e x i s t i n g h i e r a r c h i a l m o d e l s .  H e r e , we  The  Christaller  o f w h i c h a r e needed  structure within a set of interrelated The  by  comprehensive  drawbacks o f t h e  d i f f e r e n c e s between  emphasis i s p l a c e d on  we  specific  link  o f economic  i s given to i n t r o d u c i n g  ( t h e most c h a r a c t e r i s t i c  concern  i n the  city  s i z e d i s c u s s i o n ) w i t h i n the c e n t r a l p l a c e  work.  Some m a t h e m a t i c a l arguments d i s p l a y the n a t u r e  a s s o c i a t i o n between t h e h i e r a r c h i a l i d e a s , w h i l e q u a l i f y i n g any  and  frameof  uni-size class  p r e s e n t l y accepted  statements  the  4 that  are  s e e n t o be  invalid.  i s a demonstration that C h r i s t a l l e r models a r e i n c o m p a t i b l e and  the  The  thrust  rank-size  p r o b a b l y , but  t h a t the  r u l e and not  methodology of e m p i r i c a l  weak w i t h r e g a r d t o  the  e f f e c t s of  to give  analyses. like  (iii)  Much o f the  imprecisely  value of inferences  improved methodology.  On  o f t h i s s e c t i o n shows how interpretations  o f the  not  in  necessarily The  but  framework how B e s i d e s , the the  thought  frequency  evolving  statistical questions  become i n c r e a s i n g l y aware  made from e m p i r i c a l the  to  study areas,  o t h e r hand, the  deterministic  and  study  with  latter  portion  stochastic  skewed f r e q u e n c y d i s t r i b u t i o n s  are  opposition.  i t b u i l d s upon the  i d e a s o f the  an  structure.  upon growth are  fundamentals of item d i f f u s i o n .  made toward d e v e l o p i n g a f l o w c h a r t  reasoning.  assumptive  growth f a c t o r s a f f e c t i n t e r c i t y e f f e c t s of structure  the  intervening  a l s o concerns simple micro-economic  I t s p r i m a r y purpose i s t o examine w i t h i n  within  structure.  f i f t h chapter completes a c i r c u i t w i t h  second, i n t h a t discussion  may  well.  noticeably  next chapter i s devoted to  t h e s e i n hope t h a t we  o f the  as  is  c a r e and  ( i i ) b l e n d i n g d i f f e r e n t means o f c o m p a r i n g and  existing  may  included  research  (i) arbitrarily defining  distributions,  the  necessarily,  study of interurban  Observers c o n s i s t e n t l y f a i l the  discussion  chances o f c o i n c i d e n c e  r i s e when o t h e r i n d e p e n d e n t systems a r e The  of t h i s  No  suggested  attempt  o r f e e d b a c k model,  is  5 even of the simplest k i n d ; r a t h e r ,  e f f o r t s at t h i s stage  are t o t a l l y d i r e c t e d to d i s p l a y i n g impact tendencies A characteristic  feature  alone.  of the t h e s i s i s the  adherence to a systems framework f o r studying the i n t e r r e l a t i o n s of population c l u s t e r s  i n a spatial setting.  It  seems a b s o l u t e l y necessary to evoke t h i s framework when t r y i n g to integrate  the various f a c e t s of the l i t e r a t u r e  i n t o a more meaningful whole.  Being aware that no r e a l  world system i l l u s t r a t e s the p r e c i s e q u a l i t i e s of the c e n t r a l place system, h i e r a r c h i a l notions may, of course, be somewhat relaxed  (see M a r s h a l l ;  The geographical l i t e r a t u r e  I969). i s r e p l e t e with systems  t h i n k i n g but only r e c e n t l y do we f i n d the concepts f o r m a l l y applied.  To be b r i e f , the h i s t o r y o f systems t h i n k i n g  i s t i e d up with f u n c t i o n a l and e c o l o g i c a l approaches, the organismic analogy, and the i d e a of r e g i o n a l s y n t h e s i s (Harvey, 1969). E x p l i c i t to the d e f i n i t i o n of a system i s that we are concerned not only with a sum of elements whose attributes  are d i r e c t e d by causal laws, but by a sum of  r e l a t i o n s among those u n i t s and some environment. c r i t i c a l p o i n t , then, i s that a system possesses  The properties,  f u n c t i o n s , or purposes that are d i s t i n c t from i t s c o n s t i t u e n t objects, 1956).  r e l a t i o n s h i p s , and a t t r i b u t e s  ( H a l l and Fagen,  In our immediate study there i s some i n t e n t o f  complementing e x i s t i n g l i n e s of argument with simple fundamentals o f general systems  theory.  I n any  case, the  systems framework i s e s p e c i a l l y  f l e x i b l e with regard  to our  as a r e m i n d e r o f t h e  ever-present scale  t a l k of regional c i t y s y s t e m s , the  level  o f a b s t r a c t i o n and  systems as  problem.  value of a consistent  approach should  On  the  one  i s , the  question  hand, we  of  "explanation"  have an, e v e r - i m p r o v i n g  city  crystallize.  city per  size se.  equilibrium  theory d e a l i n g w i t h f u n c t i o n a l a l l o c a t i o n s i n space, whose domain i s t y p i c a l l y r e s t r i c t e d t o a c t i v i t i e s input  prices vary l i t t l e  over distance.  Most  include  centers  I n a d d i t i o n , we  of s p e c i a l s i t e  have an a p r i o r i  e s s e n t i a l l y a v o i d s the here t h a t d e s p i t e may  describe  place the  The  f a c t the  t h e o r y l a c k s , p r o m i s e s t o be  explanation  i n the  the  a t t e n t i o n we  scheme, combined w i t h our  future.  features.  to  that  I t i s argued  s o - c a l l e d entropy  same degree as  increased  situation  s p a t i a l dimension.  approach  satisfy  c e n t r a l place  give  where  than t h i s  s t o c h a s t i c model  a l a r g e r domain, i t f a i l s  c u r i o u s i t y t o the does.  the  and  but  empirical  s t u d i e s , however, c o n c e r n a domain much g r e a t e r and  we  must comment  the most t r o u b l e s o m e a s p e c t o f t h e  topic; that  When  opposed t o n a t i o n a l  B e f o r e c l o s i n g t h i s i n t r o d u c t i o n , we b r i e f l y on  serves  t o the  our approach  central  g r o w i n g awareness o f what the  best  route f o r s u i t a b l e  Chapter 2 THE CENTRAL PLACE SYSTEM A Review of C e n t r a l Place Theory Our d i s c u s s i o n of c e n t r a l place theory pursues a synthesis of the fundamental 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 , Losch, and more recent advocates of the s u b j e c t .  We p l a n  to e f f e c t i v e l y defend the n o t i o n of a c e n t r a l place  system,  while developing a strong framework f o r t r e a t i n g the o f c i t y s i z e models.  topic  Only i n t h i s l i g h t may the methodology  of theory extension be p r o p e r l y understood. Introductory Remarks The route to comprehension of the s p a t i a l economic systems of C h r i s t a l l e r (I966) and Losch (1938, 195 *)  is  2  through the independent study of s i n g l e goods or  services.  While t h e i r i n i t i a l assumptions are not e n t i r e l y i d e n t i c a l , we can nevertheless  i s o l a t e f o u r general postulates  that  appear e i t h e r i m p l i c i t l y or e x p l i c i t l y commons (i)  A homogeneous p l a i n with uniform r u r a l densities ;  (ii)  A system of f . o . b . p r i c i n g ;  (iii)  Equal demand by a l l consumers  (consuming  u n i t s ) at any r e a l p r i c e ; (iv)  Free entry of producers i n t o the market. 7  8 A clear  interpretation  of these p r i c i n g  restrictions  i s v i t a l to a n a l y s i s i n terms of cost and demand f a c t o r s . F . o . b . p r i c i n g i s simply the case where the consumer pays the p r i c e f o r a good at the production s i t e price)  plus the cost of t r a n s p o r t a t i o n  (the t o t a l being the r e a l p r i c e ) .  (spatial  f.o.b.  to h i s l o c a t i o n  Such a p o l i c y seems  s u i t a b l e f o r firms d e a l i n g with ( i ) customers and ( i i )  (the  large numbers o f  goods and s e r v i c e s whose distance  elasticity  decay  of demand) i s h i g h .  Losch puts forward h i s argument i n a more r i g o r o u s manner, while i n c l u d i n g settlement geography as only a p o r t i o n of the general l o c a t i o n problem.  By p r e s e n t i n g  h i s case w i t h i n the confines of formal economic he attaches a strong t h e o r e t i c a l  tone to the  theory  settlement  p r i n c i p l e s of C h r i s t a l l e r . Case of the S i n g l e Good Let's  imagine the world as defined by the assump-  t i o n s of C h r i s t a l l e r and Losch.  F i r s t we consider an  i n d i v i d u a l good or s e r v i c e that i s o f f e r e d at site " 0 " on the p l a i n . production  The d e s i r e s of a consumer r e s i d i n g at  the  s i t e are i n d i c a t e d by the usual convex  downward-sloping demand curve that i n t e r s e c t s both the p r i c e and the quantity axes.  Since demand " q " i s a  continuous f u n c t i o n of the f . o . b . p r i c e , we may consider how demand changes f o r d i s t i n c t f . o . b . l e v e l s the i n t e r v a l p  m i n  £ P  A  f P  m a x  (where " P  m i n  "  "p^" i n  represents  9  " 0 " w i l l purchase a  the p r i c e at which a consumer at  maximum q u a n t i t y of the good and " P  m a x  " i s that p r i c e  where the same consumer w i l l purchase a zero q u a n t i t y ) . L e t ' s consider now an i d e n t i c a l consumer who resides  "x" u n i t s d i s t a n t from the production p o i n t .  T h i s customer must pay an a d d i t i o n a l "xt" the transport  cost per u n i t distance)  of the commodity to h i s l o c a t i o n .  (where  "t"  is  to cover the movement  In other words, demand  " q " i s a continuous f u n c t i o n of the r e a l p r i c e "p^ +  xt"  i n the general case. With t h i s knowledge we can determine the  distance  " r ^ " to the l a s t customer e x e r t i n g e f f e c t i v e demand f o r the good or s e r v i c e  s u p p l i e d at  " 0 " . This d e f i n e s a market  area of r a d i u s " r ^ " f o r any f . o . b . p r i c e  "p^"?  I t should be apparent that l i n e a r demand i s , then, f u n c t i o n of marketing ( f . o . b . (cost "t")  p r i c e "p^")  a  and t r a n s p o r t  technologies.  Using our f i r s t assumption, we may compute the a r e a l demand f a c i n g the f i r m at  " 0 " . By r o t a t i n g  the  distance-demand response curve ( f o r given " p ^ " and  "t")  about a v e r t i c a l a x i s through " 0 " we can trace out a demand curve f o r the t y p i c a l consumer.  Now, when we  m u l t i p l y the volume beneath the demand cone by a " D " representing population (consumer)  constant  d e n s i t y , the  total  demanded q u a n t i t y " Q . " i n the area about " 0 " i s given i n  10 i n t e g r a l form:  Y  2fT/ Q. = D  r. i f r  )  f(p  J  i  + xt)  x dx dO  .  I f t h i s c a l c u l a t i o n i s repeated f o r a v a r i e t y of prices  ( i n the  i n t e r v a l P j_  different levels  m  f.o.b.  — P^_ — pmax) we c a n d e r i v e  o f t o t a l demand "Q^" as t h e c o n e s  i n h e i g h t and r a d i i . versus those of  n  . . (2.2)  When we p l o t t h e v a l u e s  of  vary  "p^"  " Q ^ " , an a g g r e g a t e demand c u r v e i s  con-  s t r u c t e d f o r t h e m a r k e t a r e a d e l i m i t e d b y some r a d i u s "r  max  "(where v  p.= p . ) . * i *min'  w i t h an i n i t i a l  S i n c e we a r e i n f a c t  dealing &  p r o d u c e r and c o m p e t i t i o n i s a b s e n t ,  p a r t i c u l a r demand c u r v e i s t e r m e d t h e f r e e  this  s p a t i a l demand  curve. Although i n the o r i g i n a l l i t e r a t u r e Losch r e p r e s e n t s t h i s c u r v e as b e i n g c o n c a v e be shown t h a t , w i t h o u r i n i t i a l c u r v e must be c o n v e x  t o t h e o r i g i n i t may  postulates,  t h e demand  ( D e n i k e and P a r r , 1 9 7 0 ) .  a g g r e g a t e demand c u r v e  "D^" f a c i n g our i n i t i a l  With  the  producer  the next s t e p i s to determine the p r o f i t m a x i m i z i n g p r i c e and o u t p u t r e l a t i v e t o t h e Production costs cost curve  curve.  a r e r e p r e s e n t e d b y an a v e r a g e  "AC" t h a t d e s i g n a t e s  the c o s t of p r o d u c t i o n  p e r u n i t o f o u t p u t , w h i l e a m a r g i n a l c o s t c u r v e "MC" shows t h e i n c r e m e n t s i n t o t a l c o s t as o u t p u t i s  extended.  L o s c h i l l u s t r a t e s t h e c o s t c u r v e s as f a l l i n g a t e a c h  output  11  AC  Q  Q,  2  Demand/Output  F i g . 1. P r i c e and Output C o n d i t i o n s f o r the I n d i v i d u a l P r o d u c e r w i t h no C o m p e t i t i o n a n d w i t h Free E n t r y (from P a r r and D e n i k e , 1970).  level seem  under to  monopolistic  hinder  the  Marginal to  the  small  production  profits in  other  generality  revenue  increments  conditions,  of  of  "MR",  revenue  the  investments)  rate are  his  If of  we  other into  in  does  hand,  the  assume  return  present  this  not  argument.  the  brought  expansions.  (including  on  hut  that  refers  firm  that could  production  through  normal be  earned  costs,  12  then the p r o f i t maximizing p r i c e " p " and output " Q " 1  1  are determined by the i n t e r s e c t i o n of the marginal revenue and marginal cost c u r v e s . Losch and Berry (1967) argue that the  price  charged w i l l he determined by the i n t e r s e c t i o n of the average cost curve and demand curve but t h i s i s c l e a r l y not a p r o f i t maximizing i n t e r p r e t a t i o n . price l e v e l " P £ m  n  n  (where P ^ m  n  This p a r t i c u l a r  V-^) allows the maximum  number of customers to be provided from " 0 " and may w e l l improve t o t a l revenuer u n f o r t u n a t e l y , these gains  are  more than o f f s e t by climbing operation c o s t s . The s i t u a t i o n changes somewhat when we c o n s i d e r f r e e entry i n t o production a c t i v i t y .  The p o s s i b i l i t y of  a t t a i n i n g excess p r o f i t s encourages new e n t r i e s i n t o the market while d i s r u p t i n g the i n i t i a l e q u i l i b r i u m s i t u a t i o n . New producers continue to enter u n t i l each can only earn normal p r o f i t s .  Now the competitive demand curve " D " 2  f a c i n g the s i n g l e i n i t i a l producer (and a l l new producers) i s s h i f t e d to the l e f t of " D ^ " . This i s the case because with u n r e s t r i c t e d entry the s i n g l e o r i g i n a l f i r m l o s e s customers along the edges of i t s i n i t i a l market The new output e q u i l i b r i u m i s i n d i c a t e d by the  area. tangency  of the new demand curve with the average cost c u r v e . We n o t i c e , too,  that the demand curve s h i f t s even  f a r t h e r to the l e f t i f an excessive enter the market.  number of new producers  Now the demand curve l i e s below the  average production cost at every output l e v e l and v a r i o u s  13  s e l l e r s axe forced out of b u s i n e s s .  Hence, the p o i n t  of tangency i n d i c a t e s the minimum or t h r e s h o l d s i z e the  of  firm. By observing F i g . 1, we see t h a t with u n r e s t r i c t e d  entry the e q u i l i b r i u m o u t p u t " Q  2  n  I  s  lower and the  e q u i l i b r i u m p r i c e " p " i s higher r e l a t i v e to the p r i o r 2  monopolistic c o n d i t i o n s . the f a l l i n g c o s t c u r v e :  T h i s change i s explained by as entry i n t o the market c o n t i n u e s ,  production at any one s i t e i s l i m i t e d and p r i c e s as the o p p o r t u n i t i e s f o r scale economies are  rise  lost.  Two general c o n d i t i o n s , then, a r i s e as a consequence of u n l i m i t e d e n t r y J  (i)  the l o s s of excess p r o f i t s shows  that the number of firms i s maximized and ( i i )  e a c h producer  seeks a l o c a t i o n as d i s t a n t as p o s s i b l e from h i s neighbours' i In the i d e a l case where a l l s u p p l i e r s are e q u a l l y spaced over the homogeneous p l a i n , persists,  Christaller  a uniform t r i a n g u l a r  arrangement  and Losch argue t h a t t h i s i s  the  most favourable s p a t i a l e q u i l i b r i u m p a t t e r n and t h a t , a result,  as  a net of hexagonal market areas i s p r o v i d e d .  The monopolistic state d e f i n e s , i n C h r i s t a l l e r ' s terms,  the i d e a l range of the good or s e r v i c e being  offered.  The new i d e a l range i d e n t i f i e d by the higher  competitive f . o . b . p r i c e cannot be a t t a i n e d ,  however,  since the extent of each s e l l e r ' s market i s  restricted  by h i s adjacent competitors* market areas.  The new  boundary that d e l i m i t s the competitive market area i s c a l l e d the r e a l range of the good.  14 Obviously t h i s r e a l range i s not equal i n a l l directions,  (since i t d e f i n e s the extent o f i d e n t i c a l  hexagonal c e l l s ) and f o r t h i s reason we define i t being o n e - h a l f the distance of the same good.  between adjacent  as  producers  As Parr and Denike (1970) point  out,  what C h r i s t a l l e r terms the upper l i m i t on the range r e f e r s to e i t h e r the r e a l or the i d e a l form, depending on whether or not s p a t i a l competition e x i s t s . There a l s o e x i s t s a minimum l i m i t on the range of a good which C h r i s t a l l e r c a l l s the lower l i m i t .  Getis  and G e t i s (1966i222) state that t h i s encloses " . . .  the  number of consumers necessary to provide the mimimum s a l e s volume f o r the good to be produced and d i s t r i b u t e d profitably .  . . ."  For the s i n g l e o r i g i n a l producer  t h i s threshold range i s equal i n a l l d i r e c t i o n s . competition,  With  however, the f i r m earns only normal p r o f i t s  and only a minimum l e v e l of aggregate demand determines the t h r e s h o l d range.  Now the lower l i m i t of the range i s  c o i n c i d e n t with the r e a l range and i s not equal i n a l l directions. The fundamental 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 and  Losch toward a general understanding of the s i n g l e good p a t t e r n are roughly i d e n t i c a l .  Basically,  the  former  r e l i e s upon the concept of threshold range while the l a t t e r stipulates i s paramount.  that the attainment of normal  profits  Since the demand and cost f a c t o r s u n d e r l y i n g  the two concepts are e s s e n t i a l l y the same, we u s u a l l y  15 consider L o s c h ' s treatment as only a more e x p l i c i t or s o p h i s t i c a t e d approach to the same problem that confronted Christaller. Case of Many Goods Both C h r i s t a l l e r and Losch develop schemes i n t e g r a t i n g the features  for  of the v a r i o u s s i n g l e good n e t s .  We consider L o s c h ' s a n a l y s i s f i r s t ,  since i t i s the more  general of the two, and then go on to summarize C h r i s t a l l e r ' s ideas. Losch*s d e r i v a t i o n s r e s t upon a m o d i f i c a t i o n of our f i r s t postulate.  He f u r t h e r assumes that the r u r a l pop-  u l a t i o n i s d i s c o n t i n u o u s l y d i s t r i b u t e d over the p l a i n and that i n h a b i t a n t s r e s i d e i n b a s i c u n i t s (farmsteads or hamlets) uniform t r i a n g u l a r l a t t i c e .  isotropic  settlement  that are arranged on a Reasoning t h a t the d u a l i t y  of a g r i c u l t u r e and i n d u s t r y i d e a l l y leads to t h i s punctiform d i s t r i b u t i o n (compromises between proximity to food and i n d u s t r i a l p r o d u c t i o n , s u i t a b i l i t y to most aspects of a g r i c u l t u r a l p r o d u c t i o n ) , he s t i p u l a t e s that these b a s i c settlement u n i t s l i e at the centre of hexagonal farms. The s i g n i f i c a n c e of t h i s approach unfolds when he demona t r a t e s t h a t , with t h i s discontinuous r u r a l stratum population,  (i)  of  the p o s s i b l e s i z e s of the complementary  areas f o r d i f f e r e n t goods and s e r v i c e s  and ( i i )  the  number of basic settlement u n i t s these areas e n c l o s e , l i k e w i s e grow d i s c o n t i n u o u s l y .  16  To i l l u s t r a t e t h i s c o n d i t i o n , a concept that  is  fundamental to a l l c e n t r a l place d i s c u s s i o n s i s i n t r o d u c e d . Losch formulates a method f o r determining the number o f equivalent "basic settlements i n any market a r e a .  This  number equals the sum of the f o l l o w i n g t h r e e :  (i)  number of u n i t s (or p r e f e r a b l y l a t t i c e  interior  to the c e l l ,  (ii)  of the c e l l ,  and ( i i i )  at v e r t i c e s  points)  o n e - h a l f the number of u n i t s on edges o n e - t h i r d the number of u n i t s  of the c e l l (on a t r i a n g u l a r l a t t i c e ,  Using t h i s concept,  that  is).  Losch derives the p o s s i b l e market  area s i z e s i n terms of how many b a s i c settlement are p r o v i d e d .  the  In a s i m i l a r v e i n ,  units  s p a t i a l extent of the  "nA market areas i s given by  /3"» where " A " i s the area  of the smallest hexagonal c e l l and " n " represents  the number  of equivalent settlements enclosed. The r e s u l t s of t h i s r e s t r i c t i v e be obvious:  approach should  Losch i s arguing that minimum demand f o r  commodities o f f e r e d at various farmstead l o c a t i o n s  is  u s u a l l y met by market s i z e s that o f f e r an u n n e c e s s a r i l y large number of b a s i c consuming u n i t s .  The i n f l e x i b i l i t y  of Losch's d e r i v a t i o n means that moderate surplus p r o f i t s cannot be eliminated by f u r t h e r entry and some producers are c e r t a i n to b e n e f i t . that " .  He (1954:120) emphasizes f u r t h e r  . . not a l l p o s s i b l e market areas need occur i n  r e a l i t y . , . but conversely,  every a c t u a l market area  must be on the l i s t of p o s s i b l e o n e s , "  1? From h i s f o r m u l a t i o n of market s i z e s ,  Losch proceeds  to d i s c u s s i n t e g r a t i o n of the d i f f e r e n t market n e t s . combines them by ( i )  He  ensuring that each good has one  common supply center (the metropolis) and ( i i )  rotating  the nets so as to y i e l d a cogwheel p a t t e r n of s i x  sectors  with few and s i x sectors with many production s i t e s .  He  (1954:124) s t a t e s t h a t : . . . with t h i s arrangement the g r e a t e s t number of l o c a t i o n s c o i n c i d e , the maximum number of purchases can be made l o c a l l y , the sum of the minimum distances between i n d u s t r i a l l o c a t i o n s i s l e a s t , and i n consequence not only shipments but a l s o t r a n s p o r t l i n e s are reduced to a minimum. E s s e n t i a l l y he i s applying r a t i o n a l  agglomeration  assumptions i n order to derive a r e l a t e d set  of market  nets i n hope of d e f i n i n g some reasonable n o t i o n o f an economic r e g i o n .  The u n d e r l y i n g theme of h i s  entire  a n a l y t i c argument i s , i n f a c t , that t h i s derived arrangement i d e n t i f i e s the most o r d e r l y and s p a t i a l l y confined c l o s e d system of market areas.  As Losch ( 1 9 3 8 7 5 ) points outs !  "How many of these s e l f - s u f f i c i e n t systems w i l l come i n t o existence on our p l a i n depends merely upon the commodity which has the l a r g e s t  shipping r a d i u s , as long as there  are no economic l i m i t s to the s i z e of the c e n t r a l  city."  While the t h r u s t of Losch*s approach i n v o l v e s the concept of r e g i o n a l i n t e g r a t i o n , a very important p o r t i o n of the d i s c u s s i o n concerns the numbers of c o i n c i d e n t settlement u n i t s at various points on the lattice.  triangular  Losch does not d i s c l o s e , however, h i s i n t e r p r e -  t a t i o n of the s i z e of these aggregate settlement  units  18 except through a l i s t i n g of the f u n c t i o n s they p r o v i d e . I t seems,  though, that s e v e r a l i n t e r e s t i n g  properties  a r i s e when we superimpose the various market nets i n t h i s way: (i)  Some l a t t i c e  f u n c t i o n s than o t h e r s ; among farmsteads, (ii)  p o i n t s possess more economic  hence there i s d i f f e r e n t i a t i o n  towns,  etc.;  Some l a t t i c e  points possess the same number  of functions but these f u n c t i o n s may be d i f f e r e n t ;  hence  there i s s p e c i a l i z a t i o n among c e n t e r s ; (iii)  A l l lattice  points possess at l e a s t one  f u n c t i o n but there are few with many f u n c t i o n s ;  hence,  a numerical pyramid i n the number of m u l t i p l e - g o o d supply centers i s  suggested.  In c l o s i n g o f f the Loschian c a s e , we should emphasize that he c o n s i s t e n t l y r e q u i r e s that only one producer of a given good i s l o c a t e d i n the center where that good i s o f f e r e d .  This characteristic  limitation is  based s o l e l y on the r a t i o n a l scheme used to combine the independent market area n e t s . C h r i s t a l l e r ' s approach to the m u l t i - g o o d system i s l e s s general  and we present a summary of h i s i n t e r -  p r e t a t i o n i n a considerably more r i g o r o u s manner so as avoid r e p e t i t i o n at a l a t e r  to  time.  Generally we might consider a large r e g i o n i n which " y " d i f f e r e n t goods and s e r v i c e s are p r o v i d e d . Designating the f i r s t of these as " t ^ , B  we may rank these  19 c e n t r a l goods from  " t " to  " t " i n ascending order  1  t h r e s h o l d need? a c e n t e r  offering  "t  then,  the g r e a t e s t amount o f consumer p u r c h a s i n g to p e r s i s t  i n the  l e v e l center it  long run.  and,  We  according  i s associated with  the  of  requires  power f o r  term such a p l a c e  t o our  introductory  supply-  an  "M"  postulates,  l a r g e s t complementary a r e a  on  the homogeneous p l a i n . Of c o u r s e ,  o n l y as many "M"  i n the r e g i o n as t h e r e support  that supply  the  production  implicit  on the  "M"  plain.  l e v e l places  are  these  firms  s i t e s become a r r a n g e d sustained.  words,  offering  the  spatial  a triangular lattice  b o u n d a r i e s between the v a r i o u s  d e t e r m i n e d by the  so  In other  decay o r g a n i z e  l e v e l centers, The  Since  assumption t h a t f i r m s  good o f minimum d i s t a n c e  pattern of  and  "t  i s most e f f i c i e n t l y  e n f o r c i n g an  emerge  t h r e s h o l d markets a v a i l a b l e to  those f i r m s o f f e r i n g  compete s p a t i a l l y ,  by  are  l e v e l centers  r e a l range o f  develops "M"  "t "  form hexagon shaped market a r e a s about each c e n t r a l  place. If total thresholds profits  s a l e s l e v e l s are  f o r good " t  an e x a c t  multiple  these f i r m s e a r n o n l y normal  ( s i n c e t h e y l o c a t e so as t o m i n i m i z e  movement). r e g i o n are  E x c e s s p r o f i t s may s l i g h t l y greater  As we goods and  be  consumer  e a r n e d i f s a l e s i n the  than t h i s exact  noted e a r l i e r ,  the r a n g e s f o r  multiple. different  s e r v i c e s d e c l i n e with lower t h r e s h o l d  ments; t h e r e f o r e , g r e a t e r  of  and  g r e a t e r numbers o f  requiresurplus  20  consumers l i e b e t w e e n t h e t h r e s h o l d m a r k e t a r e a s l e v e l centers f o r these some g o o d " t  same c o m m o d i t i e s .  nate centers evolve t o supply  "ty_^"  "M"  T h e r e may  i " f o r which the i n t e r s t i t i a l  power r e a c h e s t h r e s h o l d volume i t s e l f .  of  be  purchasing  I n t h i s case,  alter-  (and a l l o t h e r  goods and s e r v i c e s o f l o w e r t h r e s h o l d need) a t p r i c e s b e l o w t h o s e a t t h e "M" l e v e l p l a c e s .  These " M - l " l e v e l  c e n t e r s s e r v i c e t h e a r e a s between the t h r e s h o l d ranges of those  goods s u p p l i e d e x c l u s i v e l y f r o m " M " l e v e l  B e r r y and G a r r i s o n (1958d) c a l l marginal  "t  centers.  ." a h i e r a r c h i a l  good. A s i m i l a r argument c a l l s f o r t h e emergence o f  1-2"  l e v e l c e n t e r s where some commodity " t  i s t h e new h i e r a r c h i a l m a r g i n a l good.  These  (j i) centers  s e r v i c e the areas between the t h r e s h o l d ranges o f those goods s u p p l i e d o n l y f r o m h i g h e r o r d e r c e n t e r s  ( i . e . "M"  and  ."  to  "M-l" l e v e l places).  L i k e w i s e , goods " t  through  "t-^", are provided a t these lower order c e n t e r s . A c o n s i s t e n t property of the C h r i s t a l l e r  i s t h a t a center of a given order develops from i t s n e i g h b o u r i n g  scheme  equidistant  c e n t e r s o f the next h i g h e s t  G e t i s and G e t i s ( 1 9 6 6 : 2 2 4 ) add t h a t " I n t h i s way,  order. consumer  movements a r e k e p t t o a minimum, and a maximum number o f demands a r e s a t i s f i e d f r o m a mimimum number o f c e n t e r s . " Therefore,  j u s t as i n t h e L o s c h i a n case, a l l c e n t r a l  p l a c e s a r e l o c a t e d on a t r i a n g u l a r  lattice.^  ^ L o s c h and C h r i s t a l l e r a r e aware o f d i f f e r e n t geometr i e s a s w e l l as d i f f e r e n t m a r k e t n e t s on t h e same l a t t i c e .  21 C e r t a i n fundamental c h a r a c t e r i s t i c s of the p a t t e r n we  have o u t l i n e d seem t o  (i)  spatial  exist:  A l l c e n t e r s h u t t h e s m a l l e s t have  other  c e n t e r s dependent u p o n them f o r t h e p r o v i s i o n o f g o o d s and  s e r v i c e s ? hence, the s e t of c e n t r a l p l a c e s d i s p l a y s  interdependencyi (ii) and  Each c e n t r a l p l a c e o f f e r s a l l the  s e r v i c e s t h a t dependent c e n t e r s s u p p l y p l u s  goods additional  ones? h e n c e , t h e c r i t e r i o n o f i n c r e m e n t a l b a s k e t s  o f goods  s u g g e s t t h a t t h e s e c o m m u n i t i e s show d i s c r e t e s t r a t i f i c a t i o n of  centralityi (iii)  While the s c a l e of the p a t t e r n i s changeable,  the i n t e r s t i t i a l  placement of orders i s a  distinctive  formi (iv)  There e x i s t s a d e f i n i t e n u m e r i c a l  a c c o r d i n g t o the o r d e r s o f the  centers.  B a s i c a l l y C h r i s t a l l e r i s forwarding a geometric  pyramid  simple  argument i n w h i c h t h e m a r k e t a r e a s i z e s  i n e x t e n t by a f a c t o r "q".  increase  T h i s scheme c o n t r a s t s w i t h  L o s c h ' s where t h e r e i s a c o n s i d e r a b l y s m o o t h e r p r o g r e s s i o n of p o s s i b l e market area s i z e s .  I f we  f u r t h e r assume a  discontinuous r u r a l population, then i n a  Christaller  q = 3 s y s t e m (where "q" r e p r e s e n t s t h e n e s t i n g f a c t o r for  market a r e a s ) , the p o s s i b l e market area s i z e s i n  terms o f e q u i v a l e n t b a s i c s e t t l e m e n t s are 3 t S i m i l a r l y , i f we m a r k e t a s "A",  9,  denote the a r e a l e x t e n t o f the  then the m u l t i p l i e r  "q A" z  27,  81,  etc.  smallest  (z = 0, 1, 2 ,  ,,.)  22 represents  the progression of a l l p o s s i b l e market  sizes.  D i f f e r e n c e s Between C h r i s t a l l e r and Losch Some of the basic d i f f e r e n c e s between the two approaches have already been mentioned. divergence between the schemes a r i s e s  The c r i t i c a l  out of the d i f f e r e n t  methods employed i n combining the market networks of i n d i v i d u a l goods.  Losch considers f i r s t the commodity  with the smallest market area and then introduces commodities with p r o g r e s s i v e l y l a r g e r t h r e s h o l d requirements. I n other words, L o s c h ' s approach i s  analytic!  i t develops i n stages from the most general ideas o f Chamberlinian economic theory (where the d i f f e r e n t i a t i o n of the producer's l o c a t i o n i s but one type differentiation).  of product  C h r i s t a l l e r ' s case i s r e l a t i v e l y  inductive as he argues from the most p a r t i c u l a r to  2  most g e n e r a l .  the  Since C h r i s t a l l e r begins with the most  " n a t i o n a l commodity" while Losch begins with the most " l o c a l commodity", von Boventer (1963:171)  suggests:  ""Central place theory has a l u c i d deductive structure f o r the general arguments proceed from a p r i o r i premises to statements concerning p a r t i c u l a r instances ( f o r example, the number of functions c o i n c i d e n t at a c e r t a i n l a t t i c e point). Furthermore, C h r i s t a l l e r ' s i n t e r p r e t a t i o n r e a l l y assumes that a community system e x i s t s and that a p a r t i c u l a r community (the "M" l e v e l place) i s dominant t h e r e i n , Losch, however, does not r e l y upon the f i r s t of these assumptions i n the same sense. As he envisages a g g l o meration from the most general case while C h r i s t a l l e r approaches i t from the most p a r t i c u l a r , we f e e l the l a t t e r has an added g r a i n of inductive reasoning.  23  In e c o n o m i c - h i s t o r i c a l terms, C h r i s t a l l e r ' s method o f d e r i v i n g h i s system may he thought o f as d e s c r i b i n g the population growth i n an area which at the beginning i s very t h i n l y populated. L o s c h ' s system would appear to be a more adequate d e s c r i p t i o n of a l a n d scape i n which a c e r t a i n dense ground s t r u c t u r e e x i s t s , w i t h , i n the beginning, e n t i r e l y s e l f - s u f f i c i e n t small s p a t i a l u n i t s ( i f new commodities with e v e r i n c r e a s i n g i n t e r n a l economies of production are introduced). I t i s s o l e l y t h i s d i f f e r e n c e i n the d e r i v a t i o n of the systems which has the e f f e c t t h a t L o s c h ' s system becomes much more complicated than Christaller's. As a consequence  of these opposite approaches we  may i d e n t i f y numerous s i g n i f i c a n t d i f f e r e n c e s between the two  schemes: (i)  The d e v i a t i o n s from the optimal l a y o u t  the i n d i v i d u a l goods and s e r v i c e s  for  are smaller i n the  Loschian system since a g r e a t e r number of p o s s i b l e market area s i z e s exist} the idea that r e l a t i v e l y few market area s i z e s are permissible i n the C h r i s t a l l e r system provides the opportunity f o r i n i t i a l excess (ii)  While the general geometric  (triangular  lattice)  arrangement  of centers i s at v a r i a n c e .  profits?  appearance  i s i d e n t i c a l f o r both, the  spatial  The Loschian  system has one e x t r a degree of freedom l e f t a f t e r  the  metropolis i s s p a t i a l l y f i x e d (hence the c i t y - r i c h and the c i t y - p o o r sectors) while C h r i s t a l l e r * s system i s  entirely  symmetrical? (iii)  Losch does not consider the a d d i t i o n a l  demands of a supplying population i n a c e n t r a l place nor does he include the possible e f f e c t s of multi-purpose trips.  Concomitantly, e i t h e r we must expect s i g n i f i c a n t  24 v a r i a t i o n i n the s i z e s of hexagonal c e l l s f o r the same commodity ( I s a r d , geometric  1956s2?0-273) o r ,  regularities,  i n order to  preserve  we must compose new assumptions  to eliminate such change i n the s p a t i a l demand f u n c t i o n (von Boventer,  I9631I7I-I72),  C h r i s t a l l e r (1966:50-55)  appears to i m p l i c i t l y include these f e a t u r e s i n h i s  scheme.  Therefore, the p h y s i c a l extent of the market areas f o r most commodities i n a multi-good system w i l l be smaller i n the C h r i s t a l l e r case than i n the (iv)  Loschian}  The l e v e l of urban c o n c e n t r a t i o n  also  v a r i e s c o n s i d e r a b l y between the two approaches.  I f we  suppose that the smallest market area s i z e i s the same i n b o t h , then we f i n d fewer c e n t r a l places  (and  therefore  a greater concentration of urban population) i n the C h r i s t a l l e r formulation.  Besides,  t h i s l e v e l of urban  concentration i s d i r e c t l y r e l a t e d to the value of the "q" factor  i n the simpler system?  (v)  Losch takes b e t t e r account of the  partial  s p e c i a l i z a t i o n of production i n smaller centers? C h r i s t a l l e r assumption that each higher order central place s u p p l i e s a l l the commodities (plus some a d d i t i o n a l ones) i s a r a t h e r restrictive  one.  In the r e a l world,  smaller communities  f r e q u e n t l y supply l a r g e r places with s p e c i a l i z e d goods and  services? (vi)  L o s c h ' s approach i s more r e s t r i c t i v e  c o n s i d e r i n g entry of competitors  at the same p l a c e .  in While  the l o c a t i o n of more than one producer of the same commodity  1  25 at  t h e same c e n t r a l p l a c e r u n s  a n a l y s i s , P a r r and D e n i k e entry  (at least  contrary to Loschian  (1971:572)  suggest  that  further  i n t h e s h o r t r u n ) may be t a k e n as  another  q u a l i f i c a t i o n o f the i n c o n s i s t e n c i e s r e l a t e d t o i n ( i i i ) above.  On t h e o t h e r hand, C h r i s t a l l e r ' s more i n d u c t i v e  d e r i v a t i o n does a l l o w f o r e n t r y o f c o m p e t i n g  producers  where e x c e s s p r o f i t s may be g a i n e d ; (vii) suggest  Christaller's  incremental baskets  c e r t a i n n a t u r a l agglomerative  o f goods  tendencies  (inter-  i n d u s t r y ) among f i r m s , b u t t h i s c o n d i t i o n i s n o t c h a r a c t e r istic  o f the L o s c h i a n (viii)  in  landscape;  A c l e a r l y i d e n t i f i a b l e h i e r a r c h y i s found  t h e C h r i s t a l l e r scheme b u t n o t i n t h e L o s c h i a n  fiable,  (identi-  t h a t i s , i n terms o f o r d e r s and n o t i n d i v i d u a l  functions); (ix)  I t i s much more d i f f i c u l t  t o make i n f e r e n c e s  about the p o p u l a t i o n s i z e s o f c e n t e r s i n the L o s c h i a n model; C h r i s t a l l e r ' s r i g i d population levels be  (that  h i e r a r c h y suggests  i s , different  of and  discrete  s i z e c l a s s e s ) may  assigned to centers o f d i f f e r e n t order.  d i f f e r e n c e even more a p p a r e n t ,  that  To make  we c a n r e l a x t h e  this  assumption  u n i f o r m p u r c h a s i n g power i n t h e C h r i s t a l l e r model still  develop  Garrison,  1958d).  discrete  Nevertheless  stratification  ( B e r r y and  i t i s s i g n i f i c a n t t h a t both  observers  r e a c h q u i t e s i m i l a r c o n c l u s i o n s w h i l e u s i n g somewhat d i f f e r e n t l i n e s o f reasoning.  I n both cases  complete  26 systems of networks are derived from an i n d e f i n i t e number of goods and s e r v i c e s through p a r t i a l e q u i l i b r i u m suggestions. In both systems the t r i a n g u l a r arrangement of production s i t e s and the hexagonal shaping each commodity are found to be  of market areas f o r optimal.  Perhaps von Boventer (1963*173) best sums up consequences of the two  the  formulations:  . . . ( i ) Losch*s system i s r e a l i s t i c and capable of an extension i n t h a t on a homogeneous p l a i n a s p e c i a l i z a t i o n of production i n d i f f e r e n t c e n t e r s , an i n t e r r e g i o n a l or inter-urban exchange of i n d u s t r i a l goods and a complicated network of markets i s derived, (ii) In i t s f i n a l r e s u l t , as f a r as the o v e r a l l system of a h i e r a r c h y of c i t i e s or c e n t r a l places i s concerned, where the i n d i v i d u a l economic a c t i v i t i e s are neglected, C h r i s t a l l e r ' s system gives both a b e t t e r d e s c r i p t i o n of r e a l i t y - at l e a s t with regard to Southern Germany i n the ' t h i r t i e s - and has the advantage of being simpler, or more elegant, whereas the Losch system i s very d i f f i c u l t to t e s t at a l l . Scope and Nature of the C l a s s i c a l Argument The c l a s s i c a l l i t e r a t u r e a t t r a c t s considerable comment from geographers f o r diverse reasons.  I t i s not  our purpose here to r i g o r o u s l y q u a l i f y the c e n t r a l place approach but to make c e r t a i n that the reader i s f u l l y aware of the more important  drawbacks of the  theory.  A f i r s t group of c r i t i c i s m s deals with the a c t u a l assumptions employed by C h r i s t a l l e r and Losch.  Isard  (1956:274), f o r one, points out that Loschian a n a l y s i s i s l i m i t e d only " . . .  i n s i t u a t i o n s where raw m a t e r i a l s  are not r e q u i r e d (as i n s e r v i c e i n d u s t r i e s ) or are ubiquitous and everywhere a v a i l a b l e at the same c o s t s . "  In other  2? words, the c e n t r a l place approach i s o m i t t i n g too many production s i t e s (whether market or m a t e r i a l  oriented)  that are s t r o n g l y i n f l u e n c e d by the nature of input prices. A l s o , the theory i s r e s t r i c t e d to those a c t i v i t i e s not a f f e c t e d by more s e l e c t i v e p r i c i n g p o l i c i e s , the i n t r o d u c t i o n of which s u b s t a n t i a l l y a l t e r s the s p a t i a l extent of markets f o r many goods. However, t h i s only means that c e n t r a l place theory has a somewhat smaller domain of economic operations than some o p t i m i s t i c observers would give i t .  The theory does  possess an a n a l y t i c framework that allows i t to e x p l a i n c e r t a i n hypotheses, with a t t e n t i o n being devoted to the a p p l i c a b i l i t y of i t s assumptions, \  On the other hand, some c r i t i c i s m i s devoted to the s t a t i c or d e t e r m i n i s t i c nature of the theory.  As  Pred (1967*99) s t a t e s : In order f o r the fundamental precepts o f c e n t r a l place theory to be f a i t h f u l l y reproduced i n a r e a l world s i t u a t i o n i t would be necessary f o r every t e r t i a r y - a c t i v i t y s u p p l i e r (entrepreneur, f i r m ) to make an optimal l o c a t i o n d e c i s i o n ( s i t e and s i t u a t i o n s e l e c t i o n ) and every t e r t i a r y - a c t i v i t y consumer ( s e r v i c e c l i e n t ) to make a t o t a l l y r a t i o n a l journeyto-consume d e c i s i o n . This d i f f e r e n c e seemingly a r i s e s from disagreement as  to  what e n t a i l s a s a t i s f a c t o r y framework f o r e x p l a n a t i o n . C l a s s i c a l theory r e s t s s t r o n g l y upon economic assumptions and E u c l i d i a n geometric n o t i o n s .  E f f o r t s continue to be  channelled along f u n c t i o n a l ( r o l e s of phenomena w i t h i n organizations) and morphometric ( s p a t i a l structure and  28 form) l i n e s  (see H a r v e y , 1 9 6 9 : 7 8 - 8 3 ) .  c e n t r a l place  t h i n k i n g i s t h a t the  networks i s c o n s t a n t l y run optimal  spatial  The and  adjusting  system o f market  itself  toward some  a p p r o a c h e s r e s t upon  s o c i o l o g i c a l postulates  view o f decision-making.  so as t o a v o i d  Decisions  where t o buy  are v a r i a b l e due  i n actors'  (firms, buyers) information information.  psychological a  mechanistic  concerning  l o c a t e and  to  and  where  c a p a b i l i t y to  I n many r e s p e c t s ,  improvement o f the  seems t o o c c u r w i t h t i m e . m a t r i x a p p r o a c h e s may short run. is  still  On  the  production  confined Besides,  a l s o t e n d t o be routes,  I n t h i s way,  (see C u r r y ,  the more t e m p o r a l o r g e n e t i c d e s c r i p t i v e i n the  sense t h a t  assigned.  However, w i t h s t a g e - b y - s t a g e  simulation  studies  1962).  approaches transport  e t c , are  randomly  qualifications,  a f f o r d e x c e l l e n t p i c t u r e s of  reality  1963).  I n any the  the  hand, consumer d e c i s i o n - m a k i n g  economic a c t i v i t i e s , m i g r a t i o n ,  (Morrill,  behaviorial  to d e s c r i p t i v e t h i n k i n g  do  effect  location situation  a f f o r d good p r e d i c t i o n s i n  other  to  differences  d e c i s i o n - m a k i n g does i n v o l v e n o n - o p t i m a l cause and b e h a v i o r , but  long  layout.  behavioral  employ t h a t  Inherent i n such  c a s e , no  other  c l a s s i c a l v i e w i n the way  p r o c e s s e s and a synthesis  the  interpretation i t l i n k s the  governing  r e s u l t i n g s p a t i a l s t r u c t u r e and  t h a t Harvey ( I 9 6 9 s l 2 7 )  geographic theory.  challenges  Or  form,  suggests i s c e n t r a l to  t o take a n o t h e r v i e w , " . . .  the  29 synoptic use  as  feasibility  a basis  o f a model i s enough t o  for empirical  research.  e n q u i r e whether a s i t u a t i o n w h i c h c o u l d exist,  even i f one  has  no  Christaller  Also,  (1966:111-112)  and  Losch  to  e x i s t , does account  chaos."  (Marshall,  i t i s c e r t a i n l y to t h e i r c r e d i t  e n t i r e l y aware o f the place  It is valid  a i r - t i g h t l o g i c to  f o r i t s emergence from n o n - e x i s t e n c e o r 1969s4o).  justify its  that  ( 1 9 5 ^ » * i i i ) seem  shortcomings i n t h e i r  central  derivations. We  s h o u l d v i e w t h e i r models as  t o complement the  (that  s o l e l y ) with those of  i s , the  market l o c a t i o n s , routes, etc.).-^  first  attempts  ideas of s p a t i a l d i f f e r e n t i a t i o n  t o economic f a c t o r s equilibrium  the  intraregional  simultaneous determination  production  centers,  B a s i c a l l y , they are  the  essentials  of  transportation  deriving  a  system  o f economic o r d e r t h r o u g h a minimum o f a s s u m p t i o n s hope o f e x p l a i n i n g  (due  in  of s p a t i a l d i f f e r e n -  tiation. E x t e n s i o n s o f the We  have o u t l i n e d  p a r t i a l t h e o r y o f the  Classical Literature  c e n t r a l p l a c e t h e o r y as  l o c a t i o n , s i z e , n a t u r e , and  t i o n of a c t i v i t y c l u s t e r s .  I n many i n s t a n c e s ,  the distribu-  functional  -'Depending upon the s c a l e a t w h i c h we examine the c e n t r a l p l a c e system, e q u i l i b r i u m may be c o n s i d e r e d i n t r a r e g i o n a l ( a l l a c t i v i t i e s ) or i n t e r r e g i o n a l ( s u b s e t s of a c t i v i t i e s ) .  30 interdependence  amongst c e n t e r s and  t h e i r market  areas  has "been s t r e s s e d : t h e r e f o r e , i t i s n o t u n n a t u r a l t h e term  "system" has  combination we  "become l o o s l y a s s o c i a t e d w i t h  o f market n e t s .  e x p l i c i t l y develop  some o f t h e more b a s i c f e a t u r e s o f  f o r m u l a t i o n of c i t y  profitable,  then, to f i r s t  by s e e i n g how systems  the  I n the f o l l o w i n g d i s c u s s i o n  a c e n t r a l p l a c e system i n o r d e r t o have added f o r the  that  s i z e models. expand on t h e  c e n t r a l place theory t i e s  rationale  I t may term into  be  "system" general  theory:  A system i s a s e t o f o b j e c t s ( f o r example, c e n t r a l p l a c e s ) , a t t r i b u t e s o f the o b j e c t s ( p o p u l a t i o n , e s t a b lishments, business types, t r a f f i c generated), i n t e r r e l a t i o n s among t h e o b j e c t s ( m i d p o i n t l o c a t i o n s f o r l o w e r l e v e l c e n t e r s , u n i f o r m s p a c i n g a t any g i v e n l e v e l ) and among t h e a t t r i b u t e s ( t h e g r a p h s o f l o g l o g r e l a t i o n s h i p s ) and i n t e r d e p e n d e n c i e s o f o b j e c t s and a t t r i b u t e s ( t h e c e n t r a l p l a c e h i e r a r c h y ) . ( B e r r y ,  1967:76-77).  A g g r e g a t e R e l a t i o n s and Considerable  Elemental  effort  Components  i s d i r e c t e d toward  the fundamental interdependencies  summarizing  (that i s , empirical  s t r u c t u r a l r e l a t i o n s h i p s ) o f c e n t r a l p l a c e systems i n a closely knit  set of equations,  B e r r y , Barnum, and study areas  Tennant, 1962;  f o r these  place theory to a reasonable  facilitate  Barnum,  Marshall, I969).  empirical i n v e s t i g a t i o n s are,  r u r a l r e g i o n s so as t o f u l f i l l  is also carried  ( B e r r y and  on a t the  the p o s t u l a t e s o f  degree.  The typically,  central  Empirical research  intraurban l e v e l  i n order  integrating r e g u l a r i t i e s at d i f f e r e n t  (Berry, I967),  1962;  To d e s c r i b e t h e s e b a s i c  to  scales,  relationships  31 we  must p r o v i d e  definitions  for several variables!  p:  the p o p u l a t i o n o f a c e n t r a l  r:  t h e p o p u l a t i o n o f the complementary  place; area  s e r v e d by a c e n t r a l p l a c e ; P:  the t o t a l p o p u l a t i o n s e r v e d by a c e n t r a l  A:  the  s p a t i a l extent  o f the  place;  complementary  areas being s e r v i c e d ; Qpj  the p o p u l a t i o n d e n s i t y o f the e n t i r e  area  s e r v i c e d by a c e n t e r , i n c l u d i n g t h e o u t l y i n g a r e a and Qn  the  itself;  p o p u l a t i o n d e n s i t y o f the o u t l y i n g  trade y:  the c e n t e r  area;  t h e number o f c e n t r a l f u n c t i o n s business  (separate  t y p e s ) o f f e r e d by a c e n t r a l  place;  hence, the h i g h e s t l e v e l c e n t r a l f u n c t i o n p e r f o r m e d by t h e Dyt  center;  the maximum d i s t a n c e t h a t c u s t o m e r s  travel  to a c e n t r a l p l a c e ; t h e r e f o r e the range good  "y";  f:  reads  logt  i n d i c a t e s base 10  To b e g i n w i t h , we  of  may  "some f u n c t i o n o f " ; logarithms.  d e f i n e s e v e r a l e q u a l i t i e s as  welli  P = p+r  , . . (El)  A = f(Dy)  . . .  r = AQr  = f(Dy)Qr  . . .  (E3)  p = AQp  = f(Dy)Qp  . . .  (E4)  (E2)  32 One g e n e r a l l y e x p e c t s t h a t l a r g e r c e n t r a l p l a c e s have more c e n t r a l f u n c t i o n s , more e s t a b l i s h m e n t s , areas  than smaller centers.  and l a r g e r m a r k e t  L o g l i n e a r r e l a t i o n s h i p s seem  t o p e r s i s t between t h e number o f f u n c t i o n s p e r f o r m e d i n c e n t r a l p l a c e s and ( i ) t h e p o p u l a t i o n s ( i i ) the t o t a l populations Barnum, and T e n n a n t , 1969:163-164). the  i n those  served b y those  places.  Besides,  a l i n e a r p a t t e r n seems t o i l l u s t r a t e  t o t r a v e l t o a c e n t r a l place  of functions o f f e r e d there.^ 1962:100-101; B e r r y ,  (Berry,  1962*69; B e r r y , 1 9 6 8 : 3 7 - 3 8 « M a r s h a l l ,  a s s o c i a t i o n between t h e maximum d i s t a n c e  are w i l l i n g  places o r  consumers  and t h e number  ( B e r r y , Barnum, and T e n n a n t  1968:28).  These and s i m i l a r  may be f o r m a l i z e d i n s t r u c t u r a l  arguments  equations:  l o g p = a-j^ + b ^ y  . . . (2.3)  log P = a  + b y  . . . (2.4)  Dy = a ^ + b ^ y  . . . (2.5)  2  2  where a > a^> a ^ 2  Various from  i m p l i c a t i o n s may be drawn f r o m t h e s e  (2.3)  & (2.4)  l o gp = a  ^ - a b  (2.3)  & (2.5) l o g P = a b 1  3  2  ( 2 . 3 ) & ( 2 . 4 ) l o g P = ai2b1  b  x  b  3  - a^bg + b \  b  l o gP . .  ,(2.6a)  2  - a^j, + \ b  "  ^ + b  equations:  Dy  . . .(2.6b)  3  2  l o gp . . .(2.7a)  l  T h i s c o n t r a d i c t s a s t a t e m e n t i n B e r r y and Barnum, 1962 b u t seems j u s t i f i e d by t h e e m p i r i c a l e v i d e n c e r e f e r r e d t o above.  33 (2.4) & (2,5) l o g P = a b 2  - a b  3  3  t "  (2.6a) & (E4) l o g A = a ^  b  2  b  3  a-jbg  b,  (2.6b) & (E4) l o g A = a b 2  b  3  b  £  b~  "  (2.6a) & (E3)  log r =  a  b 2  r  b  b  bT™  +  1  "  (2.6b) & (E3)  l o g r = a,b^  -  l  o  s  p  +  l  bT  o  (2.8b)  g  Q p  . . .  (2.9a)  a-b b Dy + l o g Or 2_£ + _ £ Qp 9  supplement t h o s e  f o r establishments,  . . .  1  *3 These s t a t e m e n t s  2  (2.8a)  Dy - l o g Qp  £  3  l 2  a  . . .  b_  3  (2.7b)  l o g p - l o g Qp  2  1  - a b  3  . . .  3  + b,  1  "  Dy  £  9  *3  t h a t may  .  be  . .  (2.9b)  formulated  functional units, etc, i n a  similar  way. The r e s u l t s  of these  equations,  however,  suggest  e m p i r i c a l f e a t u r e s o f c e n t r a l p l a c e systems t h a t a r e s i m p l e r than those 1962}  Berry, (i)  c i t e d e l s e w h e r e (Berry, Barnum and T e n n a n t ,  1964): C e n t r a l place populations are c o n s t r a i n e d  o n l y by the t o t a l p o p u l a t i o n s t h e y s e r v i c e . t a t i o n strengthens  an e l e m e n t a r y  This  economic base  interpre-  rationale  f o r c e n t r a l p l a c e systems s i n c e i t a v o i d s g r o s s d e n s i t y as an e x p l i c i t v a r i a b l e . and " b " d e t e r m i n e how 2  Besides,  the r a t i o  P  the c o e f f i c i e n t s  "b^"  /P " changes as c e n t e r s  34 take  on more and more f u n c t i o n s .  Empirical  evidence  ( B e r r y , Barnum, and T e n n a n t , 1962: F i g s . 5 & 6) s u g g e s t s that  " b " i s s l i g h t l y g r e a t e r t h a n "b^" and t h a t , a s a 2  c o n s e q u e n c e , community p o p u l a t i o n s  assume a  p r o p o r t i o n o f t o t a l market p o p u l a t i o n s (ii)  The s p a t i a l e x t e n t  a r e a about a c e n t r a l p l a c e p o p u l a t i o n and g r o s s area  decreasing  as t h e y  grow l a r g e r .  o f t h e complementary  i s c o n s t r a i n e d by t h e t o t a l  density.  This suggests that the  i s a f u n c t i o n o f t h e number o f b u s i n e s s  types  by a c e n t r a l p l a c e b u t t h a t t h i s a r e a d i m i n i s h e s densities  offered  as o v e r a l l  increase.  (iii) complementary  The n o n - c e n t r a l area  places o f the trade  place population  o f the  (that i s , residents i n smaller c e n t r a l area or r u r a l  on t h e number o f b u s i n e s s  types  inhabitants)  depends  i n t h e market c e n t e r and "Or /  the n a t u r e populations  o f the d e n s i t y r a t i o  /Qp".  external  a c c o u n t f o r an i n c r e a s i n g p r o p o r t i o n o f t o t a l  p o p u l a t i o n as market a r e a s  expand  (other things  I n s h o r t summary, t h e i m p o r t a n t of c e n t r a l place  systems a p p e a r t o be  t r i b u t a r y area populations, and t h e p h y s i c a l e x t e n t  areas  equal).  aggregate r e l a t i o n s exponential:  t o t a l market a r e a  o f these  f u n c t i o n s o f the p o p u l a t i o n Also,  These  populations,  are a l l exponential  sizes of central places.  (2,6b) s u g g e s t s t h a t t h e range o f t h e h i g h e s t  good p r o v i d e d  by a c e n t r a l place  t o t h a t community's  population.  level  i s exponentially related Moreover, the r e l a t i o n s h i p  between t h e growth r a t e s o f t h e s p a t i a l components  and t h a t  35 o f the  a s s o c i a t e d m a r k e t c e n t e r depends upon  constraints  i n each  Empirical be  i n v e s t i g a t i o n i n d i c a t e s that there l a r g e r trade  l a r g e r c e n t e r s a r e more w i d e l y  centers.  areas  statements of c e n t r a l place  size,  spacing,  B e r r y and  spaced than  and  theory  discontinuities  (1962) add  o f a r e a and I f we  recall  Hierarchial  identify a t any  the e x i s t e n c e  gross the  of discrete  l i m i t s express  o f communities a t p a r t i c u l a r l e v e l s  the  of  to density c o n s t r a i n t s .  Structure  Those r e a d e r s l i t e r a t u r e may  w e l l be  f a m i l i a r with  the c e n t r a l p l a c e  q u e s t i o n i n g the  "hierarchy" to t h i s point.  erable confusion and  original  t h a t major t h r u s t o f  of c e n t r a l p l a c e s , then these  c e n t r a l i t y with regard  term  smaller  to t h e i r d e r i v a t i o n s  population served  C h r i s t a l l e r model c o n c e r n i n g  maximum s i z e  that  f u n c t i o n s of urban c e n t e r s .  Barnum  population density.  must  concerning  a s e t of e m p i r i c a l l y based i n e q u a l i t i e s t h a t  orders  and  Such p r o p e r t i e s e n t i r e l y r e - i n f o r c e the  analytic the  case.  fewer l a r g e r c e n t e r s w i t h  these  particular  a r i s e s o v e r the  i t r e m a i n s the a u t h o r ' s  avoidance of  I t i s clear that common use  consid-  of that  contention that proper  p r e t a t i o n c a n o n l y come a f t e r a r e v i e w o f the  the  term inter-  theoretical  literature. The  most i m p o r t a n t  n o t i o n t o remember a b o u t  " h i e r a r c h y " i s t h a t i t i s a s p a t i a l term when employed to describe  features of a c e n t r a l place  system.  Therefore,  I 36 it  confines to a c i t y  system o n l y s p a t i a l l y r e l a t e d c e n t e r s Lukermann ( 1 9 6 6 )  among any s e t o f c e n t e r s . must be e x p l i c i t and  s t a t e s t h a t we  about the d i r e c t i o n s o f p h y s i c a l  movement when d i s c u s s i n g h i e r a r c h i a l c o n t r o l ?  words, i t i s n o t s u f f i c i e n t populations,  circulation i n other  t o o n l y enumerate f u n c t i o n s ,  e t c , i n a s e t o f c i t i e s and e x t e n d o u r knowledge  of h i e r a r c h i a l structure. Therefore, functional  " h i e r a r c h y " i m p l i e s b o t h s p a t i a l and  (order) r e s t r i c t i o n s .  T h i s s h o u l d be i m m e d i a t e l y  apparent, since the h i e r a r c h y bridges  the  interdependencies  o f a t t r i b u t e s and o b j e c t s f o r t h e e n t i r e c e n t r a l p l a c e system,  Functional r e s t r i c t i o n  number o f c i t i e s h a v i n g  i s measured b y ( i ) t h e  the f u n c t i o n , ( i i ) the s i z e o f  the p o p u l a t i o n s e r v e d b y t h e f u n c t i o n , and ( i i i ) t h e area o f the p o p u l a t i o n served by the f u n c t i o n  (Lukermann,  1966). Spatial restriction, by  ( i ) the interdepedence  on t h e o t h e r hand, i s d e t e r m i n e d  o f c e n t e r s and ( i i ) t h e i n t e r s t i t i a l  placement o f o r d e r s . The city  h i e r a r c h y determines the o r g a n i z a t i o n o f a  system i n s p a c e .  specialization,  Seen as a consequence o f t e r r i t o r i a l  f u n c t i o n a l d i f f e r e n t i a t i o n , and degree  o f i n t e r a c t i o n among a c t i v i t y nodes, i t emerges with  some m a t u r i t y  only  i n the r e g i o n a l urban s t r u c t u r e .  Once  t h e r e , though, i t t e n d s t o d e f i n e t h e l i m i t s o f i n d i v i d u a l growth among t h e u r b a n p l a c e s .  37 The C e n t r a l  P l a c e System R e c o n s i d e r e d  The s p e c i f i c a t i o n o f a c i t y the  delimitation  the  hierarchy  o f an i n i t i a l  c o n c e p t , we  lower order t h a t  system r e s t s  upon  center f o r inquiry.  Using  can i d e n t i f y those c e n t e r s o f  are commercially l i n k e d t o the c e n t r a l  place. Recalling  our d i s c u s s i o n  o f the C h r i s t a l l e r model,  we began w i t h t h e emergence o f an "M" the  set^t^, tg, . .  the  interstially  {t  t  1 #  2  , , . ., t  difference city  by  t ^ of functions.  situated  "M-1"  ^} where  offering  On t h e o t h e r hand,  centers o f f e r the s e t  function  between t h e two s e t s .  system i s d e v e l o p e d w i t h  control  l e v e l center  "M"  type  "M"  i s the  I n t h i s manner,  a  h i e r a r c h i a l l e v e l s and  i s m a i n t a i n e d by the p r o p e r t y t h a t  functions  provided  a c e n t e r a t one l e v e l a r e p r o p e r s u b s e t s o f t h o s e  functions  given at higher h i e r a r c h i a l l e v e l s .  Besides, a concomitant feature place This  system i s i t s c l o s u r e economic i n t e g r a t i o n  o f any  central  or f u n c t i o n a l wholeness.  i s d e t e r m i n e d by t h e l i n e s o f  i n t e r d e p e n d e n c e and t h e o r d e r s i n t h e h i e r a r c h y .  It is  a c r e d i t t o C h r i s t a l l e r and L o s c h t h a t  central  p l a c e models t h a t if  combine f u n c t i o n a l  only i n a p a r t i a l sense.  they o f f e r  and s p a t i a l c o n t r o l ,  3  Chapter CITY S I Z E MODELS AND  DISTRIBUTIONS  Review o f t h e H i e r a r c h i a l M o d e l s The h i e r a r c h i a l a p p r o a c h i s e x p l i c i t derivation of existing c i t y  s i z e models.  to the  These models  a r e p r e s e n t l y c o n f i n e d t o t h e s i m p l e r b u t more  plausible  Christaller  interesting  if  interpretation?  a model b a s e d  developed.  indeed  i t would be  upon t h e L o s c h i a n l a n d s c a p e  As a r e s u l t ,  the c i t y  were  s i z e models evade t h e  p o s t u l a t e o f even p u r c h a s i n g power d i s t r i b u t i o n we r e t a i n i t f o r i l l u s t r a t i v e to  cases o f d i s c r e t e  similarly  (although  ease), but are r e s t r i c t e d  functional ordering.  Terms and N o t a t i o n Beckmann (1958) p r o v i d e s the i n i t i a l s i z e s but the r a t h e r debatable  model o f c i t y  p r o p e r t i e s o f t h i s approach  c o u p l e d w i t h t h e more r e c e n t e f f o r t s Beckmann and o t h e r s , r e q u i r e s t h a t we  i n the s u b j e c t by first  study a g e n e r a l -  i z e d model. However,  b e f o r e d e p a r t i n g on a r a t h e r r i g o r o u s  d i s c u s s i o n , t h e r e a d e r s h o u l d be a c q u a i n t e d w i t h t h e t e r m i n o l o g y and n o t a t i o n o f t h e s u b j e c t . t h a t p r o v i d e s t h e "m"th bundle s e r v i c e s i s s a i d t o possess  place  ( b a s k e t ) o f goods and  f u n c t i o n type  38  A central  "m"  (where  "m"  39 represents "1" and "m"  one  o f the d i s t i n c t  "M");  but not  f u n c t i o n subsets  a l s o , i f that place provides "m  + 1",  area, i t i s said to that surrounding  f u n c t i o n type  i t i s s a i d t o have o r d e r  t h e c e n t e r p r o v i d e s the  "m"th  basket  "m"-dominate the  area  for a  between  "m".  complementary  entire population i n  ( i n c l u d i n g the r u r a l p o p u l a t i o n  t h e u r b a n p o p u l a t i o n i n t h a t c e n t e r and  "q" i n d i c a t e s the n e s t i n g f a c t o r  f u n c t i o n type  t h a t are  "m"  system,*  The  2;  "m-1"  p l a c e ) and  d e n o t e s t h e t o t a l number o f f u n c t i o n t y p e s out the  system  (see C h a p t e r  t h a t i s , t h e number o f p l a c e s w i t h "m"-dominated by an o r d e r  and  a l l smaller centers).  Dacey (1966) r e f e r s t o the c e n t r a l p l a c e where  Since  offered  "M" through-  f o l l o w i n g n o t a t i o n i s common i n the  literatures mt  The  f u n c t i o n p r o v i d e d by a p l a c e s  the l e v e l (m = 1,  2,  i n the h i e r a r c h y as w e l l . , ,, M);  only function nt  The  size class  smallest centers  offer  one; (n = 1,  2,  s i n g l e l a r g e s t c e n t e r and market a r e a , n = M:  hence,  , . ., M) t f o r t h e i t s associated  1;  the  total  number o f f u n c t i o n s p r o v i d e d  the  system o r the number o f l e v e l s on  hierarchy; notice m = M  - n + 1  in  the  and  ^ " I m p l i c i t t o the c e n t r a l p l a c e scheme i s t h a t these f u n c t i o n subsets remain r e l a t i v e l y constant i n n a t u r e 1 t h e r e f o r e we u s u a l l y r e f e r t o them as s i m p l y "functions".  n = M - m + 1? r : m  Pjjjt  t h e p o p u l a t i o n o f t h e complementary the  "m th l e v e l o f t h e h i e r a r c h y ?  the  population i s entirely rural?  level Pj^x  when m = 1  M  the p o p u l a t i o n of a center  a r e a on  on t h e "m"th  of the hierarchy?  the population of the l a r g e s t center  i n the  system? P « m  the t o t a l the  k  m  :  population  "m"th l e v e l  served  by a c e n t e r  on  of the hierarchy?  a service or technology m u l t i p l i e r  that  denotes the p r o p o r t i o n o f the p o p u l a t i o n i n an "m" l e v e l complementary a r e a  p l u s an  (m 6. m* - M) l e v e l c e n t r a l p l a c e the  complementary  a n "m"  area  "m*"  (servicing  i n the c a p a c i t y o f  l e v e l place) that i s required to  r e s i d e i n t h e "m*  n  provide  l e v e l place  i n order to  f u n c t i o n "m* " t o b o t h ; a n e c e s s a r y H  condition exists that  > "  £.—i  k < m  1;  m = 1 ki  a simple the  proportionality factor that r e l a t e s  p o p u l a t i o n o f a c i t y t o the t o t a l  served  by t h a t c i t y ;  population  a necessary c o n d i t i o n  e x i s t s that 0 < k < 1; q«  t h e n e s t i n g f a c t o r f o r market  areas;  si  t h e e q u i v a l e n t number o f c e n t e r s  o f the  "m - l " s t l e v e l t h a t a r e dominated by a n  41 order  "m"  placej  place  systems r e q u i r e s  where " s " and Model I :  The  model t h a t i n t e r e s t s us o f h i s model i s r a t h e r  not  hy  an  constants.  o u t l i n e s the  general  several complicated  The  reader that  are  (1970)  McPherson  fashion.  d e r i v a t i o n of urban populations  first  size  development  the  formulations  Beckmann and  city  The  s k e t c h y , though, and  postulates, besides  model.  "q" are "both  1  s = q -  i d e n t i c a l model i n a more e l e g a n t  The three  that  i n this discussion.  e x p l i c i t l y derived.  evolve  geometry o f c e n t r a l  G e n e r a l Case  Dacey (1966) f i r s t  i s greeted  the  r e s t s upon  those e s s e n t i a l to  a s s u m p t i o n s t a t e s t h a t the  Christaller's amount  of  employment a s s o c i a t e d w i t h a f u n c t i o n depends on the population  supporting  that function.  The  second  states that population  i n a c e n t r a l place  is a  f u n c t i o n o f employment  (see Dacey, I 9 6 6 ) .  The  of these p o s t u l a t e s characteristic that  "k  m  P  combination "k^  A t h i r d assumption i s  for a l l centers  o n l y the  r u r a l population X  linear  offering function  "m".  b e g i n d e s c r i p t i o n o f the model w i t h t h o s e  providing  dispersed  assumption  to a s e r v i c e m u l t i p l i e r  o f each f u n c t i o n .  " is identical We  centers  leads  entire  = k =  x  (p  h?l l-k  x  +  first  f u n c t i o n to a  uniformly  r^»  r ) x  ... x  (3.D  42 In t h i s  "Tz^"  case  denotes the p r o p o r t i o n o f the  p o p u l a t i o n demanding f u n c t i o n one center providing i t .  Now,  c o n s i d e r the  center that provides both P  2  =  k  i (P  +  i)  r  2  the  +  k  2  T h i s s i m p l y means t h a t t h e c e n t e r i s determined (i)  in  first  ( p  2  r  case  and  second f u n c t i o n s :  2^  ' • *  t o the  t h e c a p a c i t y o f a "p^"  order  s u p p l y o f f u n c t i o n two  (note  "P " 2  serves " r ^ "  the  t o the s e c o n d l e v e l c e n t e r and  level  a  area.  f a s h i o n , we  "m"th  a  center):  s e c o n d l e v e l complementary  in a  the  s e c o n d l e v e l c e n t e r and  A p o p u l a t i o n group t h a t i s r e l a t e d t o  Reasoning i n t h i s  ^  ( 3 , 2  A p o p u l a t i o n group t h a t i s r e l a t e d t o  (ii)  the  of a l a r g e r  p o p u l a t i o n o f a second  l e v e l complementary a r e a  resident  +  population of  hys  s u p p l y o f f u n c t i o n one first  t o the  total  may  d e t e r m i n e the  population  center:  m Pm  =  l?i  k  i  T h i s premise r o o t e d for generating  (  P  m  +  r  i  in Christaller  next  order  "m"  have " s " s a t e l l i t e  cities  ,  3  )  rural  center.  s t e p i s t o d e t e r m i n e the n a t u r e  complementary a r e a p o p u l a t i o n s  3  complementary  a r e p r o p o r t i o n a l t o the b a s i c  p o p u l a t i o n s e r v e d by a f i r s t  (  thinking i s sufficient  a model i n which c e n t e r and  area populations  The  • • *  )  "r^",. of order  Centers  of  of  the  order  "m-1", each o f  which i s s u r r o u n d e d by a complementary a r e a o f  population  43 B  r  m  ^".  I n o t h e r words, t h e p o p u l a t i o n o f t h e complementary  a r e a a b o u t a "m"  level  market a r e a o f o r d e r  center consists of "r  "m-l" s u r r o u n d i n g  a p o p u l a t i o n o f s ( p _ 2 + _i^ r  m  t h e i r t r i b u t a r y areas.  m  r  =  s p  m-l  +  (  i  1  cities  plus and  That i s ?  +  s  )  r  m-l  • • •  To s i m p l i f y t h e s u b s t i t u t i o n method, we and McPherson's  the c e n t e r  satellite  n  m  " i n the  (  3  '  4  )  employ Beckmann  definitions!  m K  D  so t h a t  22  = m  i  m  = p ^m  =  k m  1  - p i ^m-1  (3.3) becomesi m P *m  d-Kj m  W  =  m  =  S k.r. ^_2_ x i  . . . (3.5)  where:  and  " Pm-1  1  " m-1> K  • • •  ( 3  - > 6  s i n c e P„ = p + r » m m m m  D  = m k  P  m  .  m-l But  (  from  . . (3.7)  (3.4)!  P  m  -  (  1  +  s  )  P  m-1  +  D  m  • • •  ( 3  ' > 8  44 or, P  , (1 + s ) (1 -  K ^ )  m-1  (1 -  m  . Through r e p e a t e d  . . (3.9)  substitutions*  m-1  P  =TT  M  i=l  (1 -  r, 1  and  -  k  P  (1 + s ) (1 - K.)  J J ^ l  K  i + 1  )  (1 + s )  i-1  X  (1-K.)  (1 - K  i + 1  )  . . . (3.10)  from the d e f i n i t i o n o f D t m m P  = Pi  m  S  +  k  i=2  i  1-K _ i  i  P 1  . Through s u b s t i t u t i o n s  . . (3.11)  i n (3.7)« m  m  ?  =  * l l  S  r  ^  ^  = k-.r, "l'l l-  +  kl  ,  1-2  ^ i - l  rv  J L  * l - ^-v  l-k  x  2^  1  P  i  k.  ~ i  -K . 1  1  i-pr l l (1+s) _U  (1-Kj  d-K. .  -1  + 1  )  . . (3.12)  It  should  "m"  he  o b v i o u s from  i s depicted  "...  o r segments s u p p l y i n g d e f i n e d by  the  (k^),  the  the  The  how  first  population  o f any  supplied".  I n the  central  community i s  deter-  service multipliers  the r u r a l d e n s i t y  p r o p e r t i e s o f the  t u r n to the  general  d i s c u s s i o n o f the  (r-^).  o f t h e s e i s Beckmann's o r i g i n a l  i s a constant  proportion  o f the  city  size  simple models.  hierarchial  scheme w h i c h employs an a s s u m p t i o n t h a t the center  layers  Aggregate Approach  G i v e n the model we  services  n a t u r e o f d e c l i n e o f the  The  of  order  a nested set of markets, each  geometry ( s ) , and  Model l i t  of  constructed  b u n d l e o f goods and  framework the  mined by  as b e i n g  that a c i t y  McPherson, 1 9 7 0 i 2 7 - 2 8 ) .  (Beckmann and place  (3.12)  size of  any  population i t  serves} t h a t i s i  p  The  m  =  k  P  a  model i s a p r i o r i  . . .  since  i t r e s t s more upon  t h a n development from a t h e o r y .  Therefore  we  (3.13)  intuition must  be  wary o f making p r e d i c t i o n s w i t h t h i s model, a t l e a s t u n t i l we place  u n d e r s t a n d b e t t e r how  theory.  i t r e l a t e s to c e n t r a l  Beckmann's i n i t i a l model, however, d i s p l a y s a g l a r i n g i n c o n s i s t e n c y with c e n t r a l place r e l a t i o n s h i p s . On i n t e r p r e t i n g the geometry of the system, he o v e r s t a t e s the t o t a l p o p u l a t i o n served by a c i t y on the  "m"th l e v e l .  (Since he seems to equate  appears  this error arises  " s " with " q " ) .  It  from the d i f f e r e n c e between the  that  total  number of settlements i n an economic r e g i o n and the apportioning of those settlements among v a r i o u s h i e r archial levels. In any case, Beckmann (1968) and Parr  (I969)  r e c t i f y the m i s i n t e r p r e t a t i o n i n independent c o n t r i b u t i o n s . By adding " p " to both s i d e s of ( 3 . 4 ) m  i t should be  obvious t h a t 1  P  m  =  p  m  +  s P  m-l  +  r  m-l  Using ( 3 . 1 3 ) and (3.14) together i t to demonstrate served increase  .  . .  (3.14)  i s a simple matter  t h a t both c i t y s i z e and t o t a l p o p u l a t i o n e x p o n e n t i a l l y with the h i e r a r c h i a l  leveli  p  m =  (3.15)  47 . Parr  (I969) i l l u s t r a t e s t h e n a t u r e  . . (3.16)  o f the e r r o r i n t h e  e a r l y Beckmann model by c o n s i d e r i n g the change i n t h e basic  p r o g r e s s i o n component from  " q  " to  "s+1".  1-k  1-k  A t t h i s p o i n t i n the d i s c u s s i o n i t may  be  profitable  t o compare t h e a t t r i b u t e s o f t h i s s i m p l e model and  the  more complex a p o s t e r i o r i model o u t l i n e d e a r l i e r .  To  b e g i n w i t h , the r a t i o n a l e  "k"  f o r the  factors  "k "  and  m  r e s t on q u i t e d i f f e r e n t c e n t r a l p l a c e  relationships.  The  proposal of a d i s t i n c t  f o r each o f  "m"  f u n c t i o n s seems t o be  "k "  value  m  a reasonable  derivative  the of  C h r i s t a l l e r i a n t h e o r y i n t h a t i t f o c u s e s upon the  changing  r o l e s o f ( i ) e m p l o y m e n t - f u n c t i o n and  ( i i ) center-tributary  a r e a a s s o c i a t i o n s as we  the h i e r a r c h y .  o t h e r words, w h i l e we  move t h r o u g h  In  suppose t h a t the t e c h n o l o g y  used i n  providing i d e n t i c a l functions at d i f f e r e n t l e v e l s  remains  unchanged, we  are  introducing systematic  s i z e s through  the u n i q u e s e r v i c e mix  changes i n c i t y  a t each l e v e l .  the o t h e r hand, t h e p o s t u l a t e o f a c o n s t a n t i s t o t a l l y a r b i t r a r y , though i t may empirical merit. n o t i c e t h a t as  2  a p p r o a c h e s "b^"  r e l a t i o n s h i p between c e n t e r and i s neared  ( t h a t i s , as k — * - l o g ~ "  value  i n d e e d have some  F o r i n s t a n c e when we  "b "  "k"  On  recall  (2.6a)  i n value, a  we  constant  t o t a l market p o p u l a t i o n L  (a^ - a ) ) . 2  48 In a d d i t i o n to t h i s variance we  note t h a t  that  "k  nt  that  two  " itself  t i o n o f the For  the  f a c t o r s c a n n o t be  i s a constant, since  a comparison of  would e q u a l  contradictions  "k^",  f a c t o r s have no  therefore,  = k . 2  the  assuming  repeated  but  (3.13)  and  applica-  t h i s would  introduce (3.13)  and  I t s h o u l d be  indicates  are  clear that  o b v i o u s i n t e r r e l a t i o n s h i p and  (1966:31)  Dacey  (3.1)  c a s e when ( 3 . 2 )  i n the  compared w i t h k = k^ two  compared by  f a c t o r s a f f e c t s the models i n d i f f e r e n t ways.  instance, "k"  i n terms o f r a t i o n a l e  is unjustified in  the  that,  criticizing  Beckmann*s r e s u l t . However, M o d e l I I c a n p a r t i c u l a r c a s e o f the decreasing  "k^"  necessary  that:  be  general  factors.  For  shown t o be  model by this  the  t o be  only use  true,  a  of i t is  only  m . which i m p l i e s ,  as we  n o t e d above, t h a t k =  d e t e r m i n a t i o n of remaining step  2  ~  k  l 2 p  or,  in  "  r  2  +  k  r  = k,r_  -  (3.17)  The  v a l u e s i s performed  l  r  l  .  2  general: k  .  one  instancet  at a time; f o r K  "k^"  k-^  .  m-l 52  k,r.  m-l - 2D  k,p„  . .  (3.18)  Apparently,  then,  the a p r i o r i  model and (3.1)  a p o s t e r i o r i model have a f u n d a m e n t a l p r e m i s e common.  The  flexibility  f u r t h e r premises.  study u n i t s i n the  population subsets  o f a n a l y s i s w i t h the a d d i t i o n  Rather than u s i n g urban centers  c e n t r a l p l a c e system, the a  model i s e f f e c t i v e l y  employing f u n c t i o n a l l y  ( t h a t i s : k,p i  of  those  Therefore  counterpart  the f a c t o r  o f the  system f o r one  determined  , . , ., k p  d^m  m  t h a t have the  set ( k  1 #  "k"  k ,  II r e a l l y apply  same h i e r a r c h y and  ( i n t h a t we  spatial  emerges as the  The  )  mm  . . ., k  2  p a r t i c u l a r case.  t h a t M o d e l s I and  cident  , k„p  as  posteriori  c e n t e r s as e l e m e n t s i n a more complex  system.  in  o f the g e n e r a l model, however,  comes from a h i g h e r l e v e l of  the  aggregate  ) i n the  simpler  essential notion i s  to d i s t i n c t t h a t are  systems  spatially coin-  d e p i c t c e n t e r s as nodes i n a  geometric  network), Model I I I - The  Geometric  Multiplier  Dacey (I966) s u g g e s t s exponential  "k "  t h r o u g h the  service multipliers.  to  to reasonably  interpreting  ra  o f f e r any  analytic  "k " m  as  account f o r s p e c i a l i z a t i o n U n f o r t u n a t e l y he  immediate  d e r i v a t i v e o f the Beckmann-McPherson f o r m u l a t i o n .  " m" P  From  i  n  c  r  e  a  s  '(3,10)  e  f  p r o p o s a l t h a t market a r e a r  o  m  i t s h o u l d be  obvious  i s that:  that a  It is  populations  l e v e l t o l e v e l by a c o n s t a n t  condition for this  fails  interpretation f o r his choice.  However, a v a r i a t i o n o f t h i s scheme i s an  b a s e d upon the  an  factor.  sufficient  50 1-K 1  ,  _^~  =  x  m  Defining k  constant  1 + h  =  . . . (3.20)  = 0, (3.20) h o l d s f o r a l l m  Q  0: however  this indicates that:  l-k  =  1 + h  (3.21)  . . .  x  or, k  h- =  l-k  . . . (3.22)  h  x  t h e m e a n i n g o f (3.20) i s :  Besides,  •*» ^xkj "* 1  1  k  i  =  <  1  -  l  t  i  >  m  "  1  k  i  • • •  <  3  -  2  3  )  where t h e s e r v i c e m u l t i p l i e r d e c r e a s e s i n a g e o m e t r i c f a s h i o n f o r t h e s e c o n d and h i g h e r h i e r a r c h i a l l o a d s . f o r m u l a t i o n a n d t h e Dacey s u g g e s t i o n o n l y one v a l u e , v i z .  = £.  This  are i d e n t i c a l f o r  Now, u s i n g  (3.10)s  m-1  P  m  =  P  = P and  TT  l  ( 1 + s )  ( 1 + h )  i=l  x  (l+s)  1 0 - 1  (l+h)*-  1  . . . (3.24)  since:  D  m  =  . . . (3.25)  51 it  follows  thati -  m  r,  1-k, 1  (l+h)  [ ( l - k ) (1+hr x  (l+s) (1+h)  m + 1  m+1  - (1+s) (1+s) - 1  2  (1+s)J  (1+h)  2  (3.26)  w h i c h s i m p l i f i e s ( s e e 3.21 o r 3.22) t o :  P  m  =  r  l i  k,  l 1-k,  k  TT^7  2  | |{rik~)  m  (1+s)  1  .(nq)  (1+s)  m  -  "  1  . . . (3.27) The  r a t i o n a l e f o r t h i s g e o m e t r i c m u l t i p l i e r model  i s n o t c l e a r though. however, t h a t that  Beckmann and M c P h e r s o n s u g g e s t ,  t h e growth f a c t o r i n (3.24) i s t h e same as  i n (3.16).  Unfortunately,  i t i s e a s i l y demonstrated  that t h i s interpretation i s i n error. assuming  For instance,  that: (1+s)  (1+h)  =  . . . (3.28)  _§_ + 1 1-k  means: 1-k 1-k, But and in  =  ,  . . . (3.29)  _k_ " 1+s  since  k = ^  i f" P ^  i s identical  s i n c e k , k^ > 0 we have a c o n t r a d i c t i o n  (3.29):  i n (3.16)&  (L.S. >  R.S.)  i n o t h e r words, Model I I and Model I I I c a n n o t  g e n e r a t e i d e n t i c a l market a r e a p o p u l a t i o n s must he c o n s i d e r e d  distinct.  (3.24)  and t h e r e f o r e  \  \  52 (3.25)  Besides, differences constant levelj  indicates that the population  between c e n t e r s  on adjacent  proportion of the total  i nfact  this  levels  are a  p o p u l a t i o n on t h e higher  necessitates that the populations o f  u r b a n c o m m u n i t i e s become a n i n c r e a s i n g p r o p o r t i o n o f t h e t o t a l market area While  this  populations  a s we a s c e n d t h e h i e r a r c h y .  i sa derivative of the a posteriori  m o d e l , we h a v e n o r e a s o n arbritrary  proposal.  t o expect  Therefore,  literature  completely  interpretation  e m p i r i c a l evidence  a g g r e g a t e m o d e l a more v a l i d  M o d e l I V - The C o n s t a n t A third  i snot a  since this  seems i n c o n s i s t e n t w i t h a v a i l a b l e consider the simple  (3.20)  general  we  approach.  Multiplier  elementary  model i s suggested  i n the  b u t i s nowhere d i s c u s s e d e x p l i c i t l y .  Dacey  (1966) i n t r o d u c e s t h e i d e a o f c o n s t a n t s e r v i c e m u l t i p l i e r s but  we h a v e a l r e a d y d e m o n s t r a t e d t h a t t h i s c o n t r a d i c t s  the  a s s u m p t i o n s o f t h e g e n e r a l model.  inquire  what e f f e c t  of centers the  suggest:  multipliers  all  m u l t i p l i e r were t o emerge a t  Beckmann and M c P h e r s o n  (197003)  clear pattern i nthe higher  has appeared  levels  above t h e f i r s t  i s  service  a constant  multiplier  for  i s n o t unreasonable.  assumption f o r t h i s model i s t h a t  be e x p r e s s e d a s :  2  . . ."? n e v e r t h e l e s s , t h e y d o  data that indicate  The may  distribution  " . . . t h a t t h e l a r g e g a p b e t w e e n k^ a n d k  common, b u t n o  provide  t h e r e w o u l d be o n t h e s i z e  i f a constant  second l e v e l .  H o w e v e r , we m a y  (3.3)  53 m  . . .  i=2 ^  > k',  (3.30)  or, m p  m  (1-^  - {m-l} k') = V l  +  *' £  r 2  i . . .  which  leads to  Pm  3A)t  (see  88  i l - (k k  1  f  r  While  . 4\m-j} k9 x  x  „m-i m-i  k  E  j=0  m-l m-l  .  £ p ,  !•!  1  s (l+s)3  +  those o f Model I I .  v a l u e s i n the i n t e r v a l k d e t e r m i n e d by  m  (3.19)•  <  as determined  equate  -  1  J J  to  multipliers  i t s results  are  I t seems t h a t f o r c e r t a i n  k'Ckg,  t h i s new  where k , 2  city  k^,  s i z e model  i m a t e s t h e use o f a b a s i c p r o g r e s s i o n component. i n s t a n c e , i f we  1 0  . ' . . (3.32)  t h e p r o p o s a l f o r t h i s model i s s i m i l a r pattern of service  "k'" t o t h e mean o f k,,  k~,  . . . »k  Moreover,  m  approxFor . ,  i n Model I I , t h e n Model IV u n d e r e s t i m a t e s  t h e i r populations at higher l e v e l s .  -A  (l+s) - -!  I  8  begins a t the second h i e r a r c h i a l l e v e l ,  are  (1+B)  r  l<*l+{»-W  Model I I I i n t h a t a r e l a t e d  more l i k e  (3.3D  central  .,k  places with  s m a l l t o medium p o p u l a t i o n s f o r m a l o w e r  p r o p o r t i o n o f t h e i r t o t a l market p o p u l a t i o n s first  than the  l e v e l c e n t e r s do, h u t t h e l a r g e r c e n t e r s t e n d t o  become a g r e a t e r p a r t o f t h e t o t a l p o p u l a t i o n s  they  service. Unfortunately, model w i t h t h e l i m i t e d t h i s time. evidence  i t is difficult  / H  e x i s t s , we have  little  t h a t i t emerges a t t h e s e c o n d  The b e s t we c a n do i s h y p o t h e s i z e  level.  t h a t a l a r g e gap between  "k^" and " k " b r i n g s some s o r t o f s t e a d y 2  state into  On t h e o t h e r hand, t h e e m p i r i c a l e v i d e n c e f o r the support  this  e m p i r i c a l i n d i c a t i o n s we have a t  A l s o , i f such a " k  to stipulate  t o defend  we have  o f t h e a g g r e g a t e model c o v e r s  being.  cited  o n l y a number  o f t h e l a r g e r s i z e c l a s s e s , and i t r e m a i n s t o be e m p i r i c a l l y substantiated  (though i t seems i n t u i t i v e l y  t h a t t h e v e r y l a r g e s t c e n t e r s assume  reasonable)  a smaller  proportion  o f t h e i r t o t a l market p o p u l a t i o n s . All  in all,  t h o u g h , i t seems i m p r o b a b l e t h a t  can d i s c a r d the constant  we  p r o p o r t i o n a l i t y model i n f a v o r  o f e i t h e r o f t h e two r e m a i n i n g  elementary models.  f a c t that i t i s not completely  unsubstantiated  s t u d y p l u s i t s extreme s i m p l i c i t y s u g g e s t s  The  by e m p i r i c a l  t h a t the e a r l y  Beckmann model ( i n r e v i s e d form) i s t h e most p r a c t i c a b l e of the t h r e e .  We  s a y p r a c t i c a b l e because t h e a  posteriori  model i s n o t so f i r m l y a t t a c h e d t o t h e o r y t h a t we c a n suggest  t h e n o t i o n o f d e c l i n e i n the " k " v a l u e s m  and,  t h e r e f o r e , we  do r e q u i r e some i n t u i t i v e  to a systematic  3.20, 3.30).  d e c l i n e (see 3 . 1 3 ,  o t h e r words, t h e g e n e r a l model has, unknowns t h a t c a n n o t be and  s p e c u l a t i o n as In  a t t h i s time,  extra  deduced f r o m c e n t r a l p l a c e  t h e r e f o r e i t i s not workable i n g e n e r a t i n g  theory  center  populations. H i e r a r c h i a l M o d e l s and The  the Economic Base  economic base c o n c e p t  c i t y s i z e models w i t h l i t t l e  may  be  attached  s e r v i c e s f o r consumers  o u t s i d e t h e u r b a n community w h i l e n o n - b a s i c d i r e c t e d t o the r e s i d e n t s o f the c e n t e r .  production i s  With the  t i o n t h a t a l l employment i s b a s i c o r n o n - b a s i c , d e v i s e r a t i o s between the two o f t h e h i e r a r c h i a l models. economic base c o n c e p t o f the  city  evidence  types  should c l a r i f y our  s i z e m o d e l s t b e s i d e s , we  o f the basic/non-basic  ratio  we  may  that  gain  the c h a n g i n g  significant  character  ( a t l e a s t w i t h i n the  zr? i^i  m  1  as  The  (3.3)  population i n  t h a t s e r v i c e s i t s complementary a r e a i s  k.r., while 1  confines  r e c a l l the b a s i c p r e m i s e  o f t h e g e n e r a l h i e r a r c h i a l model. "P "  the  or d e c l i n e i n s i z e .  To b e g i n w i t h , we  center  may  interpretation  o f a c t i v i t i e s e x p l a i n e d by c e n t r a l p l a c e t h e o r y ) urban centers r i s e  assump-  o f employment f o r e a c h  I t i s intended  toward u n d e r s t a n d i n g  the  (see Dacey, 1 9 6 6 ) .  difficulty  B a s i c a c t i v i t i e s p r o v i d e goods and  to  the p o p u l a t i o n f u l f i l l i n g  l o c a l need i s  56 m ^ k.. ...  p  This i n d i c a t e s s e v e r a l p r o p e r t i e s of the in m ^ r?^  basic/non-basic r a t i o : (i)  The  ratio  k  T:  I i i/ m r  i=l  p  k  i *  i s maximized a t i = l and  minimized  a t i=m > (ii) 1  The  v a l u e o f the r a t i o  =j i - m depends upon the n a t u r e  i n the  interval  of decline i n service  multipliers; (iii)  The  ratio  i s a f u n c t i o n o f the  geometry  o r t r a n s p o r t t o p o l o g y o f the c e n t r a l p l a c e s y s t e m . Table 1 i n d i c a t e s the nature  of these p r o p e r t i e s  i n f o u r c e n t r a l p l a c e systems, each w i t h d i f f e r e n t  char-  a c t e r i s t i c s but a l l h a v i n g seven h i e r a r c h i a l l e v e l s . first  and  second  systems use  ( g e o m e t r i c a l l y d e c l i n i n g and f a c t o r ) but generate t o t h e q=3 1970)  different constant  multipliers proportionality  data that i s t o p o l o g i c a l l y  C h r i s t a l l e r data  level.  v a r i a n t o f t h e second aggregate  model.  The  comparable  (see Beckmann and McPherson,  with a r e l a t i v e l y constant m u l t i p l i e r  a t t h e second  The  "k  /n  beginning  f o u r t h system i s a g e o m e t r i c a l  i n t h a t i t i s d e p i c t e d by t h e  simple  57  Table 1 Service M u l t i p l i e r s and Basic/Non-Basic of Four C e n t r a l Place Systems (1) l  "  m  =  r  k  m  k  m  q = 3 2000  ratio  (2) q = 3 r = 2000 x  1^  ratio  (3) r  l  q = 3 = 2700  Ratios (4) r  l  q = 4 =  2000  e m p i r i c a l k'  1 k =• 6"  k  k  m  ratio  m  :r a t i o  7  .000  1.00  .053  .65  .034  1.30  .054  .56  6  .001  1.01  .060  .81  .030  1.49  .062  .71  5  .004  1.01  .067  1.02  .028  1.64  .072  .91  4  .012  1.02  .076  1.34  .031  1.93  .082  1.18  3  .037  1.08  .085  1.86  .037  2.22  .095  1.69  2  .111  1.25  .098  2.79  .045  2.67  .109  2.63  1  .333  2,00  .16?  5.00  .228  3.38  .167  5.00  ! 58 The  nature  o f the  the most s i g n i f i c a n t ratio.  The  initial  service multipliers  determinant  o f the  basic/non-basic  c o n s t r a i n t i s induced  case but the v a r i a b i l i t y  is certainly  by  "k^"  of d e c l i n e b r i n g s out  i n each  some v e r y  interesting patterns. F o r i n s t a n c e , when c e n t r a l p l a c e f u n c t i o n s become extremely  specialized  i n t e n s i v e p e r h a p s ) and we  might expect  (advanced t e c h n o l o g y , r e l y very l i t t l e  a system s i m i l a r t o the  on  capital employment,  first.  In  c a s e , the e x p o n e n t i a l l y d e c l i n i n g f a c t o r l e v e l s basic/non-basic r a t i o very quickly. l a r g e communities a r e r e s t r i c t e d capture  a s m a l l e r and  o f f the  Urban p o p u l a t i o n s i n  i n the sense t h a t  o f m a r k e t s f o r h i g h e r o r d e r goods and  brings i n l i t t l e  this  employment} i n f a c t ,  the  services  the community assumes  s m a l l e r p r o p o r t i o n o f the t o t a l m a r k e t  p o p u l a t i o n as b o t h grow l a r g e r .  Employment becomes i n c r e a s -  i n g l y b a l a n c e d between t h e b a s i c and n o n - b a s i c m since 2 ^ k. —•*:§• as "M" becomes g r e a t e r . i=l  sectors  1  The gradual  second and  f o u r t h systems a r e c h a r a c t e r i z e d by  functional specialization.  S i n c e employment does  n o t t a p e r o f f r a p i d l y f o r h i g h e r o r d e r goods and a v a r i e t y of basic/non-basic r a t i o s  services,  i s permitted.  In  b o t h systems, c e n t r a l p l a c e s form a c o n s t a n t  proportion  of  ascend  t h e i r t o t a l market p o p u l a t i o n s b u t , as we  h i e r a r c h y , both It  s e r v i c e m u l t i p l i e r s and  seems t h a t the p e r c e n t a g e  the r a t i o s d e c l i n e .  increases i n basic  b r i n g f o r t h even g r e a t e r p e r c e n t a g e  the  increases i n  activity non-basic  endeavours u n t i l the  smaller size classes.  population  i n c r e a s e s f a v o r the l a t t e r i n The g r e a t e s t  increase are apparently  (i) the  absolute  The c a p t u r e  highest  order  (ii)  increments i n  determined b y i  o f the a d d i t i o n a l markets f o r  f u n c t i o n ; and  The a d d i t i o n a l demands p l a c e d  on t h e f i r s t  o r d e r goods and s e r v i c e s by t h e new members  o f the b a s i c  sector. The  Christaller  system i n t r o d u c e s  m u l t i p l i e r v a r i a t i o n , where k » k^, . . 2  constant.  Nevertheless,  H  another type o f k  m  are r e l a t i v e l y  the basic/non-basic  ratio  d e c l i n e s as we move up t h r o u g h t h e h i e r a r c h y .  steadily  T h i s seems  f u r t h e r proof that the p r o v i s i o n o f f i r s t - o r d e r  commodities  i n r e s p o n s e t o demands made by a d d i t i o n a l b a s i c  employees  i s an e x t r e m e l y i m p o r t a n t  determinant o f the s i z e o f the  u r b a n community.  We n o t i c e , t o o , t h a t i n t h i s  empirical m  example t h e r a t i o  always exceeds u n i t y ( t h a t i s ,  k. < i=l  The the n a t u r e  geometry o f t h e c i t y  cities,  system a l s o i n f l u e n c e s  o f the r a t i o but i n a l e s s s p e c t a c u l a r  I t appears t h a t with  an i n c r e a s e  fashion.  i n t h e number o f s a t e l l i t e  b a s i c a c t i v i t y g i v e s way t o l o c a l  services i n a  more r a p i d f a s h i o n as we a s c e n d t h e h i e r a r c h y .  Higher  m u l t i p l i e r s a r e needed t o meet t h e demands o f more centers  1  i n t h e q=4 system; l i k e w i s e , t h i s  smaller  introduces the  need f o r f u r t h e r e x p a n s i o n o f t h e n o n - b a s i c l o w e r o r d e r goods have h i g h e r m u l t i p l i e r s ) .  sector  (where  I t seems  t h a t as t h e i n d i v i d u a l m u l t i p l i e r s converge a t h i g h  levels  i).  o f the h i e r a r c h y , the  resultant  basic/non-basic  ratios  remain s i g n i f i c a n t l y d i f f e r e n t . The patterns  c i t y s i z e models i n d i c a t e  i n the v a r i a b i l i t y o f b a s i c / n o n - b a s i c  l e a s t w i t h i n the theory  some r e l e v a n t ratios,  domain o f a c t i v i t i e s t h a t c e n t r a l  seems t o c o v e r .  Besides,  the  size  and  i s i n f l u e n c e d by t h e  the  central  we  place  see more c l e a r l y  d i s t r i b u t i o n o f u r b a n communities b o t h  I t seems r e l e v a n t ,  s h o u l d be more aware o f the c h a r a c t e r i s t i c s  central  place  Models and  the R a n k - S i z e R u l e  Geographers d i r e c t c o n s i d e r a b l e frequency  distributions  e f f o r t toward  of urban  centers  as b a s e d on c i t y s i z e models o r e m p i r i c a l e v i d e n c e . l e a v e d i s c u s s i o n o f t h e l a t t e r i s s u e u n t i l the n e x t and  here we  the  s i z e and  central  examine the frequency  place  may  where "R"  =  R  b  continues  in a  p  R  the  dominant It  . . . (3.33)  i s the r a n k o f the  c i t y , and  t o be  i n the f o l l o w i n g f o r m t  o f the c i t y o f r a n k " R i , " p ^ largest  chapter  concerning  i n r e l a t i o n t o c i t y s i z e models.  represented  %  o f arguments  d i s t r i b u t i o n of centers  rank-size rule  of interest  be  course  We  hierarchy.  The topic  of  distributions,  Hierarchial  d e s c r i b i n g the  through  then,  t h a t we  size  how  constrains  i n d i v i d u a l u r b a n economies  place hierarchy.  at  'V  city,  'p^"  i s the  i s the  population  population of  i s a derived constant.  I f we  the graph  this  f u n c t i o n on double l o g a r i t h m i c p a p e r , we  have a  straight  line: l o g PR As  originally  u n i t y and t i p l i e d by  the  =  log P  - b log R  . . .  (and u s u a l l y ) i n t e r p r e t e d , "b" has population  i t s r a n k "R"  Hoover  (1955)  seriously question  o f the  "R"th  equals  one  o f the  arranged  to  notes yields  f a i l s to suggest and  to  a the  1955'196).  Beckmann ( 1 9 5 8 )  i s a g a i n the f i r s t  comment on t h i s r e l a t i o n s h i p . has  been shown t o be  new  issue merits  rected  He  according  scheme t h a t l i n k s c e n t r a l p l a c e p o p u l a t i o n s  recall  first  system a u t o m a t i c a l l y  t h e r a n k - s i z e r u l e . . . ", but  We  to  explicitly  While h i s o r i g i n a l model  f a u l t y , h i s c l e v e r approach to  this  praise. t h a t Beckmann's e a r l i e s t model ( i n c o r -  form) employs a c o n s t a n t  basic progression  component  s "  1-k  +  1".  However, i f we  consider t h i s m u l t i p l i e r  a random v a r i a b l e about t h a t s t a t e d c o n s t a n t , cities  on the  have i d e n t i c a l  same h i e r a r c h i a l l e v e l populations.  g r e a t e r v a r i a t i o n s as city  "m"  Besides,  increases?  s i z e s approach a continuous  distribution.  of  l a r g e s t c e n t e r mul-  the r a n k - s i z e p r i n c i p l e .  a s e r i e s of c i t y t r i b u t a r y areas  (Hoover,  a value  "Pj^".  a p p e a r s t o be  . t h a t the C h r i s t a l l e r  principle  (3.3^)  the r e l a t i o n s o f C h r i s t a l l e r ' s c e n t r a l  p l a c e h i e r a r c h y and .  M  as  then a l l  do not n e c e s s a r i l y the  component  has  i n o t h e r words,  the  r a t h e r than s t e p l i k e  Beckmann i s e s s e n t i a l l y a l t e r i n g the  rigid  62 Christaller  system t o random d i s t u r b a n c e s  the midway c e n t e r o f any representative  "R  n  on t h a t  R  level.  d e m o n s t r a t e s t h a t the  " o f a c i t y midway i n the  expressed  "n"th  overall  s i z e c l a s s can  rank be  as: =  n  q° +  (q +  1  - q°) + ( q ^  (q  - q ~  1  n  2  - q )  2  1  +  . . .  +41 .  U = 0 or 1,  for n > l , centers  only  given h i e r a r c h i a l l e v e l i s  of a l l c i t i e s  (1969)  Parr  so t h a t  i n the  where  Since  o n l y the  s i z e c l a s s can possess  an odd  "*<"  "R "  i s u s u a l l y one  and  n  .  (3.35)  i s u n i t y i f t h e number o f  s i z e c l a s s i s even and  number i s odd.  .  second  "t>C" i s z e r o i f t h a t (that i s f o r n  number o f c e n t r a l  i s w r i t t e n more  >  1)  places,  conveniently  as: R  =  (o""  n  1  + Q ~ 2 n  2  + 1)  "  (l+s)"-  + (l+s) 2  1  n  . Let's f i r s t w i t h the  .  particular b = 1  o f the  case  of ( 3 . 3 3 ) .  Now  system.  Hence,  (3.36)  i f Model I I  o v e r a l l r a n k o f a midway c i t y on a  u l a r h i e r a r c h i a l l e v e l and must e q u a l the  +  o f a l l c o n s i d e r the a g g r e g a t e model  c a n accommodate the r a n k - s i z e d i s t r i b u t i o n , t h e n product  .  2  the p o p u l a t i o n o f t h a t  p o p u l a t i o n o f the l a r g e s t c i t y  the particcity  i n the  1  63 or, (l+s)""  But  since  +  1  1 + s  the  component) and  n  2  ^  demonstrate t h a t exceeded by  (l+s) "  j^ /l-kj s  the  left  / s l l - k  + l j  side  r i g h t side no  + 1  2  , i t i s simple  (rank) of  (3.38) i s a l w a y s  (power o f the  progression  c o m p a t i b i l i t y e x i s t s between a  place  system b a s e d on a c o n s t a n t c e n t e r / m a r k e t  ratio  and  a rank-size  Parr o f the exist  to  d i s t r i b u t i o n w i t h an  also considers  a g g r e g a t e model and for b ^ l j in this  the the  central  population  exponent o f  p o s s i b i l i t y that rank-size  one.  coincidence  principle  may  case:  or. b  =  (log  R  n + 1  - log  R ) n  . By  demonstrating that  the  (3.40) i s v a r i a b l e , he of  "b"  i n (3.39) and  therefore,  the  violated.  In Parr's  therefore  be  b a s e d on the  denominator on  i s able  the  r i g h t side  to s t i p u l a t e that  (3.40) v a r i e s w i t h  "n"  and  i n i t i a l assumption of a constant ( 1 9 6 9 « 2 4 9 ) words:  concluded that  ".  a c e n t r a l place  . . (3.40)  the  . . it  is may  system  constant p r o p o r t i o n a l i t y f a c t o r i s  value  that, "b"  not  of  I  64 compatible value  w i t h a r a n k - s i z e d i s t r i b u t i o n even where  of the c o n s t a n t  "b" assumes a v a l u e  other  the  than  2 unity." Beckmann and McPherson f e e l  that a  sufficient  condition f o r rank-size c e n t r a l place coincidence i s that market a r e a p o p u l a t i o n s i n c r e a s e by a c o n s t a n t from l e v e l t o l e v e l  (see 3.2k),  m a i n t a i n t o d e v i s e Model I I I . c u s s i o n we  present  the  assumption  multiplier they  At t h i s p o i n t i n the  dis-  an argument t h a t seems t o r e f u t e  this  this assertion. To b e g i n w i t h , we (3.33).  But  Using  employing  (3.1)  (3.36),  and  c o n s i d e r the b = 1 ( 3 . 2 ? ) we  see  i f compatibility  case  of  that:  occurs  then:  Note our s u b s t i t u t i o n o f "b" f o r "q" i n P a r r ' s a r t i c l e ; t h i s d i s c r e p a n c y i s due o n l y t o a d i f f e r e n c e in notation.  65 or, (l+s)* " 1  +  . where  the l e f t  side  ^  k  M +  ( l + s ) ^  x  (i+s)  are  . <  1  (3.^3)  equals:  £(l+s)  Therefore,  -  2  t o state  that  -  1  ( l -  k  l  (l+s) ~  )  M  -  2  + d - ^ y j  the left  and r i g h t  sides  of  (3.43)  i d e n t i c a l means:  (l+s)  + k  M  (l+s) ' ' 1  x  -  1  ( 1 - k ^ ( l + s )  ((ri^f'  1  l\  (1-*!)  1  ( l  +  M  _  -  2  (1+s) +  (l 8)  s ) " -  +  •  • «  3.44)  or, k  (I+s) " 1  x  -  1 - 1  2  ( 1 - ^ ) (l+s) " M  2  +  (1+s) +  f o ra l l s  > 0, 0 < k  2(l+s) M  1  x  2  .  But  (l-k )  +  1  < 1 , we  2(l+s) " M  2  +  . . (3.45)  know:  . . . +  2(l+s)°  ^ 2 ( l + s )  M  66 Hence,  by noting  k  thatt  ( l + s )  x  that  _  1  2  4 and  M  the factor  -  (l-k )  (l+s) *' 1  0,  right  side;  the  then  model  ment  the left  being  f o rb =  1.  Next  we  exponent  coincident  with  case just  just  taking in  i s exceeded factor  by the disallows  the rank-size  compatibility  arrange-  o f the second  p r i n c i p l e when t h e  do n o t a l l u d e  t h e i r argument  unity f o r  We  should  to this  note  that  general  to the particular  case  i s not as elegant  as the  refuted. The  one  with  i snot restricted to unity.  but direct  exceeds  growth  the rank-size  Beckmann and McPherson  2(l+s)° Q  - 2 )  expanding  ( l - k ^  I - _ , „„<+,,  1  o f (3.45)  side  (1+s) +  +  2  . . . +  \  consider  elementary model  +  1  I  the rapidly  from  M  M_1  / fl < |  M  (l+s) "  x  proof  outlined  limits  simpler  i  i n t h i s case since  w h e n M >^  several  0.  recall  involve  (3.41), which  gives  forrai  •fAj-'-w.d-v s  Therefore t  We  o f i t s statements  +  k  l  . . .  (3.46)  Now  i f the r a n k - s i z e  r u l e h o l d s f o r the model and b ^ 1,  thent l  P  R  M  =  P  2 M-1  Here we have two c a s e s ©  R  2  M*  = • • • =  R  (among  PM  "M-l") o f immediate i n t e r e s t :  % p  l  p  l  . . .  (3.48)  . . .  (3.^9)  meaning: b  ©  =  f y p j  log L O G  R  M  . . . (3.50)  b (2) = l o g (P /PI) 2  l o g  Considering b  Q  (V  (M-l)  R  O  G  R  M  . . . b  @  /  > b  (£)  M  l o g (iZiT-) L  the f i r s t  m-l)  , we c a n d e f i n e  L 0 G  Now  R  (3.5D  where:  . . . (3.52) M log L  0  G  R  M  o f t h e s e terms i s l e s s t h a n  (l±^ . . . M-  1  M-2  L O S  (3.53)  ( AT)  l o g (1+s)  68 M l o g |" (1+s)^  and t h e s e c o n d term i s l e s s t h a n  M-2 l o g (1+s) (by  substituting  As M  >^  (3.36)  i n t o the denominator  -,. _ / 1  0, t h e s e terms c o n v e r g e toward  l o g  log In  \  (i-v  (1+s)  log and l o g C^^-)  (3.53)).  of  respectively,  (1+s)  o t h e r words, b ^ ( p  i t s e l f c o n v e r g e s a t t h e sum 1  fe)  +  1-  M  ^ (1+s)  log or,  *'©  -  .l io go g r r ^ ) ( i l 0  S  <  <  l  f  o  r  >  1  = log^  s  log  However, when we compare the  numerator  the  latter.  2  +  k  (1  i s not v a l i d  (3.54)  kl)  l  -  forM  0  . . . (3.55) and  (3.55)  we f i n d  that  i n t h e f o r m e r i s always exceeded by t h a t i n Therefore, f o r s ? 0 , 0 4 ^ 4 1 ,  Q  " °  as:  (1+s)  (3.54)  M  . . .  {(rar)*'* - -  ©  \  we have b  s )  (3.51)  On t h e o t h e r hand, we may r e w r i t e  b  1 + 8  +  * b' ©  *  b  f o r a n exponent  ®  M  »  and t h e r a n k - s i z e  "b" u n e q u a l t o u n i t y .  0, rule  69 The model from in light  d i s a s s o c i a t i o n o f the g e o m e t r i c  multiplier  the r a n k - s i z e a p p r o a c h s h o u l d come as no  of Parr's e a r l i e r a n a l y s i s .  I n t h a t case, market  p o p u l a t i o n s grow a t a c o n s t a n t r a t e b u t c i t y too q u i c k l y f o r c i t y rank  d e c l i n e s when t h e s e  expand a t the same r a t e .  In t h i s  and  s i z e cannot The  third  c i t y growth e x c e e d s  be  "k  " and we  s i m p l e model, a s t y p i f i e d by a  to r e l a t e  cannot  second  an argument s i m i l a r  III.  N e v e r t h e l e s s , we  t h a t s i z e s expand t o o r a p i d l y f o r r a n k  high l e v e l s  depends on the sum  2  k^,  . . . ,k  m  i n Model I I ) .  c o n s i d e r Model IV and compatible  / n  T h e r e f o r e we  As  is we  nature i n form  intuitively  declines,  the a g g r e g a t e  ( t h a t i s when " k  constant  the  s i n c e t h i s model o v e r e s t i m a t e s  k ,  of  level,  to rank-size t h i n k i n g .  develop  t o t h a t f o r M o d e l s I I and expect  will  realized.  n o t e d b e f o r e , no e x p l i c i t r a t i o n a l e d e t e r m i n e s of  size  that a constant product  s e r v i c e m u l t i p l i e r t h a t emerges a t the more d i f f i c u l t  centers  can expect a g a i n t h a t c i t y  o u t w e i g h the r a n k v a l u e and rank  s i z e s grow  l a t e r c a s e , market  p o p u l a t i o n s grow a t a c o n s t a n t r a t e b u t t h a t r a t e j hence, we  surprise  model a t  do  the r a n k - s i z e p r i n c i p l e as  of not being  concepts.  Dacey i s u n s u c c e s s f u l i n d e f i n i n g a sequence  of  s e r v i c e m u l t i p l i e r s t h a t p e r m i t s the g e n e r a l model t o conform t o a r a n k - s i z e d i s t r i b u t i o n , b u t of h i s a r t i c l e  he may  from  the  tone  w e l l be r e s t r i c t i n g h i s s e a r c h t o  a set of f u n c t i o n a l l y r e l a t e d  "k^'s".  Nevertheless, i f  70 we  can  i d e n t i f y any  bility, place  t h e n we  set of m u l t i p l i e r s that gives  cannot accept  system are a t v a r i a n c e Beginning  a sufficient  with  the  that populations with  lowest  c o n d i t i o n f o r "p^"  the  and  in a central  rank-size  levels "P "  rule.  o f the t o be  2  compati-  hierarchy, rank-size  related i s : R  M  %Tl  P  ^  _  l  "  Now  by e m p l o y i n g  one  can  2  . . . to define c e n t r a l place  a service multiplier  I n o t h e r words we (3.19)  statement than (one  P  (3.56)  stipulate  (3.17).  ^  at a time) are  "kg"  by  (3.56)  populations, introducing  c a n c o n s t r u c t a more  general  i n which s e r v i c e m u l t i p l i e r s  designated  so as t o g e n e r a t e a  rank-size  d i s t r i b u t i o n among u r b a n communities: m-1 k  m  -  p  m  -  ^ p  i m  k  m  +  r  (p  +  r  i  }  m  (3.57)  . . .  While t h i s a p p r o a c h i s i n d u c t i v e and  totally lacking in  t h e o r e t i c a l r a t i o n a l e , i t does e s t a b l i s h some a s s o c i a t i o n between the and  deductive  features of c e n t r a l place  the more e m p i r i c a l l y founded  principle.  On  the  o t h e r hand, t h e r e  as y e t t h a t the r a n k - s i z e r u l e may law  (see C h a p t e r 4)  s t s t e m e n t w i t h i n the  be  i s no  systems rank-size  suggestion  i n t e r p r e t e d as  a  framework o f c e n t r a l p l a c e  theory. I n t a b l e 2 we  present  i n a rather  comprehensive  f a s h i o n the v a r i o u s p r o p e r t i e s o f f o u r c e n t r a l p l a c e  71 h i e r a r c h i e s , each g e n e r a t e d from M = 7 , s = 2,  rj =  2000t  (i) rank-size  k^ = 0 , 3 3 3 ,  Model I f o r m u l a t e d t o conform t o c o n s t a n t  products; (ii)  Model I I ;  (iii)  Model I I I ;  (iv)  Model IV w i t h  service multipliers  ( k , k^, 2  "k'  . .  M  .t  e s t i m a t e d from t h e k^,)  defined i n ( i ) .  The t a b l e i s u s e f u l f o r q u a l i f y i n g any o f the s t a t e m e n t s we have made t o t h i s p o i n t  i n the d i s c u s s i o n . Table 2  Fundamental P r o p e r t i e s o f Midway C i t i e s i n R e l a t e d C e n t r a l P l a c e Systems v i a D i v e r s e M o d e l l i n g Approaches Model I : m  rank  The G e n e r a l Case - R a n k - S i z e P a t t e r n  Pm  r  m  Pm  k  Rank X  m  m  Size  7  1  486,500  4,223,940  4,710,440  .021  .103  486,500  6  2.5  194,700  1,278,180  1,472,880  .029  .132  486,500  5  6.5  75,000  376,060  451,060  .044  .166  486,500  4  18.5  26,300  107,820  134,120  .060  .196  486,500  3  54.5  8,940  29,980  38,920  .084  .229  486,500  2  162.5  2,990  8,000  10,990  .120  .273  486,500  1  486.5  1,000  2,000  3,000  .333  .333  486,500  72 Table 2 (Continued) Model l i t  The A g g r e g a t e  Approach  7  1  4,096,000 8,192,000 12,288,000 .040  6  2.5  1,024,000 2,048,000  5  6.5  256,000  4  18.5  3  .333  4,096,000  3,072,000  .053  .333  2,560,000  512,000  768,000  .070  .333  1,664,000  64,000  128,000  192,000  .094  .333  1,184,000  54.5  16,000  32,000  48,000  .125  .333  872,000  2  162.5  4,000  8,000  12,000 .167  .333  650,000  1  486.5  1,000  2,000  .333  486,500  Model I I I :  3,000  The G e o m e t r i c  Multiplier  10,450,000 14,143,000 24,593,000  7  1  6  2.5  2,330,000  3,161,000  5  6.5  532,000  699,000  4  18.5  115,500  156,000  3  54.5  25,500  2  162.5  1  486.5  .333  5,491,000  .029 .428 10,450,000 .043 .428  5,800,000  1,231,000 .067 .428  3,460,000  271,500  .099 .427  2,140,000  35.000  60,500 .148 .422  1,390,000  5,500  8,000  13,500 .222 .406  892,000  1,000  2,000  3,000  Model I V : The C o n s t a n t M u l t i p l i e r  .333 .333  486,500  - "k'" e s t .from Model I  1,025,000 3,727,300 4,752,300  .060  7  1  6  2.5  5  6.5  64,000  315,700  379,700 .060  .169  416,000  4  18.5  17,300  93,700  111,000 .060  .156  320,000  3  54.5  5,150  27,800  32,950  .060  .155  281,000  2  162.5  1,900  8,000  9,890  .060  .191  309,000  1  486.5  1,000  2,000  3,000  .333  .333  486,500  251,000 1,075,100 1 ,326,100  .222  .060 .190  1,025,000 628,000  73 H i e r a r c h i a l Sets We  and  apparently not  elegance.  ( i i ) existing empirical  a s u f f i c i e n t reason  support  size d i s t r i b u t i o n of  o f the b a s i c p r o g r e s s i o n  the r a n k - s i z e r u l e .  On  the  i n f e r r e d as  o f whether o r n o t  centers  the  such.  two  T h i s r a i s e s the  aligned to  he  notions question  a g g r e g a t e model, t h r o u g h  m o d i f i c a t i o n , can be  component  o t h e r hand,  r i g h t f u l l y n o t e s t h a t c o m p a t i b i l i t y o f the  reasonable  nature  of each h i e r a r c h i a l l e v e l g i v e s d e s c r i p t i v e  to coincidence  c a n n o t be  i n the  is  products.  endpoints  model and  that  service multipliers  to expect s i m i l a r i t y  P a r r s u g g e s t s t h a t the on the  evidence,  However, t a b l e 2 i l l u s t r a t e s  s i m i l a r d e c l i n e s o f the  of rank-size  aggregate  the most s u i t a b l e a p p r o a c h i n l i g h t  ( i ) e x i s t i n g theory, (iii)  the Rank-Size Rule  have a l r e a d y i n d i c a t e d t h a t t h e  model a p p e a r s t o he of  and  any  rank-size  thinking? S u r e l y , though, i n t e r n a l m o d i f i c a t i o n the  purpose o f a model whose s t r e n g t h l i e s  For  i n s t a n c e , v a r i a t i o n o f the  l e v e l adds more unknowns t o the i n the number o f s a t e l l i t e a general the  "k"  defeats  in i t s simplicity.  f a c t o r from l e v e l  argument, w h i l e  c i t i e s erases  progression m u l t i p l i e r .  the  S i n c e we  d i s t r i b u t i o n o f u r b a n s i z e s t h r o u g h one  changes  concept may  to  of  interpret  rather  intricate  a p o s t e r i o r i model, i t seems u n r e a s o n a b l e t o m a n i p u l a t e an e l e m e n t a r y model h a v i n g  i t s own  distinct  advantages.  74 However, an e x t e n s i o n allows  a new  principle  a s s o c i a t i o n between Model  to a r i s e .  c e n t r a l place  I f we  hierarchies.  size  We  may  compute t h e " k ^ ' s "  .  may  ,,  2,  c o n s i d e r the  rank-size  overall  independent  pursue the a p p r o a c h t h a t we  used to  f o r t h e g e n e r a l model so as t o r e l a t e (see ( 3 . 5 6 )  and  g e n e r a t e a h y p o t h e t i c a l "M"  o f c e n t r a l p l a c e s and demonstrate  ,  I I and t h e  formed by the v a r i o u s  t o the r a n k - s i z e p r i n c i p l e  of various  framework  c o n s i d e r a s e t of independent  systems, t h e n we may  pattern of c i t y  t h i s c a s e we  o f o u r s i n g l e system  s m a l l e r systems or 1 l e v e l s ) ,  (3.57)).  level  that, with  In  system  the a d d i t i o n M-2,  ( t h a t i s , ones w i t h M-1,  an o v e r a l l r a n k - s i z e  arrangement  evolve. C l e a r l y , t h e i d e a i s t o d e t e r m i n e r a n k s when  a constant entire  growth f a c t o r  set of c i t i e s .  We  "  s 1-k  + 1  " i s supposed  f o r an  use t h e same c o n c e p t u a l  method  o f t h e i n i t i a l Beckmann c o n t r i b u t i o n where a c t u a l  populations  v a r y about t h i s modal v a l u e  increased  number o f c e n t e r s the  and add, t o o , t h a t an  i n each aggregate s i z e c l a s s  f i r s t ) means t h a t a smoother  i s now  more The  (except  decline i n c i t y  sizes  likely. single l a r g e s t center  o f the "M"  g i v e s us t h e c e n t r a l p l a c e o f r a n k one  level  system  f o r the e n t i r e s e t .  Our n e x t s t e p i s c o n s t r u c t a h y p o t h e t i c a l s i z e c l a s s o f p o p u l a t i o n P ^ _ i so t h a t t h e m i d p o i n t o f t h e group rank are  = 1 "s" centers  Recalling (3.36),  we  know t h a t  o f t h a t second s i z e c l a s s i n t h e  has there complete  75 "M"  level  s y s t e m ; hence, we  (x  1 + s + ~~  1  c e n t r a l place the  first  +1)  2  Obviously  c l a s s where we  we  "x^" more c e n t e r s  so t h a t :  g  I ^k  =  systems  system.  add  +  a r e b u i l d i n g up o f "M-1"  Likewise,  a l r e a d y have  system p l u s  "sx^" c e n t e r s  we must add  "Xg" new  we  . . .  1  "x^"  independent  l e v e l s each to continue  supplement  t o the  "s(s+l)" centers  third  i n the  i n the s m a l l e r s y s t e m s .  centers  o f "M-2"  (3.58)  level  complete Now  systems  by s o l v i n g : 1 + s + x, + s ( s + l ) + s ( x , ) +  r  (Xp+1) —  / s =(i-k .  I n t h i s manner we place  systems  initial  +  . .  2  y (3.59)  can develop a d d i t i o n a l smaller c e n t r a l  f o r v a r y i n g s i z e c l a s s e s "n" so t h a t :  . To c l a r i f y  ^  t h i s argument,  c e n t r a l place  . .  (3.60)  "x^* s" a r e d e t e r m i n e d ffoor r 't h e  system  i n the f o l l o w i n g t a b l e :  Table 3 C o n s t a n t Rank-Size P r o d u c t s G i v e n by Independent H i e r a r c h i a l S e t s v i a Model I I Size  Popula-  Class  Rank  h  a Q  h  h  1  h~  2  h^  h-  h^  Total  tion  1  4,096,000  1  2  1,024,000  4  2  3(x )  3  256,000  16  6  6  7(x )  4  64,000  64  18  18  14  27(x )  5  16,000  256  54  54  42  54  6  4,000  1,024  162  162  126  162  206  4ll(x )  7  1,000  4,096  486  486  378  486  618  822  1  -  -  x  -  2  -  -  -  -  1  -  -  -  -  5  -  3  -  -  1 0 3 (x^)  -  5  _  1  9  .  7  7  -  307  -  1,229  l,639(x ) 6  ^ h e " h i " columns i n d i c a t e t h e t o t a l number o f p l a c e s i n each s i z e c l a s s o f independent h i e r a r c h i a l sets» " i " r e f e r s t o t h e number o f l e v e l s m i s s i n g from "M".  4,915  77 We  should  systems must be bution  emphasize, however, t h a t the  assumed t o be  sense a l o n e ,  i n t e g r a t i o n o f the to the  justify this  idea of a very  empirical  c h a p t e r and  distribution  should  hand, we  hypothetical model c a n  one  that  closure  It is  idea.  diplays We  literature  i n the  difficult  this  will  rank-size  r u l e and  summarize  i n the  argument as  but  regional  next  nature  r e a l world.  o n l y d e m o n s t r a t e s how  conform t o the  spatial  shed some l i g h t on the  must c o n s i d e r  distri-  i n a r e a l world case,  large t e r r i t o r y that  l a r g e t e r r i t o r i e s and other  assuming t h a t  a p p r o x i m a t e s the  size  this  are  i n a size  systems i s n o n - e x i s t e n t .  supposition  economic c l o s u r e the  s i n c e we  integrated  various  an  On  of the  extremely  the  aggregate  not  why  i t does.  Chapter k EMPIRICAL ANALYSIS AND  INTERPRETATION  I n t h i s c h a p t e r we have t h r e e g e n e r a l and related  inter-  objectivesi (i)  To c r i t i c a l l y  examine t h e p a r t i c u l a r t e c h -  n i q u e s employed i n e m p i r i c a l c i t y (ii)  studies;  To r e v i e w e x i s t i n g s t o c h a s t i c  so t h a t t h e y supplement one (iii)  size  interpretations  another;  To s t i p u l a t e whether o r n o t t h e s e  p r e t a t i o n s are s a t i s f a c t o r y explanations of c i t y  intersize  patterns. D i s c u s s i o n i s a r r a n g e d t o h i g h l i g h t t h e improvement o f m e t h o d o l o g i c a l concern i n the s u b j e c t , w h i l e the e x p l i c i t  role  emphasizing  o f theory i n e x p l a i n i n g the p o p u l a t i o n  d i s t r i b u t i o n amongst u r b a n p l a c e s . Background P i c k any l a r g e a r e a . I t w i l l l i k e l y c o n t a i n many s m a l l c i t i e s , a l e s s e r number o f medium-size c i t i e s , and b u t few l a r g e c i t i e s . This pattern of c i t y s i z e s has been o b s e r v e d t o be q u i t e r e g u l a r from one a r e a to another. T h a t i s , when t h e f r e q u e n c y o f o c c u r r e n c e o f c i t y s i z e s i n any a r e a i s compared w i t h t h e f r e q u e n c y o f o c c u r r e n c e o f s i z e s i n a n o t h e r a r e a , t h e two f r e q u e n c i e s a r e v e r y much a l i k e , . . Such e m p i r i c a l r e g u l a r i t i e s o f c i t y s i z e have been n o t e d many t i m e s and have l o n g posed a c h a l l e n g e t o t h o s e who would e x p l a i n o r i n t e r p r e t them. ( B e r r y and G a r r i s o n , 1 9 5 8 a t 8 3 ) 78  79 I n a few more r e l e v a n t The  words we  features  literature  of empirical c i t y  unfortunately,  i n t e g r a t e the v a r i o u s I n the  (as w e l l as  introduced size  i s c h a r a c t e r i z e d by numerous  c o n t r i b u t i o n s and,  whole.  are b r i s k l y  few  c o n c e p t s and  Berry  flexibility  r e p e t i t i o n i n terms  i n a general  interdependencies  notion  i s applied with regard  i s d e v o t e d t o the city  the  idea at a l l .  device  to  spatial  bridge  However, where  to c i t y  systems, i t i s attention  W i t h e x p l i c i t use  of  system i d e a however, e m p i r i c a l g e n e r a l i z a t i o n s  base t o r a t i o n a l l y 1969:33).  order  Besides,  since  i t affords a  sense-perception  c e n t r a l place  data  theory,  i s n a t u r a l l y couched i n t h i s framework and notion  seems advantageous  deduced p r o p o s i t i o n s tested. be  In other  ( i f not  the  i t s s t r e s s i n g of  s t u d e n t wonders why  become i n c r e a s i n g l y s u b s t a n t i v e  may  being,  systems a p p r o a c h .  empirical contributions.  o f t e n done i m p l i c i t l y and  that  a  s u g g e s t t h a t the  system i s a most adequate c o n c e p t u a l t h e o r e t i c a l and  adopt  p a r t i c u l a r l y advocates  o f systems i n q u i r y and  i n t e r a c t i o n s and  to  system.  (1964), f o r one,  grounding or urban theory  the  disparate  reviews attempt  c o n s i s t e n t model framework f o r e x p l a n a t i o n s  The  discussions.  i n h y p o t h e s e s f o r t h a t m a t t e r ) we  concept of c i t y  the  schemes i n t o a m e a n i n g f u l  i n t e r e s t s of avoiding  o f c o u r s e , the  to  the may  consistent  (Harvey,  as we  have  the  system  seen,  essential) for stating  so t h a t t h e y may  be  words, p e r c e i v i n g c i t y  empirically s e t s as  systems  d e f e n d e d as a most s a t i s f a c t o r y m e t h o d o l o g i c a l  80 device  i n that i t serves to  geographic  (i)  i n i t i a t e and ( i i )  theory.  Upon c o n s i d e r i n g s u c h a d i f f u s e systems,  t o p i c as  city  where d e b a t e c o v e r s a number o f i s s u e s ,  be more p r o d u c t i v e t o i s o l a t e  i t may  several points for  r a t h e r than attempt to u n i t e the f r e q u e n t l y ideas  substantiate  i n a c h r o n o l o g i c a l sense.  discussion  independent  The s t a t e m e n t u s e d  i n t r o d u c e t h i s s e c t i o n appears to h i g h l i g h t these (i)  What c o n s t i t u t e s  "any l a r g e  (ii)  How do we d e s c r i v e  (iii)  How do we d e m o n s t r a t e  to  issues:  area"?  a "pattern of c i t y that  "two  sizes"?  frequencies  a r e v e r y much a l i k e " ? (iv)  I n what manner do we " e x p l a i n o r i n t e r p r e t "  these e m p i r i c a l r e g u l a r i t i e s ? I t i s hoped t h a t through a n e x a m i n a t i o n o f t h e s e q u e s t i o n s we c a n d e f i n e w h i c h m e t h o d o l o g i c a l are wanting i n e x i s t i n g  basic  qualities  investigations. The S t u d y A r e a  The i n i t i a l m a t t e r i s b y f a r t h e most  neglected  a l t h o u g h i t s h o u l d be c o n s i d e r e d c r i t i c a l t o any i n t e r p r e t a t i o n of c i t y size d i s t r i b u t i o n s .  Without devoting  some a t t e n t i o n t o t h e n a t u r e o f the s t u d y a r e a a p o i n t of view i s forgone  and c o m p a r a b i l i t y o f  i n v e s t i g a t i o n s becomes i m p o s s i b l e . pretation, a "large  consistent  different  In i t s loosest  a r e a " i s an i n t u i t i v e g e n e r a l  interclass-  i f i c a t i o n i n t h a t we a r e a b s t r a c t i n g a s u b s e t o f u r b a n centers  from t h e u n i v e r s a l s e t  of a l l centers  (whatever  81 our d e f i n i t i o n o f " u r b a n " may b e ) . c l a s s i f i c a t i o n process  But such an a r b i t r a r y  seems h a r d l y a c c e p t a b l e  toward  o f f e r i n g c o n s i s t e n t s e l e c t i o n measures t h r o u g h o u t and  space  time. The  earliest  appropriate  study  studies consider entire nations as  regions.  1941), and Z i p f (1949) g i v e  Auerbach  (1913). L o t k a  original  impetus t o t h e r a n k - s i z e  t h e s i s as a d e s c r i p t i o n o f the s i z e country fact,  above some d e s i g n a t e d  there  i s a definite  ofa l l cities  (1924,  ina  population threshold.  In  theme o f n a t i o n a l i n t e g r a t i o n i n  many o f t h e r a n k - s i z e arguments ( Z i p f , 19^9; J . Q. S t e w a r t , 1947;  1961),  Berry,  Jefferson the city  principle  (1939t231),  on t h e o t h e r  hand, evokes  o f t h e p r i m a t e c i t y where "A c o u n t r y ' s  i s always d i s p r o p o r t i o n a t e l y l a r g e and e x c e p t i o n a l l y  expressive  o f n a t i o n a l c a p a c i t y and f e e l i n g . "  only the t r i o o f l a r g e s t centers n a t i o n a l u n i t s i n order  t o index  He c o n s i d e r s  t h r o u g h o u t a sample o f s i z e r e l a t i o n s and,  t h e r e f o r e , t h e domain o f h i s s t a t e m e n t i s s e v e r e l y W h i l e t h e l a r g e s t c e n t e r a p p e a r s t o be much g r e a t e r the  leading  second and t h i r d c e n t e r s ,  i s considered  restricted. than  i t i s n e v e r c l e a r how i t  disproportionately greater.  T h i s d i f f e r e n c e o f o p i n i o n i s o n l y resolved when we s t i p u l a t e what i s c o n s i d e r e d space i n each n a t i o n a l u n i t . populations  i n the f i r s t  models (see C h a p t e r 3 ) ,  an appropriate  sample  F o r example, when we g e n e r a t e d  two s i z e c l a s s e s o f t h e h i e r a r c h i a l t h e l a r g e s t c e n t e r was always much  82 greater was  t h a n t h e f o l l o w i n g two,  b u t o n l y t o a degree  d e t e r m i n e d by t h e p a r a m e t e r s o f t h e e n t i r e  A review of J e f f e r s o n ' s  system.  (1939:228) d a t a i n d i c a t e s t h e r e i s  s u f f i c i e n t r e a s o n t o doubt h i s law s t a t e m e n t on t h e samples  alone;  to also i n f e r a property  characteristic small  that  that  small  i s supposedly  o f t h e e n t i r e n a t i o n a l system from s u c h a  sample i s y e t a n o t h e r m a t t e r and must be  with a d d i t i o n a l  treated  skepticism.  However, a more e s s e n t i a l problem must be s e t t l e d before city  we  c a n even c o n s i d e r  the c o m p a r i s o n o f t h e o v e r a l l  s i z e d i s t r i b u t i o n s i n these n a t i o n a l  territories:  we must s t a t e u n e q u i v o c a l l y  whether t h e argument c o n c e r n s  city  The  calls  sets or c i t y  l a t t e r term, o f c o u r s e ,  f o r a d d i t i o n a l f u n c t i o n a l r e l a t i o n s h i p s among t h e  urban centers criteria set.  systems.  of s e l f - s u f f i c i e n c y or c l o s u r e  The  derived  and l i k e w i s e s u g g e s t s t h a t t h e r e  i m p o r t a n c e o f t h i s dichotomy  from systems may  vice versa.  e x i s t some  within a i s that  city  inferences  be c a r r i e d i n t o s e t s b u t n o t  Or, t o t a k e an example, Z i p f (19^9)  r e a l l y s t a t e t h a t t h e exponent  "b" i n a r a n k - s i z e  cannot relation-  s h i p i n d i c a t e s whether o r n o t a n a t i o n a l system o f u r b a n communities  i s i n t e g r a t e d when he s e l e c t s t h e e l e m e n t s o f  t h a t system  in a priori  It immediate  seems t h a t o u r c h o i c e  must r e s t s o l e l y upon t h e  purpose o f o u r argument.  a rather general persist  fashion.  I f we  wish to suggest  e m p i r i c a l r e l a t i o n s h i p t h a t appears t o  ( i ) i n t e r n a t i o n a l l y a t one  point  i n time o r  (iiO  n a t i o n a l l y f o r s e v e r a l time p e r i o d s ,  grouping of c i t i e s  (above some minimum t h r e s h o l d  seems a r e a s o n a b l e a p p r o a c h . On to formulate  t h e n the  the  other  level)  hand, i f we  ( i ) c o m p a r i s o n s between s u b n a t i o n a l  n a t i o n a l u n i t s or s e l v e s , o r i f we  ( i i ) amongst the plan to  (iii)  simple  subnational  wish  and  units  them-  o f f e r some economic r a t i o n a l e  f o r t h i s r e g u l a r i t y , t h e n the  systems c o n c e p t seems a b s o l u t e l y  necessary.  systems framework i s  Adherence t o t h i s  a measure f o r e n s u r i n g inferences size  s i n c e we  have no  d i s t r i b u t i o n s are  delimited.  Since  populations  are  and  consistency  the  simply  or a c c u r a c y i n  our  reason to expect t h a t  the  i n d e p e n d e n t o f how  study areas  i t i s a most common t h e s i s t h a t  d e t e r m i n e d by  functional  urban  differentiation  degree o f i n t e r a c t i o n among economic  activities  distributed  i n space, i t i s s e n s i b l e t h a t these  e l e m e n t s be  d e l i m i t e d by  the v e r y  are  factors that  urban determine  t h e i r magnitude. Spatial  i n t e r a c t i o n ( s o c i o - e c o n o m i c f l o w s ) between  c i t y p a i r s seems a d e q u a t e l y d e s c r i b e d hypotheses  ( g r a v i t y models and  in particular), cultural,  and  but  t o the  graph theory  political barriers.  i n adjacent  "...  several  interactance  applications,  d e v i a t i o n s r e s u l t from p h y s i c a l ,  that a set of c i t i e s from s e t s  by  general  Hence, when we  i n a national territory territories intuitive  f i c a t i o n , namely, t h a t the a n o t h e r as p o s s i b l e and  (and  notion  c l a s s e s be  postulate  is distinct  therefore as r e g a r d s as d i s t i n c t  corresponds classifrom  i n t e r n a l l y as homogeneous as  one  8k  (Harvey, 1 9 6 9 : 3 3 9 ) ) , we  possible"  are r e a l l y a s s e r t i n g t h a t  i n t e r n a t i o n a l b o u n d a r i e s are b a r r i e r s o f paramount i n f l u e n c e . Fortunately, this  e m p i r i c a l evidence tends to f i r m l y Hackay (1958)  supposition.  (1961),  f o r example, f i n d  between Canada and traffic  distance  the U n i t e d  N y s t u e n and  States  considerably  o f the  o f f with  (i) varying  f l o w commodities o r area.  signs that c i t y  of n a t i o n a l c i t y  s e t s and  city  Studying  same v e i n .  frontiers  d i f f e r e n t i a l l y shock s t a b l e p a t t e r n s failure  o f t h e s e new  space-economy c a n  However, r e t a i n i n g the an  implicit  r o l e would s t i l l  ception concerning  i n the  or 1966)  (Friedmann,  o f n a t i o n a l growth,  elements to e n t e r  only p e r s i s t  the  While  a n n e x a t i o n o r c e s s i o n o f a r e a s (and u r b a n c e n t e r s )  the  systems  size distributions  t h r o u g h t i m e seems t o f o l l o w i n the  development f o r c e d upon r e s o u r c e  ( i i ) the  Nevertheless  are h i g h l y c o i n c i d e n t a t the n a t i o n a l l e v e l . changing f e a t u r e s  reduces  However, i t i s h a r d l y c l e a r  p o t e n t i a l o f the g e n e r a l  are reasonable  Dacey  i n t e r n a t i o n a l boundary  i n t e r a c t i o n tapers  decay q u a l i t i e s  population there  t h a t the  between c i t y p a i r s .  a t t h i s time how  and  support  the  short  national  run.  systems e x p r e s s i o n  i n but  p r e v e n t an a p p a r e n t m i s c o n -  i n t e r n a t i o n a l comparison,  Consider  the  nature of u r b a n i z a t i o n i n today's t y p i c a l  underdeveloped  (low  r e g i o n a l dualism)  p e r c a p i t a income, t e c h n o l o g i c a l and  nations  j u s t p r i o r t o the  technical innovation, of i n i t i a l  i n t r o d u c t i o n of investment  T h e i r urban s t r u c t u r e s at t h i s  awareness ( o r a d o p t i o n ) o f the m e r i t s  and time  o f economic  85 competition  may  comparatively On  the  be  generally represented  small  other  and  rather elaborate  Now  surpluses  a l l o w ) the  advantage.  However, i n the  of administrative  markets, t r a n s p o r t a t i o n routes etc.).  h i e r a r c h y may initial  I n f a c t , the be  considered  conditions  amount o f a r a b l e c o n s t r a i n the The  and  example, l a r g e r u r b a n c e n t e r s  places,  space-economy the  stages of  initial  suggestion  social  tend to focus  original  organization  relatively l i t t l e  larger  socio-political of  (terrain  several and  c u l t u r a l v a r i a b l e s - which  long run  development.  economic h i e r a r c h y  is  spatial  (when f a c t o r s o f  production  a c t i v i t i e s whose i n p u t p r i c e s  vary  i n space become i n c r e a s i n g l y demanded)  t h e n makes p o s s i b l e and  the  on the  c o n f i g u r a t i o n o f economic  i n the  t e n d t o be m o b i l e and  political,  development,  as the most p r e v a l e n t  i s o f f e r e d t h a t the  only  comparative  concentrated  more a c o n d i t i o n o f c o n v e r g e n c e t h a t dominates  and  emergence  organization  o f f e r more  - demographic, p h y s i c a l  l a n d ) , and  that  p r i n c i p l e s o f economic earliest  a  centers.  economic space i s somewhat d i r e c t e d by  existing lines (for  large  base i s accompanied by  system d e f i n e d by  persists  hierarchy  e v o l u t i o n of a progressive  from t h i s s u b s i s t e n c e  there  lineaments of  s o c i a l and/or p o l i t i c a l  the  of  independent a g r i c u l t u r a l communities.  a c c o u n t s f o r s i z e d i f f e r e n c e s among the  a nation's  a number  hand, i n some o f t h e s e n a t i o n s  (where p r o d u c t i o n  of a c i t y  by  the  thorough i n t e g r a t i o n of  economic s p a c e s as a u n i t  (Friedmann,  social, 1961).  86 The i m p o r t a n t p o i n t h e r e , though, i s t h a t  nations  w i t h comparable low i n d i c e s o f economic development maybe r e p r e s e n t e d  by c o n s i d e r a b l e  distributions.  Also,  variety i n their city  size  i t becomes dangerous t o assume  that  i n t e r n a t i o n a l cross-time d a t a are simply  1967).  This  d a t a and n a t i o n a l  interchangeable  (Lasuen, L o r c a ,  (1961:585)  implies that Berry  justified  i n making t h e t i m e - s e r i e s  different  city  the size  relative  i n t e r n a t i o n a l nature.  e s p e c i a l l y i n the r o l e  ( D z i e w o n s k i , 1964? L a s u e n ,  and O r i a , 1967? V a p n a r s k y , 1 9 6 9 ) , suggesting  that national  units necessarily delimit city the v a l u e  of subnational  Mackay's (1958) administrative  However t h e  administrative  systems t e n d t o r e f u t e  p o l i t i c a l u n i t s as p a r a l l e l  cases.  study of i n t e r a c t i o n i n d i c a t e s that i n t e r n a l boundaries are c o n s i d e r a b l y  than i n t e r n a t i o n a l boundaries, illustrate  i s increasing  s i z e d i s t r i b u t i o n s i n sub-  n a t i o n a l or r e g i o n a l t e r r i t o r i e s ,  same s t u d i e s  Rank  when h i s  c o n t r i b u t i o n s , there  s t r e s s on i n v e s t i g a t i n g c i t y  device  that  o f a process i n which n a t i o n a l  are of a cross-time  of a crude planning  really  r e l a t e d to  economic development o f c o u n t r i e s .  I n t h e more r e c e n t  Oria,  statement " . . .  u n i t y i s e x p r e s s e d i n a system o f c i t i e s . "  Lorca,  and  i s not  s i z e d i s t r i b u t i o n s a r e i n no way  i s not the c u l m i n a t i o n  observations  time-series  that centers  N y s t u e n and Dacey  i n t h e p o l i t i c a l r e g i o n may  p e r i p h e r a l communities i n an a d j a c e n t t h e n , the e x i s t e n c e  more permeable  of a regional c i t y  region.  (I96I) dominate  Clearly,  system t h a t  displays  87 functional  structuring i s rather  independent of s t a t e  p r o v i n c i a l "boundaries, e x c e p t where n a t i o n a l advocates high that there  closure  i n these u n i t s .  e x i s t s great  v a r i e t y i n the  r e g i o n a l economies (as d e l i m i t e d by so much, i n f a c t , t h a t inferences sets  of  We  one  from o r p r o v i d i n g  policy  should  nature of  political  wonders i f we  are  stress these  boundaries)j drawing  i n t e r p r e t a t i o n s f o r comparable  cities.  C.  T.  (1958) i s  Stewart  p e r h a p s the  earliest  o b s e r v e r t o p r o p o s e a more a d a p t a b l e d e f i n i t i o n o f study area.  He  i o n t h a t may  lead to g e n e r a l i z a t i o n ,  suggests that  e x t e n d s t h i s v i e w by rank-size The  and  the  Vapnarsky  condition  units display properties "...  t o t a l l y unrelated  t o d e f e n d h y p o t h e s e s and  such a r b i t r a r y areas Similarly,  (Vapnarsky,  t o the the  closure.  of nodal  regions,  to e i t h e r  closure  , ," and  explanation  based  from the  the  regional  n a t i o n a l l e v e l without being t o t a l l y confused  s c a l e problem  (Harvey,  seems, though, t h a t we  it  1969*589).  the u r b a n system seems t o be  means o f c a r r y i n g g e n e r a l i z a t i o n s  of  subnational  or i n t e r d e p e n d e n c e i n an e c o l o g i c a l s e n s e . is difficult  of  criter-  (I969)  complementary terms  t e n o r o f h i s argument i s t h a t u n l e s s  then they are  the  s e l f - s u f f i c i e n c y i s the  f r a m i n g the  primacy w i t h i n  administrative  on  or  1969:352-353, 452-454).  s h o u l d be more c o n f i d e n t  i n t e r r e g i o n a l comparisons t h a n i n c a r r y i n g through d i f f e r e n t r e s o l u t i o n l e v e l s .  only level by It  i n making  generalizations  88  We  see w i t h i n the mainstream o f t h e c i t y  size  t o p i c a c e r t a i n change i n the purpose o f i n v e s t i g a t i o n the concomitant i n c r e a s e i n a t t e n t i o n devoted to methodology. explain city  W h i l e o p i n i o n s may size patterns  still  differ  and  refining  on how  to  i t i s c l e a r t h a t an i n t e r p r e -  t a t i o n r e s t i n g upon economic r a t i o n a l e r e q u i r e s some c o m p a r a b i l i t y i n d e f i n i n g systems.  Unfortunately,  c o m p a r a b i l i t y i s o n l y g i v e n i n terms l i k e interdependence We to  note,  and  t h a t a r e h a r d l y o b j e c t i v e l y s t a t e d as too, that there  a v o i d f o r m u l a t i n g these  c e n t r a l place theory.  i s a n o t i c e a b l e tendency  t h r o u g h an a r e a l ( h i n t e r l a n d  t h e l a r g e s t c e n t e r ) p o i n t o f v i e w , when t h e  an a p p r o a c h more i n l i n e w i t h t h e o r y , l o g i c a l the c e n t e r s i s c a l l e d be  f o r , so t h a t the l i n e s  studied (Marshall,  L o s c h i a n model  circulation  1966).  model i n t h i s r e g a r d ,  of  a city  Unfortunately,  and  so f l e x i b l e the  as the  of  the  Christaller  idea i s s t i l l  the o t h e r hand, we  system and  expect  the r e l a t e d n o d a l  dominant c e n t e r b r e a k s down i n o n l y the s m a l l e r c e n t e r s may systems.  division  o f dominance  confined  more a g r i c u l t u r a l r e g i o n s where the t e r t i a r y On  With  ( t h a t seems more s a t i s f a c t o r y f o r the  secondary s e c t o r ) i s not  paramount.  of  I n o t h e r words, c i t y systems a r e  t h a t bonds t h e system i n a whole i s c l e a r l y l i n e a r .  may  yet.  p r o p e r t i e s a l o n g the l i n e s  commonly d e l i m i t e d a p r i o r i of  closure  this  be  to  sector i s  t h a t the  region of  coincidence the  p e r i p h e r y where  evenly a t t r a c t e d to  adjacent  89 I n summary, "any  large area"  comparable base i n o n l y a q u a l i f i e d city  size distributions  ( B e r r y and  Garrison,  d e s c r i p t i v e undertaking  like  c l o s u r e and  i n our  but  o f the  f o r example, i s an  i t clouds  size,  the  Washington S t a t e  the  spacing,  interesting  case f o r r a t i o n a l  interdependency, which are  only theory  a  Comparing  O n l y t h r o u g h the use  u r b a n c e n t e r s , c a n we this  sense.  o f K o r e a and  1958a:83),  economic e x p l a n a t i o n .  i s adequate as  and  of  notions  firmly  based  functions  hope t o g i v e a c o n s i s t e n t tone  of  to  explanation.  C i t y Size Patterns:  Skew D i s t r i b u t i o n s and  A second g e n e r a l o f the  city  t h a t the t o be  i s d i v i d e d o v e r the n a t u r e and  their role  distributions.  To  While a l l o b s e r v e r s  d i s t r i b u t i o n o f c i t i e s by  h i g h l y skewed i n the  centers  add  Concepts  i s s u e c o n c e r n s the d e s c r i p t i o n  s i z e arrangement.  frequency  Related  size  shape o f a r e v e r s e - J , o f the  size  i n determining  agree appears  opinion  c l a s s e s of urban these  t o the c o n f u s i o n ,  t o a number o f p r o b a b i l i t y d i s t r i b u t i o n s  frequency  critics (each  point  showing  a  f a m i l y r e s e m b l a n c e t h r o u g h p o s i t i v e skewness) t h a t  adequately  d e s c r i b e the  The  first the  p a t t e r n o f c i t y s i z e s i n many a r e a s .  o f these  problems i s t a k e n up  d i s c u s s i o n while  here.  the  latter  at a l a t e r  i s o f immediate  time i n concern  90  The Rank-Size and Pareto D i s t r i b u t i o n s A large p o r t i o n of the l i t e r a t u r e i s devoted to the a p p l i c a b i l i t y of the rank-size p r i n c i p l e .  For conven-  ience, we r e c a l l the r e l e v a n t equations stated e a r l i e r :  % log  =  R b p  p  R  R =  . . . log  ^  -  b  log  R  have no  justification  a p r e c i s e l e a s t squares f i t to  i s f r e q u e n t l y proposed that a constant  "B"  (3,34)  and i t  (where B  a f f o r d s an improved s t a t i s t i c a l d e s c r i p t i o n of the ship.  (3.34)  . . .  In e m p i r i c a l approaches, however, we f o r expecting  (3.33)  Pj^) relation-  T h i s approximation i s defensible i n l i g h t of ( i ) the  n o t i o n t h a t populations may  be accepted  as being the same  when they d i f f e r only by chance (Thomas, 1 9 6 1 ) and  ( i i ) the  data l i a b i l i t i e s regarding the d e f i n i t i o n of the i n d i v i d u a l urban centers  (that i s , are corporate  or  metropolitan  e n t i t i e s most appropriate?), but must be determined i n a t o t a l l y o b j e c t i v e manner. By p l o t t i n g ranks versus  s i z e s on double l o g a r i t h m i c  paper, s t r a i g h t l i n e tendencies may sample follows  (3.34).  be observed where the  Goodness of f i t may  through the l i n e a r r e g r e s s i o n model.  be determined  The use of  then, i s f e a s i b l e i f ( i ) we are confident i n the c o v a r i a t i o n of the two v a r i a b l e s with center and  ( i i ) "B"  "B", overall  "p^" as the l a r g e s t  i t s e l f l i e s w i t h i n some l i m i t i n g e r r o r  band determined by the sample (Thomas,  1967).  91 Hence,  i ng e n e r a l ,  t h erank-size  equations  may be r e i n t e r -  preted as:  B  =  R \  .  .  = l o gB - b l o g R  .  . . (4.2)  . (4.1)  or, log  P  R  where  B £  (1936)  Singer a  similar  in  Pareto  t h ec i t y  A  R  R  p  (1954)  and Allen  curve  size  =  %  argue  representation,  pattern  t h ecase f o r  where  i scharacterized  regularity  by:  . . . (4.3)  a  or,  log  R  The  Pareto  of  comparing  do  suggest  =  the  relative  squares of  centers.  of  number  index  interpretation  i n (4.4)  designates  and large  centers,  metropolization"  "a" i ssmall  thegreater  comparison  populations  medium,  "index  i na given  " a "i sa u s e f u l  p l a c e systems b u t t h e a u t h o r s  theslope  o f small,  When  i nt h e i n t e r e s t s  a practical  Since  i sf l a t ,  cities  international urban  number  . . . (4,4)  R  useful  central  a satisfactory  line  large  words,  "a".  1936:254).  (Singer,  i sl e s s  ( i f ambiguously)  t h e exponent  becomes  form  theoretical  for  it  l o g A - a l o gp  and the least  becomes  the proportion  o f cities.  f o r historical  since  I n other description and  i ti l l u s t r a t e s  a r edistributed  among  how  different  total sized  On  the other  interpretation,  " h " may  (4.1)  since  i a  hand,  he  given  (4.3)  and  a related  are coincident  with  i  = b  and A  formulation  = B  .  Also,  when  of the rank-size  relationships  requires  that  a = b  1,  =  1913)  (Auerbach, "A"  and  the  simplest and  "B" a r e  Pareto  identical  constants. However, unique  and r a t h e r  relationship. in  which  related of  Since  grouped  i n an urban  member  by  concerned system  suggest  data  population  family)  or equal  a  t h e manner  individually possibility  Yet these  empirical ( o r some  distribution with  values.  i n t e r p r e t e d as t h e number than  are  data  out  rank-size  follow a Pareto  the use o f grouped  greater  bring  with  therein, the  f o r the largest population  i s really  of the  i s obviated.  of the reverse-J  characterized interval"*"  that  data  approaches  feature  i t i s solely  to the largest center  investigations  "R"  fundamental  a l l centers  employing  other  the complementary  are a l l  an open  class  In this  case,  of centers  with  to the s t i p u l a t e d class  boundary. Due and  nature,  consider continuous may  be  which  to the aforementioned and t o f a c i l i t a t e  t h e d i s c r e t e sample or countably  thought  data  l i a b i l i t i e s ,  a n a l y s i s we space  infinite.  frequently  of city  s i z e s as  In fact  (4.1)  o f as d i s c r e t e analogues  of  i n form  being  and  specific  Class interval i s not mistaken f o r size class i s a device attached to central place theory.  (4.3)  probability helpful employ  to a  density be  and  To  upper more  may in  tails  seems  discrete  where  the  sample  The because use  of  of  assumes  second  the  city  to  grouped  with  class  size  of  may  have  information To  center  "A"  realm  given  of  the  size  most  interested.  make  inferences  two  to  the  are  this  i n t e r v a l s of increasingly  assumptions:  size  interval;  from  rank  basic  city  i n  values).  of  data)  m u s t make  the  rather  supposition upper  t a i l  sparse;  A.  assumption p r i m a c y may  i s especially be  relevant  inadvertedly  clouded  interval.  postulates,  metropolization  Class over a  i n this  wish  become  open c l a s s these  we  deals  largest  estimate  logarithms  pj^s  the  frequently  we  i n the data  of  the  i f we  their  population  i t i s  (using  the  That  single  With index  from  are  that  That  (ii)  hy  we  important  when  approaches  regarding  d i s t r i b u t e d i n each  more  analysis  discrete  population  relationship, then  evenly  postulating  (since  i s , large  the  Pareto  (i)  is  the  Unfortunately,  It  size  (that  that  are  i t i s  points).  noticeably  distribution  Nevertheless,  we  approach  with,  specific,  (4,4),  what  discrepancies  deviate  the  of  not  begin  considerable  be  aware  continuous  intervals  the  functions.  we  may  i s related  intervals also function.  refer  suggest  how  to  the  usual  to  the  range  the index  a  variable  of  urbanization?  portion as  o f total  urban.  a good  and  "P  place  m a x  through  o f course,  i nan area  f o rt h e system  that  the pro-  i s classified  of cities  may be  (4.1) a n d (4.3) w h e n we a r e c o n f i d e n t  empirical  "  being,  population  Populations  calculated of  thelatter  f i t .  Forinstance,  i sthepopulation  ("R" i s m a x i m i z e d ) ,  i f a > 1, b ^  of thethreshold  (4.3) i m p l i e s  that  1,  (smallest) total  urban  population i s :  finax  x  d  ( ? ) - ** ( =  aA 1-a  x p  ~  a  d x  max  \  A  a-1  a - l max  p  . Which  converges  ( a sA  0)  a t :  lim \ A - * «°  =  ) p•max  . . (4.5)  aA  s * -= r  p m  1  a  .-a "  x  . . .  A  (4.6)  or.  1  B  which  b  1  b-1  ' max  . . .  p  arecorrected  forms  o f t h e Beckmann  (4.7)  (1958:247)  derivation. On t h e o t h e r that  total  urban  hand  i f a 4  population i s :  1, b > 1,  (4.1) s u g g e s t s  95  H  dy = B  (4.8) 0) at«  w h i c h c o n v e r g e s (M lim M  B b-1  00  (4.9)  1 (4.10)  1-a Unfortunately, not  a s i m i l a r harmonic s e r i e s d e r i v a t i o n i s  p o s s i b l e f o r a = b = 1, b u t t o t a l u r b a n p o p u l a t i o n i n  t h i s c a s e i s always l e s s t h a n A ( l o g R + 1 ) . g  (1947) p r o v i d e s  convenient  approximations to f i n i t e  mations f o r a l l three v a r i t i e s .  o f the index  i t may be a t t r i b u t e d  (4,6) o r rank (4,9) d e p e n d i n g on t h e n a t u r e  of metropolization.  I n any e v e n t , when a r a n k - s i z e o r P a r e t o ship holds  of m e t r o p o l i z a t i o n "a",  r e g i o n through  ( i i ) the constant  of the t h r e s h o l d center  population.  relation-  f o r a system o f c i t i e s we c a n f o r m u l a t e  of u r b a n i z a t i o n f o r the nodal  size  sum-  I t i s interesting to  n o t e t h a t when c o n v e r g e n c e does o c c u r to e i t h e r s i z e  J . Q. S t e w a r t  an i n d e x  ( i ) the index  "A", ( i i i ) t h e  ( a > 1 ) , and ( i v ) t o t a l  I t i s suggested t h a t t h i s frequency  parameter  ( m e t r o p o l i z a t i o n ) complements an a g g r e g a t e u r b a n i z a t i o n index  t h a t may be b l u r r e d by v a r y i n g i n t e r n a t i o n a l concep-  tions of rural-urban  distinction.  1 96 Steady-state  Distributions  Some v e r y p a t t e r n s as process. we  are  intriguing  equilibrium  The  efforts treat  states  o f an u n d e r l y i n g  related  t h e o r y may  surprising  that  i n the be  hope t h a t  implied.  the  For  system o f c i t i e s  at  be  time  c l a s s b o u n d a r i e s are  " t ^ " divided  p r e n e u r s h i p ) o p e r a t e on  likely  p o p u l a t i o n i n the The  be  a regular matrix (i)  t^,  the  a  are  not  in  given  Now,  considering class  logarithms as  individual  a  . itervals of  forces  investment, technology,  R  communities initial  entrethrough  class  assume v a r y i n g p r o p o r t i o n s o f the  total  system. o f c e n t e r s among the  d e s c r i b e d by  T h a t the  class  transition probabilities  (Adelman, 1 9 5 8 ) .  I f we  in  assume:  d i s t r i b u t i o n of percentage  o v e r a time i n t e r v a l i s the (ii)  into  i s , the  . . ., t ) , the  redistribution  i n t e r v a l s may  (that  evenly spaced).  migration,  (consider tg,  etc.)  i l l u s t r a t e d by  of equal proportionate width  intervals  of  interpretations.  arguement may  instance:  an  d i v e r s e phenomena (word f r e q u e n c i e s  identical probability The  structure  to  t h i s reason i t i s  p r o s e samples, income d i s t r i b u t i o n s ,  time  priori:  d e v e l o p i n g some r e a l w o r l d i n t e r p r e t a t i o n calculus  time  stochastic  approach t h e r e f o r e , i s t o t a l l y a  abstract  (for  city-size  same i n each c l a s s  changes  interval;  T h a t t h e s e changes r e m a i n i n v a r i a n t  over a l l  intervals;  t h e n any  initial  d i s t r i b u t i o n of c i t y s i z e s  approaches  a  u n i q u e e q u i l i b r i u m s t a t e as t the  first  the  frequency  ulation  >^  n  p o s t u l a t e as a normal d i s t r i b u t i o n distributions  depict  (that i s ,  o f p e r c e n t a g e changes i n pop-  s i z e o f s m a l l , medium, and  a p p r o a c h normal d i s t r i b u t i o n s w i t h t h e n we  When we  l a r g e communities a l l the  a r e e s s e n t i a l l y assuming the  same  law  parameters),  of  proportionate  effectt A v a r i a t e s u b j e c t t o a p r o c e s s o f change i s s a i d to obey the law o f p r o p o r t i o n a t e e f f e c t i f the change i n the v a r i a t e a t any s t e p o f the p r o c e s s i s a random p r o p o r t i o n o f the p r e v i o u s v a l u e o f the v a r i a t e . ( A i t c h i s o n and Brown, 1957:22). The  i m p l i c a t i o n here,  are l o g n o r m a l l y  of course,  distributed  average number o f c e n t e r s p e r time p e r i o d e q u a l s The o f the  total  i n the  steady  e n t e r i n g each c l a s s  the  average number  p o p u l a t i o n i n the  of i n d i v i d u a l  populations  centers  " t " as i n " t , " ) . n -L  that t h i s estimate since  given.  " t ^ " , t h e n the t h r o u g h the  We  spreading  stochastic matrix  should r e a l i z e ,  Simon (1955) a v o i d s  c l a s s are the f i r s t  i n t r o d u c t i o n o f new  1958,  (Adelman,  i s the  however,  i s based on r a t h e r i n f l e x i b l e  e x t e n s i o n beyond the h i g h e s t  For  nearly  i n each c l a s s i n t e r v a l  ( i ) f u r t h e r e n t r y i n t o the l o w e s t  a l l o w i n g a steady  estimation  having  a c c o u n t s f o r a growth i n t o t a l p o p u l a t i o n assumes t h a t the mean v a l u e  interval  departing.  system t o be  t h i n k o f a l l the  i d e n t i c a l p r o p u l a t i o n s a t time  same a t  s t a t e , when the  second assumption allows a crude  i n s t a n c e , i f we  out  i s that population s i z e s  postulates  c l a s s and ( i i ) disallowed.  restriction  by  communities o v e r  the  98 threshold  size.  The  thrust  i n t e r p r e t a t i o n f o r the f(x)  (where  w e i g h t i n g o f the  and  "a"),  "C"  as the  Beta f u n c t i o n of  i n (4.3);  that  new  centers  "f(x)"  lowest value of  where " P " the  density  making the again  (  the  conceptual leap  value  may  integrate  e s t a b l i s h the  t o the  i s z e r o but  This "x"  the  and  zero. that:  (4.12)  (4.12)  is clearly  Therefore,  by  variable"x"  assumes  p r o b a b i l i t y that i t  "x."  d i s t r i b u t i o n function =  interpreted  c o n t i n u o u s case once  " f ( x ) " between "1"  may  and  be  determined),  infinity  to  " F ( x ) " where:  0  for x <  T  ll-TA"^  for x y  T  f  F(x)  be  . . .  + 1 )  P a r e t o law.  assumes a v a l u e between "x." we  size  Simon d e m o n s t r a t e s  p r o b a b i l i t y t h a t the  "x^"  "T" may  gamma f u n c t i o n .  f u n c t i o n o f the  (where the  a given  0,  C P(/+l)x- ^  represents  of  anala-  f o r which " f ( x ) " exceeds  However, where x =  the  i s r e a l l y employed  constant.  variable  and  determines  i s the number o f c i t i e s  "x"  (4.11)  (a random  i s , urban s i z e )  i s a constant that but  . . .  "x"  > 1  i s a normalizing  f(x)  function:  f o r x 2- T  "p^" "f"  an  for x £ T  i s the  comparable t o  "x"  probability density  =  where £(x,f+l)  gously to  o f h i s argument i s t o p r o v i d e  )  i n d i c a t e s the  p r o b a b i l i t y t h a t the  (4.13)  ...  continuous v a r i a b l e  assumes a v a l u e i n a s p e c i f i e d c l a s s i n t e r v a l ,  a  99  c o n d i t i o n t h a t we may employ t o determine the number *'N(x)" o f c e n t e r s having p o p u l a t i o n s g r e a t e r than o r equal to "x" (x 7 T ) : N(x)  =  .  . .  (4.14)  where "X" i s the t o t a l number o f communities i n the system. T h i s d e r i v a t i o n i s c e r t a i n l y analagous t o the Pareto ship given i n (4.3).  relation-  B e s i d e s , t h i s i n t e r p r e t a t i o n demon-  s t r a t e s p r e c i s e l y how the simple B e r r y - G a r r i s o n (1958a:88-89) a p p l i c a t i o n may be c o n s i d e r e d a v a l i d d i s c r e t e r e p r e s e n t a t i o n o f Simon's s t o c h a s t i c model ( d e s p i t e the apparent conceptual flaw i n t h e i r e f f o r t :  see footnote 37 o f page 88 o f  B e r r y and G a r r i s o n a r t i c l e ) . The lognormal d i s t r i b u t i o n t r u n c a t e d a t p o i n t "T" ( A i t c h i s o n and Brown, 1957'87-99) was g i v e n i l l u s t r a t i o n i n m a t r i x form above and Adelman (1958:894) provides a l i n k t o the Yule i n t e r p r e t a t i o n i n a Markov process v i a ". , . a r e s e v o i r o f p o t e n t i a l e n t r a n t s . "  While i t i s  d i f f i c u l t t o s p e c i f i c a l l y compare the p r o b a b i l i t y d e n s i t i e s of the two d i s t r i b u t i o n s through t h e i r parameters, i t appears t h a t the Yule approach g i v e s a s u p e r i o r f i t near the p o i n t o f t r u n c a t i o n .  I n f a c t , by g e n e r a t i n g a p e r f e c t  h y p o t h e t i c a l r a n k - s i z e d i s t r i b u t i o n i t i s simple t o demons t r a t e by g r a p h i c a l method t h a t the best f i t t i n g  lognormal  t a i l o v e r p r e d i c t s the number o f communities i n the f i r s t class interval.  100 Without that be  the  laboring  differences  between  n e g l e c t e d i f our  For  instance,  butions  many  concave  to  While  this  improved  the  (1961)  in  do  interval  the  of  on  the  rank-size  a  certain  to  size  d i s t r i -  i s properly  are scrutinized.  achieves  factors  like  an ( i ) data  ( i n a proportionate  spaces,  largest  distribution  may  distributions  by  s m a l l sample  lognormal  city  of unequal  very  conclude  imprecise.  framework  i s obscured  f o r the  i s less  Yule  effects  intervals  appear  literature  lognormal  the  various parts of  there  of  that  ( i i i )  the  communities,  parameters.  and  ( I t  provides a  superior  f i tfor  distribution  when the  exponent  than  Comparison To  study  suggests  (4,1)  i n  i s relatively  supposed  ( i i )t h e  should  distributions  when p l o t t i n g  hypothetical "b"  two  axis  nature  that  we  size  size  seems  the  f i t , certainty  open c l a s s (iv)  of  point, the  analysis  i n Berry's  the  r e l i a b i l i t y , sense)  this  unity).  of  Distributions  extent, this the  topic  has  been  previous discussion.  exist  additional  concerning the  the  covered  However,  misconceptions  similarity  of  i n  the  frequency  d i s t r i -  qualify  super-  butions. To fluous is  begin with,  dichotomy  widely held.  of The  Jefferson's  More  recent social  hope  of  primacy  appeal  scientists  realizing  must  and  distinction,  (1939)  from  in  the  i t s e e m s we  rank-size  though,  terms  seems  for-a uniqueness emphasize  general law  a  to  that arise  thesis.  regularities  statements  of  human  instead,  101 behaviour.  I n any  case,  primate c i t y  precludes  attributes.  The  t o the with  adherence t o the  rational  persistence  explanation  and  are".  or with  c o n t r o l s i n indigenous subsistence rank-size  S u r e l y , no  (i)  size centers guished  I t may  be  By  emergence o f one  the  By  rigorous  and, intermediate  double-logarithmic or s i z e versus  plotting  are  of c i t y  inductive sizes,  This  then  whole  implies, of  course,  problem must employ  ( i n e i t h e r size versus  c u m u l a t i v e p e r c e n t a g e form) w i t h  d e v o t e d t o c o r r e l a t i o n s and  distin-  centers.  d e t a c h e d from the  t r e a t m e n t o f the  ways$  "capital"  prefer a s t r i c t l y  d i s t r i b u t i o n o f community s i z e s .  sizes  general  so t h a t a group o f l a r g e c e n t e r s  from a n o t h e r group o f s m a l l  s i o n model.  great  a d e f i c i e n c y or v o i d of  i d e a o f p r i m a c y cannot be  t h a t any  t r e a t i n g primacy  a c c o u n t e d f o r i n two  p o i n t i s t h a t even i f we  . .  systems i n  o v e r a l l pattern of c i t y  o r e m p i r i c a l a p p r o a c h t o the q u e s t i o n the  peasant s o c i e t i e s .  1961:574).  i n J e f f e r s o n ' s extreme c o n c e p t i o n j (ii)  The  and  g e n e r a l i t y i s l o s t by  a d e v i a t i o n from the  i n a system.  as  (Berry,  economies  political-administrative  r e l a t i o n s d e p i c t complex c i t y  more advanced n a t i o n s  as b u t  . . associated  superimposed c o l o n i a l  i n underdeveloped c o u n t r i e s  and  o f i t s economic  o f t h i s d i v i s i o n seems r e l a t e d  n o t i o n t h a t primate c i t i e s  overurbanization  i d e a o f a unique  r e s i d u a l s o f the  rank  attention  linear  regres-  102  B e s i d e s , i n f e r e n c e s based upon l i m i t e d sample spaces ( C . T. Stewart, 1958; B e r r y , 1961; Mehta, 1964; L i n s k y , 1965* Rosing, 1966) must be t r e a t e d with reserve.  considerable  U n f o r t u n a t e l y , Berry appears t o be alone i n  r e c o g n i z i n g the importance o f t h i s i s s u e . I n v e s t i g a t o r s a l s o seem h e s i t a n t about a c c e p t i n g the s i g n i f i c a n c e o f the exponent "b" i n the r a n k - s i z e formulation.  Since the exponent v a r i e s w i t h the type o f  data c o l l e c t e d ( A l l e n , 1954) and i t i s g i v e n some i n t e r p r e t a t i o n through the h i e r a r c h i a l and Simon models, i t would appear t h a t the acceptance a p r i o r i o f a "b" value of u n i t y (Vapnarsky, I969) i s r a t h e r q u e s t i o n a b l e . Another most meaningful p o i n t should be r e i t e r a t e d a t t h i s time.  The use o f grouped data f a c i l i t a t e s a n a l y s i s  t o a g r e a t degree but tends t o obscure the more t r a d i t i o n a l form o f primacy ( t h a t i s , the " c a p i t a l " c i t y ) through an open c l a s s i n t e r v a l f o r the l a r g e s t p o p u l a t i o n s i z e s . While we should agree w i t h Lasuen, L o r c a , and O r i a (1967) t h a t B e r r y ' s lognormal  technique weights the e f f e c t o f the  l a r g e number o f s m a l l c e n t e r s i n the system, we cannot agree t h a t i t adequately v e r y g r e a t e s t community.  d e s c r i b e s the d e v i a t i o n o f the (Notice t h a t the p l o t t i n g s f o r  the sample o f Spanish c i t i e s are done i n c o r r e c t l y i n both a r t i c l e s ) .  A l l i n a l l , i t may be best t o r e t a i n a  c o n s i s t e n t approach o f d i s p l a y i n g the frequency b u t i o n s throughout any s i n g l e study.  distri-  103 In variation  conclusion  supposedly  and  Brown  s t i l l  from gain  chapter  of  grouped  In  the  we  larities  i s a  argument  seems t o  this  i s a  certain  uni-size  and  the and  i n the  next  these  empirical  the  the  role  this  line  systems  relation  this  of  to  are  i n stochastic  thinking and  these  theory  section  found  theory,  between  general  assigned  explanation  regu-  with  ( i i i )  to  probabilistic  the  properties  of d e t e r m i n i s t i c  and  Garrison  (1958ai90)  summarize  article  thing, a  the  the  Approaches  cited  to  of  general  s t o c h a s t i c argument  refers  of  models.  Berry  often  operation  Explanation  objectives of  complement  arguments  Stochastic  estimating  this  issue  are  Aitchison  of  properties  t o p i c but  about  The  nature  from  clarify  hierarchial  one  the  of  debatable  revolve  ( i i )to  notions  hopefully  of  rather  evaluate  reasoning,  that  While  forfeit  major  and  explanation  explaining reality. to  may  overlooks  discussion.  Acceptable  (i)  systems  difficulties  studying the  Interpretation  in  among  the  data, "by  fact,  investigation  characteristic.  emphasize  insight  alone.  that  slopes)  lognormally  (1957:94)  parameters  graphs  note  ( i n t e r c e p t s and  a l l  and  we  the  i n the  about  probabilistic presence  ability  to  of  c l o s i n g pages  rank-size  infinite  p r e d i c t i n these  i n some number  terms  of  i s not  case  their  relationships!  explanation an  of  the  "For  sense causes enough;  104 we  wish  broad real  explanations  world  into  exercise,  thea priori  statement  does  model  theproperties  butthetheory i sn o t c l e a r  within  effect  that  a o f the  i s an  i s portrayed  a t a l l .  an interpretation o f the statistical  provide  (1955:43?)  Simon  Mapping  ways  thelawo f proportionate  However,  of  i ne x p l i c i t  theoretical context."  interesting by  viable  some  arethat  helpful  insights.  t h eYule  The f e e l i n g s  distribution  . . . would hold i f t h egrowth o f population were due s o l e l y t o t h e n e t excess o f b i r t h s over d e a t h s , a n d i f t h i s n e t growth were p r o p o r t i o n a l to present population. This assumption i s c e r t a i n l y s a t i s f i e d a t l e a s t roughly. Moreover, i t need n o t hold f o r each c i t y , b u t only f o r t h e aggregate o f c i t i e s i neach p o p u l a t i o n band. F i n a l l y , the equation would s t i l l be s a t i s f i e d i f there were n e t m i g r a t i o n to o r from c i t i e s o f p a r t i c u l a r regions provided t h e net a d d i t i o n o r l o s s o f population o f i n d i v i d u a l c i t i e s w i t h i n a n y r e g i o n were p r o p o r t i o n a l t o c i t y size. " Ward  (1962)  extends  (information market long  frequency  proportionate provides  (i) in  becoming  opportunities  basis  interval  Migration  requires  that  be matched  proportionately  o f aggregate  i s attained  o f urban markets.  The s t o c h a s t i c  Predian  arerealized i n the  q u a l i f i c a t i o n s o f Simon's  l a r g e r must  (ii)  that  o f occuring  a particular class  tionately  discussion  distribution  t o thesize  several  i na t y p i c a l  t o act)  opportunities  The P a r e t i a n  relative  thesis  and a b i l i t y  expansion  run.  this  when t h e  i s randomly He  also  models that  a r ebecoming  cities propor-  b y a s i m i l a r number  smaller; from  rural  areas  o r abroad  i s not  105  accomodated (more than l i k e l y t h i s i s d i r e c t e d  toward  the l a r g e s t c e n t e r s ) ; (iii)  The u s e f u l n e s s o f the approach  by the nature of the data:  i s hindered  c i t y sizes i n metropolitan  or c o n u r b a t i o n form have c h a r a c t e r i s t i c a l l y lower exponents than i n c o r p o r a t e form - the a p p l i c a b i l i t y of the Yule d i s t r i b u t i o n , though, i s r e s t r i c t e d t o those v a l u e s g r e a t e r than or equal t o u n i t y . U n f o r t u n a t e l y , t h e r e appears to be at best v e r y sketchy support f o r the type o f growth p o s t u l a t e d i n the s t o c h a s t i c matrix:  Madden's (1956) study o f the s t a b i l i t y of urban  growth i n the U n i t e d S t a t e s i n d i c a t e s t h a t the law of p r o p o r t i o n a t e e f f e c t may w e l l be approximated  i n the r e a l  w o r l d when we c o n s i d e r the growth of a l l c e n t e r s (no c l a s s i n t e r v a l s ) through equal time p e r i o d s .  Obviously,  however, the i n v e s t i g a t i o n r e f u t e s q u a l i f i c a t i o n ( i ) above: t h i s p o i n t s t o a fundamental  d i s t i n c t i o n between the f r e -  quency d i s t r i b u t i o n s of percentage changes i n s i z e o f communities and, say, f i r m s (Simon and B o n i n i , 1958)  in  an i n d u s t r y . There i s c l e a r l y l e s s tendency f o r urban c e n t e r s t o take p r o p o r t i o n a t e l o s s e s i n p o p u l a t i o n (especi a l l y the l a r g e c e n t e r s ) than f o r f i r m s t o take s i m i l a r l o s s e s i n employees, value added, e t c ,  Simon's model seems  to i n d i c a t e a b e t t e r d e s c r i p t i o n of the data than of the processes i n v o l v e d . I n any case, the essence of the problem from a s c i e n t i f i c v i e w p o i n t i s t h a t the p r o b a b i l i s t i c scheme  106  avoids i n t r o d u c i n g those f a c t o r s t h a t supposedly cause the s t o c h a s t i c mechanism t o operate.  The most c r i t i c a l  of these i s c e r t a i n l y the notion o f d i s t a n c e o r s e p a r a t i o n , f o r o n l y through study o f t h i s v a r i a b l e can we understand why and how urban communities are f u n c t i o n a l l y r e l a t e d . Or, t o phrase the p o i n t d i f f e r e n t l y , the s t o c h a s t i c argument i s a t o t a l l y aggregated one i n the sense t h a t i t cannot a l l o c a t e weightings  f o r d i f f e r e n t orders o f o p p o r t u n i t i e s .  Without knowledge o f how i n d i v i d u a l elements are r e l a t e d i t becomes i m p o s s i b l e t o make p r e d i c t i o n s about t h e i r a t t r i b u t e s a t a l a t e r time. However, such d i s c u s s i o n s t h a t simply weigh the d i f f e r e n t i a l m e r i t s o f p o s i t i v i s t ( o r r a t i o n a l i s t ) and more c o n v e n t i o n a l ( o r symbolic) v i e w p o i n t s toward theoryr e a l i t y i n t e r r e l a t i o n s (Lukermann, 1961) tend to obscure an unnoticed d i s t i n c t i o n between the s t a t i s t i c a l and s p a t i a l economic arguments.  To i l l u s t r a t e , the law o f  p r o p o r t i o n a t e e f f e c t t y p i f i e s a model d e a l i n g w i t h i n d i v i d u a l p o p u l a t i o n members ( r e c a l l Simon's 1955 i n t e r p r e t a t i o n t h a t the p r o b a b i l i t y the "k+l"st person i s found i n a c e n t e r o f s i z e " x i s p r o p o r t i o n a l t o "x f * ( x ) " , the u  t o t a l p o p u l a t i o n o f communities o f t h i s s i z e ) , w h i l e the C h r i s t a l l e r o r Loschian models deal w i t h p o p u l a t i o n groups (the C h r i s t a l l e r model i s d e p i c t e d by p o p u l a t i o n a s s o c i a t e d w i t h baskets o f goods).  subsets  I n terms of systems,  the Yule d i s t r i b u t i o n p o r t r a y s random behaviour r e s o l u t i o n l e v e l while c e n t r a l place theory  a t a high  provides  p r e d i c t i o n a t a lower l e v e l (Burton, I 9 6 3 d i s c u s s e s g e n e r a l p o i n t w i t h regard t o q u a n t i f i c a t i o n ) .  this  I n modeling  terms, the former approach i s d e s c r i p t i v e , micro, and p r o b a b i l i s t i c w h i l e the l a t t e r i s a n a l y t i c a l , macro, and deterministic.  While t h i s s c a l e dichotomy does r e l a x  p h i l o s o p h i c a l debate i t cannot r e a l l y r e f u t e the s u p e r i o r value o f the economic argument as a methodological  device.  General Systems Theory General systems t h i n k i n g e s s e n t i a l l y suggests a methodology o r viewpoint  f o c u s s i n g on p r o p e r t i e s common  t o a l l types o f systems.  I t i s an approach t h a t f a v o r s  s t u d y i n g the t o t a l i t y o f r e l a t i o n s among elements and emphasizes q u a l i t i e s l i k e wholeness and o r g a n i z a t i o n (Rapoport,  1966).  H a l l and Fagen (1956) s t i p u l a t e two macroscopic p r o p e r t i e s o f systems t h a t appear r e l e v a n t t o the present discussion.  I f we r e c a l l our simple i l l u s t r a t i v e model  of how an expanding space-economy induces an urban s t r u c t u r e , then t h i s i s c a l l e d p r o g r e s s i v e s y s t e m a t i z a t i o n , s i n c e we witness: (i)  Strengthening  o f p r e - e x i s t i n g interdepend-  (ii)  Development o f r e l a t i o n s among members  encies;  previously unrelated; ( i i i ) A d d i t i o n o f p a r t s and r e l a t i o n s t o the e x i s t i n g system.  108 B e s i d e s , c e n t r a l i z a t i o n o r dominance by a l e a d i n g member (the  p r i m a t e c e n t e r ) i s a common t r a i t  an i n c r e a s e  i n t h e sum  of r e l a t i o n s .  suggests that coincidence distribution  t h a t may  accompany  Beckmann ( 1 9 5 8 )  o f a l l o m e t r y and the  (see c o r r e c t e d form i n ( 4 . 6 )  Fareto  above)  mean some o p t i m a l a s s o c i a t i o n e x i s t s between  may  population  i n t h e l e a d i n g p a r t and p o p u l a t i o n t h r o u g h o u t the city  system.  system, we  entire  I n our r a t h e r f u n c t i o n a l view o f t h e u r b a n  stress  i t s behaviour (flows, responses, etc.)  b o t h i n t e r n a l l y and t h r o u g h t r a n s a c t i o n s w i t h i t s e n v i r o n m e n t (Harvey, 1 9 6 9 * 4 5 6 ) .  We  simply consider that  as a h i g h e r o r d e r system  ( a socio-economy  consumers) from w h i c h t h e c i t y yet  necessarily, abstracted.  i n t e r p r e t a t i o n , we may the of  spatial  system  of  environment individual  i s conveniently,  From our s t r i c t l y  determine  (at l e a s t  economic  qualitatively)  e x t e n t o f the environment v i a c o n s i d e r a t i o n  the p r o p o r t i o n o f demands e x e r c i s e d t h r o u g h l o c a l  f o r e i g n markets.  N e e d l e s s t o s a y , as p r o g r e s s i v e  a t i z a t i o n o c c u r s i n the u r b a n system, the l o c a l  or  system-  or domestic  p o r t i o n o f t h e e n v i r o n m e n t becomes r e l a t i v e l y more i m p o r t a n t . Systems  are c l a s s i f i e d  i n t o c l o s e d ^ o r open t y p e s  a c c o r d i n g t o the n a t u r e o f energy exchange commodities,  innovations, c a p i t a l ,  (information,  etc.) with t h i s  -^Note t h a t c l o s u r e i n t h i s i n t e r p r e t a t i o n i s s i g n i f i c a n t l y d i f f e r e n t from i t s meaning - e a r l i e r i n the d i s c u s s i o n ? t h e r e , i t r e f e r r e d t o s p a t i a l i s o l a t i o n which a c c o u n t s f o r o n l y a p a r t o f the t o t a l environment.  109 environment. system a  Obviously thehierarchial  denotes  an open  competitive process  needs some  interesting  tant, can  o f individual  though,  state." by  themselves,  communities  cultures  that  activities where  Urban  by  elements  is  a  that  presents  Most  impor-  a  steady  competition  o f a process  state  systems  systems,  that  (homeostasis)  prices  vary  economies), causation  relatively  mostly  independent  When a l a r g e  occurrence.  does  expansion  system  markets)  i s directed  o f those  between  proportion  of the  itself,  central-  and regional  ( a s i nt h emost  levels)  negative  feedbacks such  primitive  and concomitant  arepossible  (that  i n space).  e v o l v e t o g e t h e r b u t where  feedbacks  types  p l a c e d i na more  I n other words,  notexist  positive  l i t t l e  (foreign  (at thenational  hierarchy  t o only  environment  t o demands  i sn o t i nthe c i t y  theurban  hierarchy  t o a domestic  arerather  o r primacy  i sr e s t r i c t e d  competition and t o those  relate  i nt h e system.  a natural  and  that  such  o f t h e environment  market  ization  Spatial  example  a n argument  attributed  part  linkages  total  from  " . . . living  o r developing toward  i sa good  such  value  input  growth  distant  notion.  o r g a n i z e d open  1962^7).  city  environmental disturbances. However,  is,  i t i sderived  (196?:?6-78)  ideas concerning this  as hierarchially  o f a  t o t h e wants and  Berry  t o maintain an equilibrium  despite  of  related  consumers.  (von B e r t a l a n f f y ,  urban  tends  directly  since  i st h e contention that:  be d e f i n e d  maintaining  condition,  structure  i nparticular  space-  cumulativesubsystems  110 (Haruyama, 1 9 6 3 ) .  O n l y as the l o c a l  c o n t r o l s the t o t a l market, and convergence,  may  feedbacks  I n f o r m a t i o n i s the term we  entropy,  Entropy differ  be  checked.  employ t o d e s c r i b e the  I t s thermodynamic c o u n t e r p a r t ,  i s s a i d t o i n c r e a s e when a system  randomized  becomes more  ( i n f o r m a t i o n and n e g a t i v e e n t r o p y a r e  i s maximized i n an u r b a n o n l y by  increasingly  planning favors regional  these p o s i t i v e  o r g a n i z a t i o n o f a system,  environment  system  analagous).  when community  sizes  chance.  To c l a r i f y t h i s argument we d e f i n i t i o n of entropy  c o n s i d e r the  ( K l i r and V a l a c h ,  statistical  196?i6l):  I f o u t o f "n" e v e n t s , each c a n o c c u r w i t h the n p r o b a b i l i t i e s 0,, 0 . . . . 0 , where 0. = 1 n i=l 9  1  (i.e,  some o f t h e e v e n t s do t a k e p l a c e ) , t h e n t h e n formulation H = - 27 0. l o g 0. i s c a l l e d entropy. i-l 1  It "H"  s h o u l d be  a  1  o b v i o u s t h a t when a l l "0^"  are  identical,  i s maximized. Now  we  may  interpret  ( i n descriptive  terms a t  least)  t h e d i s t r i b u t i o n o f t h e t o t a l u r b a n p o p u l a t i o n "P"  amongst  "n" c e n t e r s i n e n t r o p y terms.  as  ratio we  By c o n s i d e r i n g "0^"  o f a community's p o p u l a t i o n "p^" t o t h e t o t a l  may  illustrate (i)  (ii) H = log  a  n.  "P"  thati  Minimum e n t r o p y o c c u r s when a l l "P" i s  r e s i d e n t o f one  community and H =  0;  Maximum e n t r o p y o c c u r s when p^ = P/n  the  and  Ill However, i n a c i t y system maximum o r g a n i z a t i o n i s t h r o u g h t h e s p a t i a l h i e r a r c h y and t h e t r i v i a l H = 0 i s dismissed.  For instance,  attained  case  i f we know t h e  where population  o f a c e n t e r and i t s l e v e l i n t h e h i e r a r c h y , we c a n compute the p o p u l a t i o n s o f a l l the c e n t e r s  i n the system.  r e a l w o r l d , h i e r a r c h i e s are never p e r f e c t it  nor complete  i s h y p o t h e s i z e d by B e r r y (196?i71) t h a t the  rule  i s the steady  state that balances  I n the hut  rank-size  h i e r a r c h i a l organ-  i z a t i o n w i t h r a n d o m i z a t i o n due t o chance  local variabilities.  O b v i o u s l y i n terms o f p o p u l a t i o n f i g u r e s  alone,  a  more o r g a n i z e d s t a t e may be a t t a i n e d t h e n t h r o u g h h i e r a r c h i a l c o n s t r a i n t but theory precludes  such a c o n d i t i o n .  This  demonstrates  p r e c i s e l y why we need a g u i d i n g t h e o r y  explanation,  for a state of perfect  primacy  represents  maximum o r g a n i z a t i o n i n a p o p u l a t i o n s y s t e m b u t p l a c e t h e o r y and i t s r e l a t e d models t e l l i r r e l e v a n t i n a n economic s y s t e m .  in  central  us t h a t t h i s  Only through theory  is are  we c e r t a i n o f e l i m i n a t i n g a b s u r d i t i e s . Therefore  i t seems more n a t u r a l t h a t we s h o u l d  speak o f a c o n d i t i o n o f d e s i r a b l e  entropy,  but not  minimum o r maximum e n t r o p y when we speak o f c i t y and t h e r a n k - s i z e r u l e .  of  hierarchies  However, i t may be u s e f u l t o  an e x p r e s s i o n o f e n t r o p y f o r a d i s c r e t e  rank-size  provide  distri-  4  bution of t o t a l population "P" : Our d e r i v a t i o n c o n t r a d i c t s a f o r m u l a t i o n d e v i s e d  by v o n F o e r s t e r  ( I 9 6 6 ) , r e s t a t e d by C u r r y  i n c l u d e d i n r e v i e w s by B e r r y ( 1 9 6 4 )  (I963)  and  and O l l s o n ( 1 9 6 6 ) .  While  112 H =  log l  log  +  b  Pi  4log 2 P D  2 + b  +  log n  b  p  F u l f i l l m e n t of t h i s steady state through time r e q u i r e s that as " P " and " n " grow, " H " remains r e l a t i v e l y s t a b l e . This,  i n f a c t , hypothesizes that the p r i n c i p l e s of e q u i f i n -  a l i t y are met and population values are independent of c e n t r a l c o n d i t i o n s ( r e c a l l that t h i s i s a n a t u r a l r e s u l t of a stochastic  interpretation).  The meaning of the entropy approach may be enhanced by c o n s i d e r i n g the d i f f e r e n c e between independent settlements i n a backward economy and interdependent communities i n a progressive economy.  The i n i t i a l case represents  a  our i n t e r p r e t a t i o n s of entropy are r a t h e r d i f f e r e n t , the i n i t i a l e f f o r t s u f f e r s i n s e v e r a l respectsJ ( i ) the combinatorial r e d e f i n i t i o n of entropy means that the sum of the logarithms i n the second term of t h e i r expression must be minimized and not maximized, and ( i i ) c o r r e c t use o f Lagrangian m u l t i p l i e r s leads to t h i s m i n i m i z a t i o n . I t i s not c l e a r at a l l how entropy i s maximized through the r a n k - s i z e r u l e . Note, too, that i n our f o r m u l a t i o n above "p " i s the same as " p „ " employed e a r l i e r i n t h i s 0  chapter and each i s not to be confused with the n o t a t i o n used i n the previous chapters! " p , " , of course, i s now the l a r g e s t center i n the system.  113 weakly  l i n k e d and  maximum with  relatively  entropy:  their  the  other  the  system  a t t r i b u t e s of  size. hand,  i s persistently  through  along the  the  these  ment  of  system,  this  integration  the  (1969}  lines, that  of  space  the  added  value.  I f a l l centers  e f f e c t s of  However,  the  to  centers  large  vice  versa,  centers  implies  may  a  unique  on  population  how  a  size  economic  intertwined. individual within for  the  may  be  an  at  growth  of  Aggregate  of  alone  i s not  community  stages  entire set. to  To  i n  entropy percentage  long  run.  centers  relative  i n entropy i n  and,  larger  Reconsidered  the  t h e s i s we  urban  centers  but  only  and  despite  may  hopefully  express  become variation  a  size  hierarchial  continuum,  justified  understanding  random  illustrate a  emphasized  c a n n o t be  through  that  strong  such  have  population  unexpected  induce  the  population  populations  system  i n the  Model  parte of  basis  considered  environ-  entropy.  indivisibilities  urban  the  same  increase  of  i n  on  the  small  i n d i c a t e an  classes  It  about  constant  decrease  different  that  on  to o b j e c t i v e l y  population  concentration  The In  grow  remains  vigorous  the  and  process.  specify  entropy  into  interesting  general  i t i s difficult  then  on  competition  an  focussing  Unfortunately,  rate  uncorrelated  information  economic  four  possesses  economy,  presents  while  depicts  that  remain  bringing  framework  Vapnarsky  system  centers  Integration of  specialization. argument  closed  how  of  continuum  qualities growth  let's  recall  114  Beckmann's simple model. The c h a r a c t e r i s t i c f e a t u r e o f t h i s d e t e r m i n i s t i c " s model i s the use o f a b a s i c p r o g r e s s i o n component  j—^  +  "  ^  t h a t relates p o p u l a t i o n s on adjacent h i e r a r c h i a l l e v e l s . W i t h i n t h i s r i g i d framework, s p a t i a l v a r i a t i o n o f p o p u l a t i o n sizes i s severely r e s t r i c t e d .  As a r e s u l t , we mentioned  i n the p r e v i o u s chapter t h a t i t i s h e l p f u l t o view the " s " component as a random v a r i a b l e about the mean  + 1 .  I n a growth c o n t e x t , l e t ' s begin by c o n s i d e r i n g a v e c t o r ^1 t h a t r e p r e s e n t s urban p o p u l a t i o n s a t time " t ^ " i n a c i t y system. interval i s  I f r e l a t i v e growth i n the f i r s t  time  , then p o p u l a t i o n s a t time " t " are 2  expressed a s : 2 = P l rfk p  +  * ' '  1  I n t h i s manner, we can show: m-l Pm (rfk ^  Values i n "p^" d i f f e r only by chance?  (ii)  "P "  1 S  1 6 )  Now, assuming t h a t :  (i)  m  '  . . . (4.1?)  +  which c l e a r l y resembles (3.15)•  ( 4  lognormally d i s t r i b u t e d , a property  t h a t i s suggested by Beckmann (1958) and s u b s t a n t i a t e d t o a reasonable (iii)  degree v i a p l o t t i n g ; The time p e r i o d s are reasonably comparable; " s "  then the random v a r i a b l e  ^ + 1  may a l s o be l o g n o r m a l l y  d i s t r i b u t e d ( A i t c h i s o n and Brown, 1957:12). t h i s d i s c u s s i o n s u f f e r s i n two r e s p e c t s .  Unfortunately,  It i s difficult  115 to i n t e r p r e t independent growth i n the s u c c e s s i v e  time  i n t e r v a l s , a c o n d i t i o n t h a t n e c e s s i t a t e s lognormal d i s t r i b u t i o n of the component.  Besides, the i d e a c o n f l i c t s w i t h  those n o t i o n s of cross-time " s " assume  +1  i s normally  analysis that e s s e n t i a l l y distributed.  Nevertheless,  the approach provides a more s u i t a b l e framework f o r d e s c r i b i n g urban growth than can be a t t a i n e d v i a e q u i l i b r i u m adjustments alone.  C l e a r l y , t h i s i s a t h e o r e t i c a l t o p i c t h a t deserves  increased a t t e n t i o n . While i t i s c e r t a i n t h a t c e n t r a l place theory  has  a somewhat l i m i t e d domain i n the r e a l w o r l d , at t h i s time we have no other a n a l y t i c statements to d i r e c t e x p l a n a t i o n of the nature of urban systems.  However, keen awareness of  the fundamentals of t h a t theory g i v e s v a l u a b l e i n s i g h t t o a r i g o r o u s methodology f o r e m p i r i c a l r e s e a r c h . I n the f o l l o w i n g chapter we take a more s e l e c t i v e viewpoint  and, a f t e r a t t a c h i n g v a r i o u s growth f a c t o r s to  the c e n t r a l place framework, attempt to e x p l a i n or r e f u t e c e r t a i n i n d u c t i v e g e n e r a l i z a t i o n s concerning primacy and rank-size relationships.  Chapter 5 CHANGING PATTERNS OF INTERURBAN STRUCTURE T h i s chapter i s l a r g e l y devoted toward s k e t c h i n g the i n t e r r e l a t i o n s o f v a r i o u s growth f a c t o r s and i n t e r urban s t r u c t u r e ,  Beckmann's simple h i e r a r c h i a l  model  provides  the framework f o r a type o f c o m p a r a t i v e - s t a t i c s a n a l y s i s , where o p t i m a l e q u i l i b r i u m c o n d i t i o n s are assumed t o p e r s i s t both before and a f t e r an impact i s i n t r o d u c e d 1968:273).  B e s i d e s , the concomitant  (Nourse,  effect of structure  upon growth i s s e t w i t h i n a more d e s c r i p t i v e d i s c u s s i o n of the d i f f u s i o n process. A graphic i n t e r p r e t a t i o n o f the simple model i s o u t l i n e d near the chapter's end. With t h i s i d e a i n mind, there are some attempts g i v e n toward a l i g n i n g o r q u a l i f y i n g i n d u c t i v e g e n e r a l i z a t i o n s w i t h the c e n t r a l place p r i n c i p l e s . (Burton, I963 emphasizes r e l a t i n g hypotheses t o a developing body o f t h e o r y ) . The methods used i n the argument, however, r e q u i r e c o n s i d e r a b l e refinement before we can e s t a b l i s h s t r o n g statements  r e l a t i n g d i s t r i b u t i o n p a t t e r n s and independent  factors. 116  117  Growth i n a T h e o r e t i c a l Context The  p r o p e r t i e s of the aggregate model l e n d i n s i g h t  t o the nature of change i n urban s t r u c t u r e as  fashioned  by p o p u l a t i o n and economic growth i n a r e g i o n .  The  partic-  u l a r drawbacks of t h i s t h e o r e t i c a l argument i n v o l v e the assumption t h a t development may  be c h a r a c t e r i z e d i n a  d e t e r m i n i s t i c model t h a t precludes  disequilibrium.  Our  procedure i s t o study each f a c t o r i n i s o l a t i o n ( t h a t i s , h o l d i n g a l l other f a c t o r s constant) and then attempt t o conceptually integrate t h e i r diverse e f f e c t s . Population To b e g i n w i t h , l e t ' s c o n s i d e r a c i t y system t h a t evolves on an i s o t r o p i c p l a i n so t h a t p o p u l a t i o n growth i s s p a t i a l l y d i r e c t e d by the e x i s t i n g s t r u c t u r e .  More  p r e c i s e l y , we suppose t h a t ( i ) centers a t t r a c t p o p u l a t i o n increments p r o p o r t i o n a l to t h e i r i n i t i a l s i z e s and ( i i ) s p a t i a l extension  (or c o n t r a c t i o n ) of the e n t i r e  system proceeds s y m m e t r i c a l l y  about the dominant c e n t e r .  Of course, our a n a l y s i s i s c h a r a c t e r i z e d by v i e w i n g  func-  t i o n a l d i f f e r e n t i a t i o n through bundles of commodities s i n c e we assume C h r i s t a l l e r i a n agglomeration. Now  we can analyse the e f f e c t s of p o p u l a t i o n growth  by r e f e r r i n g back to Figure 1 i n the second Curve "D " 2  chapter.  represents the aggregate demand f a c i n g a f i r m  t h a t i s earning normal p r o f i t s i n the c o m p e t i t i v e good case.  single  Under m u l t i p l e good c o n d i t i o n s , approximate  118  tangency o f the average c o s t and demand curves may o n l y c h a r a c t e r i z e those f i r m s p r o v i d i n g h i e r a r c h i a l  marginal  goods (except, o f course, where low order f u n c t i o n s are g i v e n i n r e l a t i v e l y high l e v e l c e n t e r s t h a t possess r a t h e r substantial  i n t r a u r b a n markets).  remaining submarginal requirements  I n any case, f o r the  goods i n the same basket, t h r e s h o l d  are l e s s and excess p r o f i t s are l i k e l y  g r e a t e r f o r the r e l a t e d  firms.  A uniform i n c r e a s e i n p o p u l a t i o n s h i f t s the demand curve "D " t o the r i g h t ( f o r i l l u s t r a t i v e purposes, say 2  to "D^") and a l l o w s excess p r o f i t s t o be a t t a i n e d f o r any p a r t i c u l a r commodity t h a t i s i n i t i a l l y h i e r a r c h i a l  marginal.  Depending upon the amount of p o p u l a t i o n i n c r e a s e , t h i s s u r p l u s o f purchasing power may or may not be s u f f i c i e n t l y l a r g e enough t o induce e n t r y of a competing f i r m ( s ) o f f e r i n g t h i s same commodity at each c e n t e r on t h i s l e v e l . However, c e r t a i n goods t h a t were f o r m e r l y  supramarginal  and only produced a t a h i g h e r l e v e l probably f i n d a s u f f i c i e n t t h r e s h o l d base i n these lower l e v e l p l a c e s w h i l e other submarginal  commodities s u r e l y s t i m u l a t e  i n t r a u r b a n competition.' ' 1  additional  Considering a l l h i e r a r c h i a l  l e v e l s t o g e t h e r , we observe t h a t bundles of goods become r e d e f i n e d a c c o r d i n g t o the emergence of new  hierarchial  marginal goods due t o i n c r e a s e s o f i n t e r s t i t i a l  purchasing  power. "*"The same good may be considered supramarginal o r submarginal depending upon which endpoint o f the basket we relate to.  The most n o t i c e a b l e e f f e c t , then, of p o p u l a t i o n growth i s a tendency toward a r e d u c t i o n of c o n c e n t r a t i o n i n the c i t y system.  I f new  functional f u n c t i o n s are  not added t o the o r i g i n a l "M" b a s k e t s , d e n s i t y i n c r e a s e s may  l e a d t o the spreading of these bundles over more than  "M"  levels.  this  More s i g n i f i c a n t , perhaps, i s the r e s u l t of  decentralization: (i)  S i m i l a r numbers of f u n c t i o n s may  shift  downward i n each h i e r a r c h i a l l e v e l , but f i r m m u l t i p l i c a t i o n i s r e l a t i v e l y more r a p i d i n the lower l e v e l s (see P a r r and Denike, (ii)  1970);  I d e n t i c a l bundles may  become c h a r a c t e r i s t i c  of the two s m a l l e s t s i z e c l a s s e s . B e s i d e s , the simple model does q u a l i f y our unders t a n d i n g of how urban s t r u c t u r e s are transformed, i n the more advanced r e g i o n s . the s i z e and frequency  at least  By s t r e s s i n g t h a t changes i n  d i s t r i b u t i o n of c e n t e r s a t h i g h e r  l e v e l s depend upon the e n t r y t h r e s h o l d s f o r p l a c e s a t a l l lower l e v e l s , i t eliminates the more i n t u i t i v e views such as: " I f the p o p u l a t i o n should only double, there would a r i s e twice as many c i t i e s i n each rank order,. „ . " (Nourse, 1968:210).  Recalling  t h a t the c o r r e c t sequence f o r s a t e l l i t e  c i t i e s i s s(s + l ) ~ n  2  ( s i z e c l a s s e s g r e a t e r than the  f i r s t ) , we can r e i n t e r p r e t  " P ^ t - ^ " - , the t o t a l p o p u l a t i o n  served i n the c e n t r a l place system at time " t - ^ " p r i o r to p o p u l a t i o n growth, i n urban and r u r a l compenents:  120 P  M  <V  = Pin +  (  V  M  + s  w  (s  2?  +  l)n"2  o V . (t ) SA  n=2  (s + I ) ' 1  1  1  n¥l  . . .  v  ±  1  (5.1)  Now i f other f a c t o r s are constant, a r e l a t i v e l y s m a l l p o p u l a t i o n i n c r e a s e leads t o the beginning o f a "M+l" l e v e l hierarchy.  Hence " P ^ ] ^ " ^ ) " d  6 1 1 0  "^  3  population  i n the same system a f t e r t h i s i n c r e a s e (Pj^+i ( )  =  v  P  M ^1^'  where " 2 P ^ ( t - ^ ) " simply means the o r i g i n a l p o p u l a t i o n i s doubled) and the number " f " o f c e n t e r s a t each l e v e l o f the transformed system i s d e r i v e d from: f  < P m + i ' V = Tiirisj-  (  f  W  . . . (5.2)  and f ( p  l»  V  =  ( s + 1 )  f ( p  2' 2 t  )  • • '  where "k" i s the p r o p o r t i o n a l i t y f a c t o r and not a l l " f * s " are o f i n t e g e r form. I n any case, p o p u l a t i o n i n c r e a s e s designate  concom-  i t a n t r u r a l d e n s i t y i n c r e a s e s so t h a t more c e n t r a l p l a c e s emerge and c e n t e r s o f the same s i z e move c l o s e r t o g e t h e r . P a r r and Denike ( 1 9 7 0 ) mention t h a t t h i s c h a r a c t e r i s t i c d e c e n t r a l i z a t i o n (import s u b s t i t u t i o n ) i s e s p e i c a l l y p r e v a l e n t among the higher l e v e l s of the urban h i e r a r c h y i n the U n i t e d S t a t e s and i s due p a r t l y t o i n c r e a s e d r e g i o n a l demand f o r c e r t a i n s p e c i a l i z e d ( p r o f e s s i o n a l , f i n a n c i a l , etc.) services.  The authors s t r e s s , on the other hand, the  well-known case of d e c l i n e i n r u r a l p o p u l a t i o n s and how  121  t h i s causes t h r e s h o l d ranges t o e v e n t u a l l y become u n a t t a i n a b l e from the s m a l l e s t c e n t e r s .  U n f o r t u n a t e l y , our model i s  not f l e x i b l e enough to s i m u l t a n e o u s l y account f o r those f u n c t i o n a l t r a n s f e r s t h a t converge a t the i n t e r m e d i a t e size classes. Income Since we are now h o l d i n g t o t a l p o p u l a t i o n an income i n c r e a s e amounts to a per c a p t i a income t h a t i s the same f o r a l l consumers.  constant, increase  Since we a l s o assume  resource use i s not handled more e f f i c i e n t l y v i a improved technology, t h i s income i n c r e a s e may a r i s e from an a b s o l u t e i n c r e a s e i n the amount of p r o d u c t i v e resources used per head o f p o p u l a t i o n .  (Lampard, 1 9 6 8 , emphasizes d i f f e r e n c e s  between growth and development). Therefore,  j u s t as i n the previous case, there i s  i n c r e a s e d purchasing power i n each a r e a l u n i t and the aggregate demand curve f a c i n g the v a r i o u s firms s h i f t s t o the r i g h t i n a f a s h i o n c h a r a c t e r i s t i c of the p a r t i c u l a r commodity and the l e v e l on which i t i s being o f f e r e d . Nourse  (1968:212-215)  argues c o r r e c t l y t h a t the  income i n c r e a s e a l l o w s each i n d i v i d u a l t o purchase more of a l l goods and t h a t fewer people are needed t o comprise a t h r e s h o l d s i z e market f o r any p a r t i c u l a r c e n t r a l f u n c t i o n . For purposes of a n a l y s i s , he s t i p u l a t e s t h a t the income i n c r e a s e does not a f f e c t the s u p p l y i n g p o p u l a t i o n needed f o r each bundle of f i r s t order goods 1 as a consequence,  122  the f a c t o r "k" r e l a t i n g community and market p o p u l a t i o n s i n c r e a s e s a t the f i r s t l e v e l .  I n keeping w i t h the p r o p e r t i e s  of the simple model he extends the p r o g r e s s i o n component " s " +1  as w e l l and higher l e v e l p l a c e s are r e l a t i v e l y  g r e a t e r than b e f o r e .  I n general terms, the number o f  c e n t r a l places of the f i r s t l e v e l a f t e r the income i n c r e a s e becomes t f'l-k(t,)  k(t ) ?  ^  J  where p ^ ( t ) = P^(t-^). 2  "V  . . .  (5.4)  R e t a i n i n g the s u p p o s i t i o n of a  constant number of s a t e l l i t e s , he proceeds to exhaust " P ( t ^ ) " v i a a p p l i c a t i o n of the M  B P m (  ( t  ) 2 )  - rs+i-k(t ) " < l-k(t )  I  i-k(t) \ s + i r a > P (t ) m  2  2  formula:  x  2  X  J  1  m  x  ...  (5.5)  u n t i l he d e f i n e s a l l the urban p o p u l a t i o n s i n a system " *" P  M  (* M  ^M).  Nourse's e s s e n t i a l t h e s i s i s t h a t a per c a p i t a i n c r e a s e of income extends the number of centers i n the system so t h a t they become c l o s e r together.  At the same  time the h i e r a r c h y c o n t r a c t s to accommodate the expanded low l e v e l urban p o p u l a t i o n t o t a l s . U n f o r t u n a t e l y , i t seems t h a t t h i s a n a l y s i s s u f f e r s i n several respects.  F i r s t of a l l , the l i n e s of i n t e r -  dependencies among the d i f f e r e n t s i z e d places are severed by c o n s i d e r i n g such a truncated h i e r a r c h y .  The  system  123 cannot be c o n s i d e r e d i n e q u i l i b r i u m s i n c e ( i ) no s a t i s f a c t o r y market e x i s t s f o r h i g h order commodities and ( i i ) the s t a b i l i t y of r u r a l populations i s neglected.  Furthermore,  h i s employment o f a "k" i n c r e a s e determined by reduced e x t e r n a l markets i s a r a t h e r debatable f e a t u r e even w i t h i n the c o n f i n e s o f our crude model.  A more thorough  examin-  a t i o n o f the case i s c l e a r l y r e q u i r e d . L e t ' s concentrate e n t i r e l y upon the demand s i d e . By u s i n g an approach t h a t i s r a t h e r more a p p e a l i n g than Nourse's, we c a n a v o i d some o f the c o n c e p t u a l f l a w s t h a t mar h i s argument.  R e c a l l i n g the fundamental assumptions  (see Chapter 3) t h a t l i n k urban and market p o p u l a t i o n s we observe t h a t supply c h a r a c t e r i s t i c s become a t best an i m p l i c i t f a c t o r i n the modeling scheme. Table 2 o f the t h i r d chapter i l l u s t r a t e s the case where a market t h r e s h o l d o f  3000  i s needed to uphold the  f i r s t bundle o f goods and s e r v i c e s .  T h i s f i g u r e may be  c o n s i d e r e d h a l v e d , f o r example, i f per c a p i t a income i s doubled throughout  the r e g i o n .  By keeping our r e a s o n i n g  more c o i n c i d e n t w i t h those n o t i o n s o f c e n t r a l place theory t h a t advocate maximum s p a t i a l c o m p e t i t i o n (see the purchasing power argument o f B e r r y and Garrisons I958d) i t seems more p l a u s i b l e t h a t t h i s new purchasing t h r e s h o l d i s met by p o p u l a t i o n s drawn evenly out o f e x i s t i n g c e n t e r s and complementary areas.  I n other words, two c e n t e r s w i t h "p-^"  equal t o 500 and " r " equal to 1000 r e p l a c e the s i n g l e 1  center.  I t i s a l s o apparent t h a t t h i s argument avoids  124  Nourse's l i m i t a t i o n t o s m a l l income increments t h a t do not induce r a p i d growth of the "k" f a c t o r .  Now, the  g e n e r a l e f f e c t o f the i n c r e a s e i s d e s c r i b e d by: f(P .t ) = v fCp^t.^ m  . . . (5.6)  2  where "v" denotes the p r o p o r t i o n between per c a p i t a income a f t e r and before the increment. The r e s u l t o f t h i s argument should be c l e a r . income i n c r e a s e spread e v e n l y over  An  a l l consumers expands  the number of p l a c e s i n a c i t y system and lowers the p o p u l a t i o n s o f p l a c e s on l e v e l s comparable case.  t o the i n i t i a l  I n f a c t , s u f f i c i e n t increments may induce two  or more i d e n t i c a l and adjacent subsystems the o r i g i n a l  to replace  system.  O b v i o u s l y , though, our argument i s somewhat weaker than Nourse's on a t t a c k i n g the i s s u e o f supply.  Nevertheless  we may t h i n k of the process j u s t o u t l i n e d as being cons t r a i n e d by some lower l i m i t of the s u p p l y i n g p o p u l a t i o n at the f i r s t (and every other) l e v e l although the d e f i n i t i o n of t h i s bound l i e s o u t s i d e our a p r i o r i  structure.  In both o f these cases, t o o , we assume away an added b e h a v i o u r i s t i c i m p l i c a t i o n .  C e r t a i n l y as incomes  r i s e , customers i n c r e a s i n g l y t u r n tov/ard i n c o m e - e l a s t i c commodities  (high s t y l e f u r n i t u r e , fashion c l o t h i n g ,  s p e c i a l i z e d medical s e r v i c e s , e t c . ) r a t h e r than items l i k e a g r i c u l t u r a l s t a p l e s and home f u e l s .  Now i t i s  e x c e e d i n g l y d i f f i c u l t t o s t a t e j u s t how t h i s new dimension  125  a f f e c t s the i n i t i a l system through an income i n c r e a s e but we i n t u i t i v e l y expect t h a t ( i ) the t r a n s f e r o f nearh i e r a r c h i a l marginal goods t o lower l e v e l s and ( i i ) the mixture  of f i r m m u l t i p l i c a t i o n i n e x i s t i n g centers with  the e n t r y o f bundles i n new c e n t e r s are s u i t a b l y pronounced. The tendency t o agglomerate seems t o go hand i n hand w i t h i n c o m e - e l a s t i c i t y and may w e l l serve t o s u s t a i n the i n e r t i a ( h i e r a r c h i a l l e v e l s and number o f communities on each l e v e l ) o f the o r i g i n a l system. A l l i n a l l , i t i s i m p o s s i b l e t o present an a c c u r a t e p i c t u r e o f p r o g r e s s i v e s y s t e m a t i z a t i o n i n a simple s e t o f equations.  While i t i s t r u e t h a t our views and Nourse's  are a t odds on c e r t a i n r e l e v a n t p o i n t s , i t i s s i g n i f i c a n t to note t h a t e i t h e r approach suggests a constancy o r c o n t r a c t i o n i n both h i e r a r c h i a l l e v e l s and f u n c t i o n a l c e n t r a l i z a t i o n may be r e a l i z e d v i a per c a p i t a income increases. Innovations Innovations  o f knowledge and technique  may be  g e n e r a l i z e d as f u n c t i o n s o f i n t e r a c t i o n p r o b a b i l i t y or i n f o r m a t i o n exchange  i n open systems (Berry and Horton,  1970).  Economic development may then be viewed from the p e r s p e c t i v e of such i n n o v a t i o n s o c c u r r i n g i n the l a r g e s t centers and spreading through time t o other communities i n the system. Lampard's ( 1 9 6 8 : 1 0 6 ) c y b e r n e t i c s framework  emphasizes  s t a b i l i t y o f i n t e r u r b a n s t r u c t u r e since " . . .  the t r a n s -  f o r m a t i o n o f human settlement p a t t e r n s (the e v o l v i n g  126  system o f c i t i e s , f o r example) i n v o l v e s the emergence and g e n e r a l i z a t i o n o f n o v e l t y w i t h i n the p o p u l a t i o n system.  . . ",  although growth may appear d e v i a t i o n - a m p l i f y i n g i n the v a r i o u s s m a l l e r subsystems. In our present d i s c u s s i o n , f o u r g e n e r a l cases appear t o he o f s p e c i a l i n t e r e s t : (i)  I n n o v a t i o n s i n t r a n s p o r t a t i o n technology,  i n c l u d i n g both new means and route improvements: (ii)  Innovations that strengthen r u r a l  (farming)  productivity; (iii)  Innovations i n marketing technology t h a t  permit the e n t r y o f s c a l e economies i n t o c e r t a i n e x i s t i n g activities; (iv)  Innovations t h a t b r i n g e n t i r e l y new a c t i v i t i e s  i n t o the d i f f e r e n t s i z e d urban  communities.  W i t h i n our c o m p a r a t i v e - s t a t i c s framework we may g i v e more s p e c i f i c a t t e n t i o n t o cases ( i ) and ( i i ) as they r e s t on the demand s i d e l i k e the f a c t o r s j u s t analysed above. N e v e r t h e l e s s , we can a l l u d e t o s t r u c t u r a l t r a n s f o r m a t i o n s f o r the remaining cases; b e s i d e s , here we emphasize how s t r u c t u r e channels economic development as w e l l . In the f i r s t i n s t a n c e , a t r a n s p o r t a t i o n i n n o v a t i o n c l e a r l y a f f e c t s o n l y demand c o n d i t i o n s w i t h i n the f.o.b. supposition.  G e n e r a l l y , we may c o n s i d e r such an improvement  as being s i m i l a r to an i n c r e a s e i n per c a p i t a income, although i t s b e n e f i t s only accrue t o the complementary area p o p u l a t i o n s on each h i e r a r c h i a l l e v e l ( s i n c e we assume  127 the s u p p l y i n g p o p u l a t i o n t o he l o c a t e d a t the p r o d u c t i o n site).  I n t u i t i v e l y , we expect t h i s impact to change the  urban s t r u c t u r e r o u g h l y along the l i n e s we for  an income i n c r e a s e .  hypothesized  On the other hand, t h i s  statement  must be q u a l i f i e d a c c o r d i n g to v a r i a b l e s l i k e ( i ) the i n i t i a l v a l u e of the p r o p o r t i o n a l i t y f a c t o r "k", ( i i ) the v e r y magnitude of t h i s b e n e f i t g i v e n t o e x t e r n a l consumers, and ( i i i ) changing behaviour  ( f o r example, multipurpose  due to new means as opposed to simple decreases friction. to  trips)  in spatial  By i t s e l f , improved customer m o b i l i t y p o i n t s  i n c r e a s e d f u n c t i o n a l d e c e n t r a l i z a t i o n i n a c i t y system. The improvement of r u r a l p r o d u c t i v i t y  through  i n n o v a t i o n s of a g r i c u l t u r a l methods, mechanization, e t c , changes the urban s t r u c t u r e i f concomitant m i g r a t i o n i s assumed.  r u r a l to urban  By v i e w i n g t h i s r e d i s t r i b u t i o n  process i n a v e i n s i m i l a r to t h a t o u t l i n e d e a r l i e r f o r p o p u l a t i o n growth, we expect a s l i g h t i n c r e a s e i n the  "k"  f a c t o r and the b a s i c p r o g r e s s i o n component to accompany the emergence of a new h i e r a r c h i a l l e v e l .  This follows  because as r u r a l d e n s i t i e s d i m i n i s h , f i r s t l e v e l p l a c e s become s m a l l e r and tend t o l o s e f u n c t i o n s (except the lowest order convenience  goods) to second l e v e l p l a c e s .  Such an upward t r a n s f e r occurs throughout  the e n t i r e h i e r a r c h y  and e s s e n t i a l l y suggests i n c r e a s i n g f u n c t i o n a l c e n t r a l i z a t i o n w i t h a new  dominant c e n t e r e n t e r i n g the system.  While  the  number of communities i n c r e a s e s , t h e i r p o p u l a t i o n s on l e v e l s comparable to the o r i g i n a l h i e r a r c h y tend to decrease.  128  Scale economies, a l l o w i n g l a r g e p l a n t s t o produce a t lower m a r g i n a l and average c o s t s , place more emphasis on the supply s i d e o f the argument.  P a r r and Denike (1970)  g i v e l u c i d i l l u s t r a t i o n o f how such s c a l e changes permit f u n c t i o n s t o he t r a n s f e r e d from lower t o higher l e v e l s o f the h i e r a r c h y .  Considered i n i s o l a t i o n , the e f f e c t s o f  s c a l e extensions  are more n o t i c e a b l e among the lower  h i e r a r c h i a l l e v e l s where ( i ) i n t r a u r b a n t h r e s h o l d s are not too s u b s t a n t i a l and ( i i ) convenience r e t a i l i n g , as c h a r a c t e r i z e d by minimal c a p t i a l o u t l a y s , dominates the basket items.  The t r a n s f e r o f f u n c t i o n s suggests a c o n t r a c -  t i o n o f the urban h i e r a r c h y as many o f t h e i n e f f i c i e n t f i r m s i n s m a l l e r communities are p r i c e d out o f t h e i r markets.  As the p o s s i b i l i t i e s o f r e a l i z i n g s c a l e economies  f o r i n d i v i d u a l goods i n the same basket are not i d e n t i c a l , improved marketing technology may w e l l evoke i n c r e a s e d f u n c t i o n a l c o n c e n t r a t i o n i n the c i t y system. Innovations  t h a t b r i n g t o t a l l y new commodities  i n t o the r e g i o n a l market cannot be r e l a t e d t o urban s t r u c t u r e i n the same e x p l i c i t f a s h i o n as the previous f a c t o r s . Obviously,  though, a h i g h incidence o f low order  innovations  r e l a t i v e t o h i g h order types may somewhat strengthen functional decentralization. On the other hand, the i n n o v a t i o n i d e a  provides  a convenient means f o r d i s c u s s i n g the other side o f the coin:  t h a t being, o f course, how i n t e r u r b a n s t r u c t u r i n g  channels the course o f economic growth.  Let's  consider  129  how items t h a t are n e i t h e r r e s o u r c e - o r i e n t e d nor r e g i o n a l s p e c i f i c spread from an i n n o v a t i o n o r i g i n .  I t appears t h a t  ( i ) the s t r o n g e r the d i s t a n c e decay, the c l o s e r w i l l d i f f u s i o n f o l l o w the c o n s t r a i n t s o f d i s t a n c e w h i l e ( i i ) the weaker the d i s t a n c e decay, the c l o s e r w i l l d i f f u s i o n f o l l o w the s i z e d i s t r i b u t i o n o f urban communities (Pederson, I n any case, without c e r t a i n h i e r a r c h i a l  1970).  aspects,  an economic r e g i o n may be t y p i f i e d by c u r t a i l e d d i f f u s i o n o f both low and h i g h order goods.  The i n t e g r a t i v e r o l e o f  p e r i p h e r a l c e n t e r s i s emphasized as a means t o o f f s e t t h i s concentrated  and f r e q u e n t l y weak economic environment.  Observers i n c r e a s i n g l y s t r e s s the f o c u s s i n g o f r e g i o n a l p o l i c y upon the l o c a t i o n and f u n c t i o n a l c h a r a c t e r i s t i c s of growth c e n t e r s (among o t h e r s , Friedmann, I 9 6 6 , L i t h w i c k and Paquet, 1 9 6 8 ) .  Though investments  are u s u a l l y a s s o c i a t e d  w i t h the p r o v i s i o n o f h i g h order goods, suggestions are forwarded t h a t r e g i o n a l convergence may be sought v i a . l o w order goods as w e l l .  P o l i c y i m p l i c a t i o n s here i n c l u d e  ( i ) the i n c r e a s e o f d i f f u s i o n sources, ( i i ) the improvement of a c c e s s i b i l i t y t o the primary  i n n o v a t i o n c e n t e r , and  ( i i i ) the speeding up o f the urban growth process i n low d e n s i t y areas (Pederson,  1970),  The important p o i n t being  s i g n i f i e d i s t h a t the extent o f urban amenities tends t o determine the very economic h e a l t h (growth,  s t a b i l i t y , etc.)  of the e n t i r e r e g i o n . Despite c o n s i d e r a b l e v a r i e t y i n how investment i s s p a t i a l l y a l l o c a t e d , the p r i n c i p l e s o f comparative advantage  I 130  evoke h i e r a r c h i a l symptoms a t some l a t e r stage.  Whether  p e r i p h e r a l growth n a t u r a l l y f o l l o w s a f r o n t i e r o r d i m i n i s h i n g r e t u r n s s e t i n a t the l a r g e s t c e n t e r s o r growth i s thoughtf u l l y r e d i r e c t e d , the r e g i o n a l p e r i p h e r y i s e v e n t u a l l y taken up by urban subsystems.  Along the path toward hypothe-  t i c a l r e g i o n a l convergense, " . . . the d i f f u s i o n o f innovat i o n s down the system o f c i t y - s i z e s i s the means by which growth and change are t r a n s m i t t e d throughout the economy and i n t e g r a t e d n a t i o n a l development i s achieved and maintained."  (Berry and Horton, 1 9 7 0 : 6 ? ) .  Differential  urban growth may i t s e l f be considered some f u n c t i o n o f the process o f i n n o v a t i o n d i f f u s i o n (Pred, I 9 6 6 ) , I n v a r i o u s p a r t s o f t h i s t h e s i s we have s t r e s s e d the manner i n which a w e l l d e f i n e d h i e r a r c h y c o n s t r a i n s the growth o f d i f f e r e n t s i z e d communities.  Our s e r i e s o f  e q u i l i b r i u m adjustments are r e a l l y taken t o i l l u s t r a t e a c i t y system a f t e r complete s a t u r a t i o n o f the p a r t i c u l a r d i f f u s i o n process.  P o p u l a t i o n members and i n n o v a t i o n s are  the c r i t i c a l items s i g n i f i e d .  Rationale for equilibrium  tendencies depends upon the maximization flows o f mobile f a c t o r s o f p r o d u c t i o n .  of i n t r a r e g i o n a l I n t h i s way  employment (and, t h e r e f o r e , p o p u l a t i o n ) increments tend t o be p r o p o r t i o n a l t o urban s i z e w h i l e economic a c t i v i t i e s f i l t e r down from i n n o v a t i n g areas (Thompson, 1968:52-59 o u t l i n e s t h i s phenomenon).  U n f o r t u n a t e l y , not even a smoothly  o p e r a t i n g market process can promise f u l f i l l m e n t o f t o t a l adjustment i n the l o n g r u n .  131  A Brief  Synthesis I t would be hazardous, indeed, t o i n f e r  but tendencies  anything  from the f o r e g o i n g d i v e r s e impact arguments.  S e v e r a l g e n e r a l p o s s i b i l i t i e s seem to e x i s t : (i)  Regions i n which p o p u l a t i o n growth  completely  o u t s t r i p s t e c h n o l o g i c a l advance and income e x t e n s i o n  may  w e l l be c h a r a c t e r i z e d by a m u l t i - l e v e l h i e r a r c h y and  high  r u r a l d e n s i t i e s ; b e s i d e s , the "k" f a c t o r w i l l be  relatively  low; (ii)  Regions i n which p o p u l a t i o n growth i s r e t a r d e d  but other growth v a r i a b l e s continue may  show s t a b i l i t y or  c o n t r a c t i o n of the h i e r a r c h i a l s t r u c t u r e over time; the  "k"  factor l i k e l y increases, e s p e c i a l l y with s i g n i f i c a n t r u r a l to-urban  migration; (iii)  Regions d i s p l a y i n g more balanced growth of  p o p u l a t i o n and economic f a c t o r s probably preserve the  gross  f e a t u r e s of the i n i t i a l h i e r a r c h y to a g r e a t e x t e n t . On the other hand, i t becomes e s s e n t i a l to q u a l i f y these statements by c o n s i d e r i n g t h a t : (i)  A f i n i t e set of community s i t e s would p o s s i b l y  a l t e r the v a r i o u s impacts i n a r a t h e r t y p i c a l f a s h i o n : f o r i n s t a n c e , p o p u l a t i o n growth may  be simply a t t r a c t e d  to e x i s t i n g p l a c e s so t h a t "k" i n c r e a s e s ; (ii) proceeding  Innovations may  independently  the simultaneous  have c l e a r t h r e s h o l d s when  but may  act q u i t e d i f f e r e n t l y w i t h  change of other f a c t o r s .  P a r r and Denike  (1970) i l l u s t r a t e the case where a s c a l e change i n marketing  i s brought about by an improvement i n t r a n s p o r t a t i o n ( o r an i n c r e a s e o f p o p u l a t i o n o r per c a p i t a income f o r t h a t m a t t e r ) , a c o n d i t i o n t h a t may w e l l l e a d t o s h r i n k i n g o f the urban h i e r a r c h y . Of course, d i v e r s e f o r c e s t h a t we cannot c o n t a i n i n our t h e o r e t i c a l argument are p e r s i s t e n t l y a l t e r i n g r e a l world i n t e r u r b a n s t r u c t u r e s as w e l l .  We should be p l e a s e d ,  t h e r e f o r e , i f we approximately account f o r the d i r e c t i o n s of r e a l w o r l d changes alone.  I n c o n c l u s i o n , then, the  confidence we place i n e x p l a i n i n g these adjustments depends l a r g e l y upon our f e e l i n g s toward the m e r i t s o f c e n t r a l place theory. Graphing the Aggregate Model The  s i m p l e s t i l l u s t r a t i o n o f the s i z e d i s t r i b u t i o n  of p l a c e s v i a the aggregate model i s g i v e n by p l o t t i n g v a l u e s f o r a h y p o t h e t i c a l system on l o g a r i t h m i c - n o r m a l p r o b a b i l i t y paper.  Comparing d i f f e r e n t systems by t h i s method makes  s e v e r a l graphic p r o p e r t i e s obvious: (i)  A r o u g h l y s t r a i g h t l i n e t h a t seemingly i n d i c a t e s  a t r u n c a t e d lognormal  d i s t r i b u t i o n i s , at c l o s e r i n s p e c t i o n ,  s l i g h t l y convex t o the s i z e a x i s f o r few h i e r a r c h i a l l e v e l s and more concave t o t h a t a x i s f o r many l e v e l s ? (ii)  The slope o f the a p p a r e n t l y s t r a i g h t l i n e  depends upon the number o f h i e r a r c h i a l l e v e l s when the "k" f a c t o r remains constant; on the other hand, a r e d u c t i o n o f "k" means a steeper l i n e , c e t e r i s p a r i b u s ;  133 (iii)  A change i n geometry a f f e c t s the slope t o o ;  as " s " i n c r e a s e s the l i n e becomes f l a t t e r . (iv)  The a r b i t r a r y p o i n t of t r u n c a t i o n above  minimum s i z e d p l a c e s a f f e c t s d i f f e r e n t systems i n v a r i o u s ways 5 a m u l t i - l e v e l e d h i e r a r c h y may  give a f l a t l i n e i f  "k" i s s m a l l (perhaps l a r g e r u r a l d e n s i t i e s ) s i n c e many c e n t e r s l i e i n the f i r s t grouped i n t e r v a l . These n o t i o n s are somewhat u s e f u l when r e l a t e d to B e r r y ' s ( 1 9 6 1 ) e m p i r i c a l study. why  They suggest, f o r i n s t a n c e ,  there i s c o n s i d e r a b l e v a r i e t y i n these c i t y s i z e  u t i o n s though many are n e a r l y lognormal.  distrib-  I n v e s t i g a t i o n along  t h i s path seems to be a l o g i c a l s t e p toward s t r e n g t h e n i n g c r o s s - c u l t u r a l comparisons of c i t y systems. Furthermore, the a p r i o r i statements may  suggest what  be the most r e l e v a n t f a c t o r s i n promoting c r o s s - t i m e  s i m i l a r i t y o f i n t e r n a t i o n a l data.  T o t a l p o p u l a t i o n of the  urban system, independent of the manner ( b i r t h r a t e s , r u r a l - t o - u r b a n m i g r a t i o n , etc.) in which i t i s d e v i s e d , seems to be the one c r i t i c a l v a r i a b l e t h a t expands the urban h i e r a r c h y .  In cases where some h i e r a r c h i a l  aspects  are presented a t a p o i n t i n time ( e i t h e r f o r economic, s o c i a l , or a d m i n i s t r a t i v e r e a s o n s ) , these aspects are l i k e l y s o l i d i f i e d by the p o p u l a t i o n growth and w i t h i n a maturing  space-economy.  redistribution  Since no r e a l world .  h i e r a r c h y approximates p e r f e c t i o n , the emergence of  new  l e v e l s simply makes the lognormal d i s t r i b u t i o n more p l a u s i b l e . To the extent t h a t p o p u l a t i o n t o t a l s and n a t i o n a l areas  134  show  some p o s i t i v e c o r r e l a t i o n , i t i s not s u r p r i s i n g t h a t  these two v a r i a b l e s are p o i n t e d t o i n most e m p i r i c a l s t u d i e s as being conducive t o low degrees o f primacy ( B e r r y , Hehta, 1964s L i n s k y , I 9 6 5 ) .  On the other hand, we  196ls  should  be somewhat h e s i t a n t , then, o f b e l i e v i n g t h a t income per c a p i t a alone always v a r i e s p o s i t i v e l y w i t h the degree o f r a n k - s i z e shown by i n d i v i d u a l n a t i o n a l systems t h a t u n f o l d over time (Lasuen, L o r c a , and O r i a ,  I967,  assume t h i s t o  be the c a s e ) . C l e a r l y , added t h e o r e t i c a l arguments and more p r e c i s e i n d u c t i v e approaches are needed before confidence variables.  sufficient  may be placed i n the r o l e o f d i s p a r a t e growth T h i s promises t o be an important  s i a l topic i n future interurban research.  and c o n t r o v e r -  Chapter  6  CONCLUDING REMARKS The p r o b l e m thesis  concerns  attention and  to  size  distribution.  f r o n t s w i t h i n an e x p l i c i t  The c o n s c i o u s  the  city  the discussion i n this  support  of this  improve t h e p r e v a l e n t methodologies size  t o p i c more  body o f g e o g r a p h i c a l l i t e r a t u r e The f i n d i n g s diverse. analysis  feature of  thesis are  now i n u s e a n d t o  firmly  into  the growing  and theory.  of the d i f f e r e n t  G e n e r a l l y , though,  t h e tone  arguments a r e r a t h e r i s that  logical  should replace i n t u i t i o n as geographic scientifically.  area there  Furthermore, such  Unfortunately, i n this  intuitive  as the d e f i n i t i o n  o f study areas  problem and w i s h f u l  a r e common. (including  and t e c h n i q u e s  analysis) severely constrains the value o f  i n f e r e n c e s t h a t many o b s e r v e r s  on t h e c i t y  tendencies  d i s r e g a r d o f r e s e a r c h methodology  of s t a t i s t i c a l  If  endeavours  i s an atmosphere o f data m a l l e a b i l i t y  t h i n k i n g t h a t suggests  points  framework.  o f t h i s framework throughout t h e  The e s s e n t i a l p u r p o s e s  anchor the c i t y  theoretical  systems  d i s c u s s i o n i s perhaps t h e most s a l i e n t  thesis.  proceed  Considerable  i s g i v e n t o v a r i o u s s u b t o p i c s on b o t h  empirical  entire  area directing  efforts size  are taken  topic  put forward.  to continue  scientific  research  t h e n t h e y must be f u n n e l l e d a l o n g  135  f i  136  two  r e l a t e d paths: (i)  Toward improved i n t e r p r e t a t i v e schemes on the  t h e o r e t i c a l s i d e ; an argument o f t h i s t h e s i s i s t h a t c e n t r a l place t h e o r y o f f e r s a s t r o n g "base f o r such c o n t r i b u t i o n s ; and (ii)  Toward c a r e f u l l y s t r u c t u r e d e m p i r i c a l  studies  on d i f f e r e n t s c a l e s that, i n c o n j u n c t i o n w i t h t h e o r e t i c a l e x t e n s i o n s , w i l l suggest those f a c t o r s t h a t  primarily  determine the form o f the frequency d i s t r i b u t i o n o f c i t y sizes i n a p a r t i c u l a r region. B e s i d e s , our review has r e v e a l e d t h a t s e v e r a l l e s s g e n e r a l s u b t o p i c s deserve i n c r e a s e d (i)  a t t e n t i o n as w e l l :  The scheme o f h i e r a r c h i a l s e t s , which d e a l s  w i t h one complete and many p a r t i a l h i e r a r c h i e s o f an independent n a t u r e , suggests a more f l e x i b l e v i e w p o i n t ( a t l e a s t where r e s o l u t i o n l e v e l s are r e l a t i v e l y low) toward c o m p a t i b i l i t y o f c e n t r a l place t h i n k i n g w i t h the c h a r a c t e r i s t i c u n i - s i z e class d i s t r i b u t i o n s of empirical research; (ii)  Concern over the L o s c h i a n model should enhance  t h i s same c o m p a t i b i l i t y ; (iii)  E x t e n s i o n s o f the economic base concept v i a  c e n t r a l place theory may w e l l provide v a l u a b l e feedback at both the i n t r a - and i n t e r u r b a n (iv)  levels;  Added e f f o r t s are needed i n the attempt t o  give c e n t r a l place arguments a reasonable temporal dimension ( t h a t i s , when a h i e r a r c h y  i s a l r e a d y assumed t o e x i s t ) ;  137  (v)  Emphasis on g e n e r a l systems concepts should  complement the d e t e r m i n i s t i c  and  stochastic  arguments;  perhaps, indeed, the entropy i d e a can a s s i s t i n  describing  n o n - e q u i l i b r i u m f e a t u r e s w i t h i n a dynamic framework ( f o r example, i t may  have promise as a device to d e s c r i b e i n t e r -  urban s t r u c t u r i n g p r i o r to h i e r a r c h a l (vi)  maturity);  A crude a n a l y t i c a l base has  i n v e s t i g a t i n g the  been set  i n t e r r e l a t i o n s of growth and  interurban  s t r u c t u r e ; model b u i l d i n g w i t h i n t h i s subtopic may i n t e r e s t i n g t h e o r e t i c a l and (vii) { s l o p e s , etc.)  practical  for  have  implications;  Emphasis placed upon the p e c u l i a r of i n d i v i d u a l d i s t r i b u t i o n s may  aspects  prove  f r u i t f u l i n the search f o r c r o s s - c u l t u r a l r e g u l a r i t i e s . 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