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UBC Theses and Dissertations

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

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STRUCTURE AND PROCESS IN THE CHRISTALLERIAN SYSTEM by GORDON FREDRICK MULLIGAN B . S c , The U n i v e r s i t y of B r i t i s h Columbia, 19&9 M. A . , The U n i v e r s i t y of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of GEOGRAPHY We accept th is thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1976 (c) Gordon Fredrick Mulligan In present ing th is thes is in pa r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make i t f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT This d isser ta t ion deals with theoret ica l central place systems of the C h r i s t a l l e r i a n type. By employing a form-function-process methodology the author attempts to embrace central place structure and process in a consistent and general manner. At tent ion i s f i r s t given to systemic structure as depicted by the general h ie rarch ia l model of c i t y s i z e . Given th is s t ructura l framework, in te res t i s then turned to modell ing within-systems adoptive processes (the issue of innovation i s not considered). F i n a l l y , the e f fec ts of d i f fe ren t types of parametric sh i f t s - both continuous ( ins tan-taneous) and d iscrete (long run) - are examined wi th in the context of the estab l ished models. By e l i c i t i n g a number of l aw- l i ke statements the author i s intending to lay some of the foundations for a general theory of i n te r -urban growth and development. The scope and content of the more relevant assert ions are presently ou t l i ned . I t i s demonstrated that cer ta in a t t r ibu tes of ind iv idua l centra l places are in t imate ly re la ted to overa l l systemic proper t ies. For instance, the inverse of the basic/non-basic ra t i o of a system's largest c i t y i s shown to be ident ica l to the urban/rural population balance for the i i ent i re system. In addi t ion a novel type of input-output model i s i n t r o -duced so as to i l l u s t r a t e the economic base underpinnings of the h ie rarch ia l model. Special concern is given to the serv ice mu l t i p l i e r s in the s t ructura l argument: these are shown to r e f l e c t employment and demand ra t ios for the various h ie rarch ia l a c t i v i t i e s . Then the e f fec ts of sh i f t s in these mu l t i p l i e rs upon central place propert ies are examined within a comparative s ta t i c s framework. The po la r i za t ion of h ie rarch ia l and wave-l ike d i f fus ionary patterns i s establ ished by showing the former ( l a t te r ) to accompany: ( i ) systemic openness (c losure) , ( i i ) area! ( l inear ) d imensional i ty , ( i i i ) slow (rapid) decl ine in the service m u l t i p l i e r s , and ( iv) low (high) f r i c t i o n a l const ra in ts on spat ia l i n te rac t i on . F i n a l l y , temporal (long run population changes) and spat ia l (a l locat ion of nonnodal a c t i v i t i e s ) var ia t ion are shown to induce char-a c t e r i s t i c changes in such d i f fus ionary patterns. i i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES x i ACKNOWLEDGEMENTS x i i Chapter 1 INTRODUCTION 1 1.1 Background 1 1.2 General Intentions and Themes of the Thesis 7 1.3 Preview. . . 9 Footnotes to Chapter 1 16 2 THE GENERALIZED HIERARCHIAL MODEL 18 2.1 Introduction 18 2.2 Terms and Notation . 18 2.3 Economic Base Considerations 21 2.4 Hierarch ia l Input-Output Linkages 30 2.5 The Problem of Evolut ion 34 2.6 A Proposal of Stage-Like Development 35 2.7 Concluding Remarks 44 Footnotes to Chapter 2 45 Chapter Page 3 CENTRAL PLACE DIFFUSION. 47 3.1 Introduction 47 3.2 Cr i t i c i sms of the Ex is t ing Models 49 3.3 An A l te rna t ive Model of Central Place Di f fus ion 56 3.4 Proposi t ions Based on the A l te rna t i ve Model 57 3.5 Addi t ional Comments on the A l te rna t i ve Model 89 3.6 Concluding Remarks 91 Footnotes to Chapter 3 93 4 PARAMETRIC INFLUENCES ON STRUCTURE AND PROCESS IN THE CENTRAL PLACE SYSTEM 97 4.1 Introduction 97 4.2 Population 100 4.3 Technology 125 4.4 Per Capita Income 136 4.5 Concluding Remarks 142 Footnotes to Chapter 4 143 5 A MODIFICATION OF THE STRICT HIERARCHIAL FORMAT 148 5.1 Introduction 148 5.2 Structure 148 5.3 Process 155 5.4 Concluding Remarks . . . 157 Footnotes to Chapter 5 158 v Chapter Page 6 SUMMARY AND CONCLUSIONS 159 6.1 Introduction 159 6.2 Structure 160 6.3 Process 163 6.4 Di rect ives for Future Research 164 BIBLIOGRAPHY 168 APPENDICES A STAGE MATURATION OF A FOUR LEVEL K=3 CENTRAL PLACE SYSTEM 174 B THE NOTION OF EQUIVALENT CENTERS IN HUDSON'S STOCHASTIC TELLING PROCESS 180 C THE ALTERNATIVE DIFFUSIONARY MODEL: PROPERTIES OF THE TABLES AND AN OUTLINE OF THE TELLING PROCESS 187 D THE IMPACT OF EXOGENOUS EMPLOYMENT SHIFTS IN A SECOND LEVEL CENTRAL PLACE 195 E THE DIVISION OF CENTRAL PLACE POPULATIONS ACCORDING TO ACTIVITY SECTORS 198 F THE IMPACT OF AN EXOGENOUS SHIFT IN A SERVICE MULTIPLIER IN A SECOND LEVEL CENTRAL PLACE 201 G A NOTE ON THE RELATION BETWEEN MONEY INCOME AND REAL INCOME IN THE HIERARHCIAL FORMAT 203 H THE ALLOCATION OF NONNODAL ACTIVITIES 208 v i LIST OF TABLES* Table Page 3.1 Adoptive Times fo r Centers in an Open K=3 Five Level Central Place System 58 3.2 Adoptive Times for Centers in a Closed K=3 Five Level Central Place System 59 3.3 Adoptive Times for Centers in an Open K=3 Five Level Central Place System 60 3.4 Adoptive Times for Centers in a Closed K=3 Five Level Central Place System 61 3.5 Adoptive Times for Centers in an Open K=3 Five Level Central Place System 62 3.6 Adoptive Times for Centers in a Closed K=3 Five Level Central Place System 63 3 .7 Adoptive Times for Centers i n an Open K=3 Five Level Central Place System 64 3.8 Adoptive Times for Centers in a Closed K=3 Five Level Central Place System 65 3.9 Adoptive Times for Centers in an Open K=3 Four Level Central Place System 66 The t a b l e h e a d i n g s p r e s e n t l y g i v e n a r e somewhat l e s s d e t a i l e d than t h e i r a p p r o p r i a t e c o u n t e r p a r t s in the t e x t . v i i Table Page 3.10 Adoptive Times for Centers in an Open K=3 Four Level Central Place System 66 3.11 Adoptive Times for Centers in a Closed K=3 Four Level Central Place System 67 3.12 Adoptive Times for Centers in a Closed K=3 Four Level Central Place System 67 3.13 Adoptive Times for Centers in an Open K=3 Four Level Central Place System 68 3.14 Adoptive Times for Centers in an Open K=3 Four Level Central Place System . . . 68 3.15 Adoptive Times for Centers in a Closed K=3 Four Level Central Place System 69 3.16 Adoptive Times for Centers in a Closed K=3 Four Level Central Place System . . . 69 3.17 Adoptive Times for Centers in an Open Ki=3, K 2=4, K3=3 Four Level Central Place System 70 3.18 Adoptive Times for Centers in an Open Ki=3, K 2=4, K3=3 Four Level Central Place System 70 3.19 Adoptive Times for Centers in a Closed Ki=3, K 2=4, K3=3 Four Level Central Place System. 71 3.20 Adoptive Times for Centers in a Closed K x=3, K 2=4, K 3 = 3 Four Level Central Place System 71 3.21 Adoptive Times for Centers in an Open Ki=3, K 2=4, K3=3 Four Level Central Place System 72 3.22 Adoptive Times for Centers in an Open Ki=3, K2=4, K3=3 Four Level Central Place System 72 3.23 Adoptive Times for Centers in a Closed Ki=3, K 2=4, K3=3 Four Level Central Place System 73 v i i i Table Page 3.24 Adoptive Times for Centers in a Closed Kx=3, K2=4, K3=3 Four Level Central Place System 73 3.25 Adoptive Times for Centers in an Open K=4 Four Level Central Place System 74 3.26 Adoptive Times for Centers in a Closed K=4 Four Level Central Place System 75 3.27 Adoptive Times for Centers in an Open K=2 Five Level Central Place System 76 3.28 Adoptive Times for Centers in a Closed K=2 Five Level Central Place System 77 3.29 Adoptive Times for Centers in an Open K=2 Five Level Central Place System 78 3.30 Adoptive Times for Centers in a Closed K=2 Five Level Central Place System 79 3.31 Adoptive Times for Centers in an Open K=2 Four Level Central Place System 80 3.32 Adoptive Times fo r Centers i n a Closed K=2 Four Level Central Place System 80 4.1 A Proposal of a One Percent Growth and Redis t r ibut ion Scheme in a K=3 Four Level Central Place System. 116 4.2 An Analys is by A c t i v i t y Sectors of the Proposed One Percent Growth and Redis t r ibut ion Scheme in a K=3 Four Level Central Place System 117 4.3 The Impact of Population Growth on Spat ia l Adoption in an Open K=3 Four Level Central Place System 123 ix Table Page 4.4 The Impact of Population Growth on Spat ia l Adoption in a Closed K=3 Four Level Central Place System 124 5.1 Var ia t ion in Central Place Populations as a Consequence of a Local ized A c t i v i t y in a K=3 Five Level Central Place System . . . 153 5.2 A Comparison of Adoptive Times (Standardized) Between ( i ) Centers in a Closed K=3 Five Level System and ( i i ) Centers in a Modif ied Version of that System 156 A. l Populations of Rural Areas and Central Places in a K=3 Four Level Central Place System which Matures in a Stage-Like Fashion 178 B. l Types of Equivalent Centers in a K=3 Five Level Central Place System. . . . 181 x LIST OF FIGURES Figure Page 2.1 A centra l place system with nesting fac tor • K = 3 2 2 3.1 Adoptive times for centers in an open K=2 four level central place system 81 3.2 Adoptive times for centers in a closed K=2 four level central place system 82 x i ACKNOWLEDGEMENTS I should f i r s t of a l l l i k e to express my considerable indebted-ness to my parents. Without the pleasant working environment that they provided for me in the i r Squamish home I fear that even yet I would be piecing together that a l l - impor tant f i r s t d ra f t . A quiet study, good food and dr ink , and winter sunsets on Gar iba ld i made the absence of cer ta in urban amenities (ones that a bachelor can best appreciate) a b i t more to le rab le . Following that I must extend special grat i tude to my advisor Ken Denike. Ken f i r s t st imulated my in te res t in the research area of present concern and was always helpful - both i n t e l l e c t u a l l y and f inan-c i a l l y - during my graduate sojourn at UBC. The inputs of the remaining members of my committee must a lso be noted. Curt Eaton's i n c i s i v e c r i t i c i s m s of the e a r l i e r draf t were extremely useful and the i r considerat ion l e d , in my opin ion, to a much t i gh te r f i na l d ra f t . John Chapman and Gary Gates gave helpful organiza-t iona l comments as w e l l . More s i g n i f i c a n t , however, was the i n t e l l ec tua l freedom that . that committee - and indeed th is could be stated for the ent i re Geography x i i Department at UBC - accorded me. By never channeling my thoughts in any one spec i f i c d i rec t ion and by never imposing s t r i c t guidel ines for my thesis schedul ing, those ind iv idua ls allowed the whole exercise to be -as i t must be - a worthwhile learning experience. I must also thank the Central Mortgage and Housing Corporation (CMHC) for the fe l lowship funding that was extended to me in the ear ly 1970's and UBC as well for the stipend that was granted in 1974. The Urban Studies (thanks Walter) and Geography teaching ass is tantsh ips were l i kewise much appreciated during th is f i na l year . Also at th is time I should l i k e to commend Sharon Hal le r for doing her usual f ine typing job and acknowledge Nick Watkins for dra f t ing up the three f igures included wi th in the text . Of course, I must also express more than a l i t t l e appreciat ion to a number of f ine fel lows that I met through the Geography Department a t UBC. With Warren, Mike, J . B . , Sparky, Ar thur , and others I shared a number of Monday-to-Friday ups and downs (but in the good old years , na tu ra l l y , they only rare ly ended on the Fr iday) . F i n a l l y , my good f r iends from the "outside world" must also be thanked: John and Trud i , Tony and D i l son , Max, S t e n c i l , Thorns, Dink, M i lne r , Pursewarden, Pret ty Boy, and Susan - i t ' s been a l l r i gh t . x i i i Chapter 1 INTRODUCTION 1.1 Background For some time now soc ia l s c i e n t i s t s have expressed concern over the s i ze d i s t r i b u t i o n of urban communities. An evergrowing body of l i t e r a -tu re , exh ib i t ing the nuances of various d i sc i p l i nes —geographers , economists, h i s t o r i ans , s o c i o l o g i s t s , among others, have a l l commented on the topic — has made the c i t y s i ze d i s t r i bu t i on issue central to an inter-urban perspective on human organizat ion. Quite na tu ra l l y , however, the intent ions of these observers have not always been co inc ident : as a r e s u l t , somewhat d i s t i n c t pools of in teres t may be discerned wi th in the mainstream of thought. The theme was i n i t i a t e d on a s t r i c t l y empirical basis by Auerbach (1913) and Lotka (1924, 1941). More s p e c i f i c a l l y , th is ear ly work was so le ly d i rected toward re la t ing the cumulative numbers of urban places (above a cer ta in population threshold, that i s ) in a designated region to the i r respect ive population s i z e s . The repet i t ion of th is exercise over space ( i . e . in d i f fe ren t regions) and through time eventual ly made i t apparent that patterns of human settlement in d i f fe ren t parts of the world could be character ized by a rather wide var ie ty of such d i s t r i b u t i o n s . 1 2 This deviat ion was e f f ec t i ve l y polar ized in the wr i t ings of Jef ferson (1939) and Z ip f (1949) who i n i t i a t e d the now commonly-made d i s t i n c t i o n between primate and rank-s ize propert ies of c i t y s ize d i s t r i -butions and who gave the e a r l i e s t r a t i ona l i za t i on — the degree to which a region was " in tegrated" — for th is dev ia t ion .^ With the growing recogni t ion that some factors (e .g . area! extent or to ta l population of the region) seemed to induce the development of one pattern or another, i t was only natural that cer ta in models expla in ing such d i s t r i bu t i ona l var ie ty should be engendered. The interested reader might re fer to Simon (1955), Berry (1961), Curry (1964), Thomas (1967), Fano (1969), Berry and Horton (1970), and Parr and Suzuki (1973) for some relevant examples. What a l l these models have in common is a decidedly aspatial perspective and a s tochast ic -ent rop ic mechanism so as to e l i c i t d i s t r i bu t i ona l change over t ime. ( i ) Models Having an E x p l i c i t Spatial-Economic Structure However, the c i t y s i ze topic has been approached from an en t i re l y d i f fe ren t vantage as w e l l . Due to the enl ightening (yet only p a r t i a l l y successfu l ) theoret ica l contr ibut ions of C h r i s t a l l e r (1966) and Losch (1954), 3 an a l te rna t ive means of modelling patterns of c i t y s ize has been r e a l i z e d . The proper modell ing of t he i r theories en ta i l s f i r s t l y a recogni -t ion of the spat ia l (geometric) and economic propert ies of ind iv idua l a c t i v i t i e s . This has been — a n d s t i l l remains — a somewhat d i f f i c u l t task: l a rge ly because the theories themselves have the i r own pa r t i cu la r short-comings in e x p l i c i t l y deal ing with the notion of a spatial-economic equ i l ib r ium. 3 As a consequence, modelling for the most part has followed C h r i s t a l l e r i a n d i r e c t i v e s , since that theory i s much the simpler and more f l e x i b l e of the two and because i t does at least seem in agreement with cer ta in domains of 4 r e a l i t y . This seems an appropriate spot to emphasize that the upcoming thes is deals so le l y with the C h r i s t a l l e r i a n case, although in places geometries are u t i l i z e d which are more general than C h r i s t a l l e r himself o r i g i n a l l y suggested. For that reason, then, the reader should confine h is th inking to the realm of C h r i s t a l l e r i a n theory whenever the term central place appears in the upcoming tex t . Beckmann (1958) was rea l l y the f i r s t to model c i t y s izes according 5 to centra l place p r i n c i p l e s . His h ie ra rch ia l st ructure was t y p i f i e d by two c h a r a c t e r i s t i c s : ( i ) c e n t r a l place populations of va r y i n g s i z e remained a constant p r o p o r t i o n of the market area populations which they s e r v i c e d ; and ( i i ) the r a t i o of the populations of c e n t r a l places (market areas) on successive l e v e l s remained constant throughout the h i e r a r c h y . Unfortunately, Beckmann at th is time erred in his in terpre ta t ion of the C h r i s t a l l e r i a n geometry — he s p e c i f i c a l l y mistook i t s p r inc ip les of domination — and several of h is assert ions became of questionable s ta tus . This postulat ional error was eventual ly r e c t i f i e d by Beckmann (1968) himself but the reader might well prefer the l uc id de ta i l i ng of the problem found in the independent statement of Parr (1969). Pa r r ' s a r t i c l e was espec ia l l y important because there he challenged not only the assumptions of the seminal e f f o r t , but the deduced resu l ts of 4 that argument as w e l l . In p a r t i c u l a r , Beckmann (1958) asserted that h is basic progression component model (see property ( i i ) above) success fu l l y re lated C h r i s t a l l e r i a n theory to the rank-s ize p r i n c i p l e ; Par r , however, took great care in demonstrating that such coincidence was not a n a l y t i c a l l y p o s s i b l e . 6 In the meantime, Dacey (1966) rep l ied with a somewhat more general model based upon h is previous experiences with the centra l place topology (see Dacey (1965) for a comprehensive axiomatic treatment of both the C h r i s t a l l e r i a n and Loschian geometries).^ Concern here was espec ia l l y d i rected toward e l iminat ing the s ingle geographic coe f f i c i en t of propor-t i o n a l i t y (see property ( i ) above) by introducing a complete set of serv ice mu l t i p l i e r s — i n th is way d i f fe ren t production technologies for diverse bundles of goods and services could be embraced wi th in the h ie ra rch ia l scheme ( i . e . there would be an ind iv idua l mu l t i p l i e r for each bundle or h ie ra rch ia l l e v e l ) . Now the central place populations were seen to be composed of h ie ra rch ia l sectors and the supply technology charac te r i s i z ing each sector was interpreted to be invar iant despite changes in the scale of operat ion. To take an example, a l l central places in a system would produce a set of f i r s t level (convenience) goods but the same technology would be employed in the largest center as in a l l the smaller centers of the system. Very recent ly , the present author (see Mull igan (upcoming 1976)) demonstrated that these mu l t i p l i e rs were ac tua l l y inherent to or i m p l i c i t in the basic progression component model so that the two models could not r e a l l y be considered d i s t i n c t (as o r i g i n a l l y surmised) and that , in 5 pa r t i cu l a r , the one-mul t ip l ie r model was jus t a special case of the more general formulat ion. Another s i gn i f i can t contr ibut ion by Dacey involved the de l inea-t ion of the aforementioned population sectors into basic (export-or iented) and non-basic ( loca l i zed) components. His i n te rp re ta t i on , although en t i r e l y co r rec t , was extremely b r ie f and the reader might well be advised to consult Parr (1970) for c l a r i f i c a t i o n . The l a t t e r i s s t i l l the f i nes t c r i t i q u e of the various central place models, in that i t covers the i r ana ly t i c p roper t ies , underlying economic pos tu la t ions , and some relevant empir ical q u a l i f i c a t i o n . In a contemporary a r t i c l e , Beckmann and McPherson (1970) phrased the general argument in somewhat d i f fe ren t terms. Par r , Denike, and Mull igan (1975) have recent ly i l l u s t r a t e d , however, that the assumptive underpinnings of the i r model were coincident with those of Dacey's, although the former did al low for a new dimension of f l e x i b i l i t y in the central place topology (through a var iab le nesting factor on the same l a t t i c e ) . In add i -t i o n , th is author in Mull igan (upcoming 1976) has questioned other un-supported a l lega t ions by Beckmann and McPherson; in f a c t , he has s p e c i f i c a l l y demonstrated how i t was unclear that a "constrained" version of the i r model might be considered coincident with ( i ) the revised ed i t ion of the o r i g ina l Beckmann (1958) model and ( i i ) the rank-s ize ru l e . Besides t h i s , even fur ther debate has been engendered by the s t ruc tura l reformulations of Dacey (1970) and Dacey and Huff (1971). For the present t ime, though, i t should be s u f f i c i e n t to say that the Par r , Denike, and Mul l igan (1975) paper has hopeful ly c l a r i f i e d the idea that the general h ie rarch ia l model — i r respec t i ve of how i t i s mathematically 6 presented — d e f i n i t e l y has an economic base in te rp re ta t i on . I t fo l lows from the above, then, that any simpler model (such as the case with the basic progression component) which can be shown to be a particular example of that general model has a va l i d economic base in terpre ta t ion as w e l l . This in b r i e f has been an overview of those contr ibut ions which the author deems were most i n f l uen t i a l in the development of h is own thought in th is spec i f i c research area. Now i t i s only f a i r to note that th i s fami ly of s t ructura l models based on C h r i s t a l l e r i a n p r inc ip les has at t racted a cer ta in amount of c r i t i c i s m — s e e Parr (1970), Henderson (1972), M i l l s (1972), and Richardson (1973), amongst others. In th is author 's op in ion, however, these assert ions must properly be construed as being di rected to the shortcomings (and inconsis tenc ies in the Loschian case) of the relevant body (bodies) of theory. As long as such a pos te r io r i models f a i t h f u l l y r e f l e c t the c r i t i c a l aspects of a v a i l -able theory — a lbe i t such models might at times seem naive v i s - c l - v i s the real world — then they do not merit c r i t i c i s m in themselves: e i ther an improvement of or a subst i tu t ion for the relevant body (bodies) of theory i s ca l l ed fo r . Natura l ly i f the reader i s acquainted with the C h r i s t a l l e r i a n and Loschian arguments then i t i s common knowledge that both contr ibut ions have been c r i t i c i z e d and qua l i f i ed at numerous times in the past . The author suggests Isard (1956), von Boventer (1963), and Parr and Denike (1970) as three good c r i t i c a l reviews of the ava i lab le mate r ia l . Neverthe-l e s s , i t should be s t i l l more than evident that no other corpus of thought ex i s t s which can even sugges ts challenge to e i ther as a " . . .theory of the l oca t i on , s i z e , nature, and spacing of . . ." human economic a c t i v i t i e s at the reso lu t ion level (scale) which interested the two seminal researchers. 7 ( i i ) Models of Process Based on a Given Structure Out of the mainstream of recent thought concerning the d i f fus ion of items through space have ar isen two models which s p e c i f i c a l l y deal with adoption in the central place s e t t i n g . Hudson (1969) and Pederson (1970) have devised s im i l a r techniques — e a c h of which bears a generic resemblance to Hagerstrand's (1965) well-known model character ized by a mean informa-t ion f i e l d — for representing adoptive patterns amongst communities given the int roduct ion (or innovation) of an item at the largest center of a central place system. By f i r s t postulat ing an appropriate geometry and then a pa r t i cu la r c i t y s ize d i s t r i bu t i on as well — that i s , by f i r s t postu lat ing a spec i f i c s t ruc tura l model — each was able to incorporate the notion of process or change (through the transmission of items) in to the otherwise s t a t i c se t t i ng . The author f e e l s , however, that there are serious inconsis tencies in both of these contr ibut ions and that the source of these inconsis tencies must be examined before accurate modelling may proceed. In add i t i on , nei ther of the two authors was able to take advantage of the rather wide d i s t r i bu t i on of c i t y s izes which may be represented by the general h ie r -arch ia l model and, as a consequence, the i r deductive assert ions would ( i f they were correct ) be necessar i ly confined by the rather i n f l e x i b l e premises of t he i r assumptive models. 1.2 General Intentions and Themes of the Thesis Given th is structure-process dichotomy, the author wishes now to speci fy his general intent ions in or motivations for undertaking the 8 wr i t ing of a thesis in th is pa r t i cu la r research area. In b r i e f form these intent ions are as fo l low ing : ( i ) to a r t i c u l a t e f u r t h e r the s t r u c t u r a l p r o p e r t i e s of the general h i e r a r c h i a l model and r e s o l v e , i f p o s s i b l e , new l a w - l i k e statements which have h i t h e r t o gone unnoticed but which are neverthe-less implicit to t h a t model; . ( i i ) to incorporate the notion of process (by d e v i s -ing a new d i f f u s i o n a r y model) i n t o the s t a t i c c e n t r a l place s e t t i n g in a manner which would appear both more c o n s i s t e n t and more general than past attempts have been; and ( i i i ) t o introduce temporal and s p a t i a l v a r i a t i o n -due to d i f f e r e n t types of s h i f t s which may occur in the a c t i v i t i e s located a t the va r i o u s c e n t r a l places - into the s t r u c t u r a l model and then d i s c e r n j u s t how such changes would be expected to r e d e f i n e d i f f u s i o n a r y p a t t e r n s . The author fee ls that while numerous pa r t i cu la r statements of a l aw- l i ke character do emerge here and there throughout the text (see the fo l lowing sect ion fo r a preview), the in tegrat ing theme or idea of the thesis i s that structure and process may now be e x p l i c i t l y re la ted in the central place context. Inasmuch as the author requires a single over-r i d i ng theme to have — by de f i n i t i on — a t h e s i s , then the argument that those two research perspectives ( i . e . st ructure and process) can now be a r t i cu la ted in a more consistent and general manner than they have been i n the past is that t h e s i s . Since the author is working at the in ter face of two such general perspectives i t i s only natural that the text i s interspersed with con-s iderable methodological comment. I t i s perhaps appropr iate, then, to c lose th is sect ion by emphasizing that concern in th is thesis i s more d i rected to modell ing ex i s t i ng theory rather than to extending the underlying 9 theory per se. There i s , of course, considerable feedback between the operations of devising theory and representing such theory in model form; in the present context, the author i s t rus t ing that by: ( i ) devoting e x p l i c i t a t t e n t i o n to the i n t e r n a l statements of the e x i s t i n g t h e o r i e s of s t r u c -utre and process ( t h i s i s accomplished by representing more of such statements in the re l e v a n t modelling format); and ( i i ) q u a l i f y i n g (to some extent a t l e a s t ) the domain of r e a l i t y which can be favorably covered by such t h e o r i e s ; as well as ( i i i ) extending the f l e x i b i l i t y (now in an added a p r i o r i sense) of the models representing the t h e o r i e s so as to suggest a p p r o p r i a t e new avenues f o r theory extension he i s playing a part — a lbe i t qui te minor — in the f i na l a r t i c u l a t i o n , at some future time, of a general structure-process theory of inter-urban growth and development. Before c los ing th is chapter the author wishes to give a preview of some of the more pa r t i cu la r arguments which are taken up in the tex t . A cursory examination should make i t apparent that more of the ana ly t i ca l working i s in fac t devoted to the s t ruc tura l — rather than processual — side of the argument but th is natura l ly fol lows from the author 's research in teres ts in the past as well as from the "form-funct ion-process" methodology present ly being advocated. 1.3 Preview The upcoming chapter of th is thesis serves to out l ine the presently accepted h ie rarch ia l formulat ion. The author, however, does not simply summarize here the ex is t ing mater ia l : ra ther , he i s a lso determined to 10 present a more general (accurate) p icture of the ind iv idual urban economies by d i s t i ngu i sh ing , amongst the general populations the re in , between those that are act ive economic agents (employees) and those that are not q (dependents). As a consequence, i t becomes possible to demonstrate that any community's r a t i o of basic to non-basic a c t i v i t i e s (as expressed in terms of employees) depends upon the service mu l t i p l i e r s typ i fy ing the economy of the ent i re system as well as the actual placement (ordering) of that center wi th in the overa l l h ie rarch ia l scheme. The reader might not ice that th is out l ine bears a generic resemblance to the Par r , Denike, and Mull igan (1975) statement and he i s advised to consult that a r t i c l e i f some concern i s f e l t over re la t i ng th is new genera l izat ion to some of the e a r l i e r con t r ibu t ions . The author then re i te ra tes his argument but takes a somewhat d i f f e ren t tack i n doing so. He complements the economic base in te rp re ta -t ion by iden t i f y ing the w i th in - leve l and between-levels input-output l inkages as re f lec ted in the set of service m u l t i p l i e r s . This input-output perspect ive is s im i l a r to that t r a d i t i o n a l l y u t i l i z e d in economics; now, however, at tent ion i s s p e c i f i c a l l y given to the l inkages amongst a c t i v i t e s which coexis t in space ( i . e . occupy the same point) but are on d i f fe ren t leve ls of the hierarchy. In the f i na l part of that chapter, the author presents some rather speculat ive views on the s tage- l i ke propert ies which are i m p l i c i t to the h ie rarch ia l model. Put s imply, i t i s demonstrated how various existing sectors of the space-economy may be af fected — given i n i t i a l l y the locat ion and number of the centers of product ion, an ava i lab le technology 11 with constant returns to scale in agr icu l tu re and industry , and the ongoing a l l oca t i on of bundles (with ever increasing thresholds) according to central place doctr ines — as the system's h ie rarch ia l a t t r ibu tes become defined over t ime. This i s suggested as an extremely simple evolut ionary exten-sion of the s t a t i c model. An important accompaniment of th is discourse i s the i l l u s t r a -t ion that the ra t io of urban to rural population for the ent i re system (or any wel l -def ined subsystem) is in t imate ly re lated to the basic /non-basic ra t i o of the largest center in the system (subsystem). The second chapter, then, may be construed as an extension of the ana ly t ic debate in the l i t e r a t u r e . But that chapter serves an addi t ional purpose as w e l l . In that the argument de l imi ts the s t ructura l propert ies of a family of central place systems, and these same systems (more accur-a t e l y , the i r rea l world mappings) are commonly associated with process and change, then the second chapter a lso serves to de l im i t the domain of structures in which inter-urban processes may be modelled and analysed. While th is might seem a b i t fas t id ious to some observers — a n d perhaps even t r i t e to some "form fashions funct ion" adherents — i t i s a point which should be well taken. As the th i rd chapter i nd i ca tes , i t s neglect in the past has been the source of numerous methodological problems in the l i t e r a t u r e . That chapter embraces process and change under the general head-ing of d i f f u s i o n . In th is t h e s i s , d i f fus ion i s simply construed as a macro-adoptive process. The author says macro because, in the present context , communities themselves are the ind iv idua ls which adopt a pa r t i cu la r item —whether that item is a new consumer product, a new technique of 12 product ion, a d isease, some new opinion or piece of knowledge, or the l i k e — a l t h o u g h , in f a c t , i t i s always groups ( ind iv idual consumers, f i rms , professional organizat ions, e t c . ) wi th in those same communities which perform the real act of adoption. The issue of immediate concern, however, i s the manner in which t h i s macro process may be conceptual ized. The author fee ls that researchers in the past — m o s t not iceably Hudson (1969) and Pederson (1970) — h a v e shown good ins ight in extending the Hagerstrand (1965) argument from the micro level to the macro level but he disagrees somewhat with a number of t he i r e x p l i c i t ana ly t i ca l statements. The author begins by qua l i f y ing these past contr ibut ions and then goes on to suggest rev is ions which are hopeful ly more consonant with — as well as being more general i n te rp re ta -t ions of — the tenets of central place thought. The purpose of th is th i rd chapter, then, is to de l im i t those factors which would seem most c r i t i c a l in e f fec t ing cha rac te r i s t i c spat ia l and temporal adoptive patterns wi th in the central place s t ruc tu re . The author contends, for instance, that the r e l a t i ve i so la t i on of the system ( v i s -5 - v i s other systems), the nature of the s ize d i s t r i b u t i o n of the system's communities ( re f l ec t i ng the r e l a t i v e rate of decl ine of the serv ice m u l t i p l i e r s ) , and the e f f i c i ency of the system's t ransportat ion f a c i l i t i e s are a l l s i g n i f i c a n t in determining whether inter-urban d i f fus ion would be wave- l ike (where the acceptance of an item would spread outward from the largest center — t h i s being the assumed source of d i f fus ion) or whether the adopted item would "jump around" down through the urban h ierarchy. I t should be emphasized that the hypotheses generated in th is chapter are achieved through the merging of thought in two d i s t i n c t research 13 areas — central place theory and grav i ty -potent ia l theory — and there are necessar i ly concomitant ana ly t i ca l problems (not jus t concerning eventual confirmation per se) involved in such an operat ion. Most impor-t an t l y , the author wonders whether or not the two theories are completely independent of one another — improved future research may suggest, fo r instance, that the parameters of those models representing the two theories are in t imate ly re la ted . In the absence of such i nd i ca t i ons , however, the author i s content to assume independence at the present t ime. In add i t i on , the parametrical var ie ty allowed by th is merging — Hudson and Pederson did not have th is advantage — i s exceedingly great so the author decided to opt for a heur i s t i c presentation which i s char-acter ized by a host of numerical examples. The fourth chapter pr imar i ly represents an attempt at incor-porating some st ructura l parametric change into the s t a t i c model. In a sense these comments form a lengthy c r i t i que and rev is ion of the statements found in Nourse (1968) —appa ren t l y the sole observer who has endeavored to re la te system-wide "before and a f te r " h ie rarch ia l a t t r i bu tes . However, in terest i s present ly def lected to the s ingular impacts of var ia t ions in populat ion, per capi ta income, and technology (marketing and transportat ion) at a point in time (Nourse d id not deal with such instantaneous sh i f t s ) as well as over the long run (Nourse deal t with long run sh i f t s — although he ca l led th is comparative s ta t i c s analys is — but the present author i s skept ica l towards many of those asser t i ons ) . In contrast to the evolut ionary argument of the second chapter, inter-urban migrations (of a c t i v i t i e s and productive factors over the long run) amongst the centers of the system are now accommodated (even to the extent that the emergence of new centers becomes theo re t i ca l l y f e a s i b l e ) . 14 In add i t i on , th is chapter pays some at tent ion to reso lv ing j us t how independent population growth amongst the communities of the central place system would be expected to a f fec t the overa l l spa t ia l and temporal propert ies of an ongoing d i f fus ionary p r o c e s s . ^ I f there i s a pers is tent underlying theme for the long run analys is in the fourth chapter i t i s that numerous voids haunt the pre-sent ly ex i s t i ng theoret ica l formulat ions. Hierarch ia l modell ing i s r e a l l y pictured as being a sor t of t rade-o f f : at the resolut ion level of between-c i t i e s issues the aggregation inherent to the service mu l t i p l i e r approach is a valuable ana ly t i ca l convenience ( i f not a necessi ty) yet when ind ica t ions of long run change in those same mu l t i p l i e r s are des i red , a much more thorough understanding of t he i r actual composition ( in terms of ind iv idua l — and not bundles of — goods and serv ices) i s r e q u i r e d . ^ The f i f t h chapter introduces a s t ructura l d i s to r t i on into the usual symmetry of the central place scheme. Af ter i n s t i t u t i n g a l oca l i zed (found at a central place s i te ) incremental s h i f t in an a c t i v i t y of a noncentral place type, the author analyses the subsequent a l l oca t i on of ;' the total population which would be needed to support (as in the c l a s s i c a l argument) th is new body. This a l l oca t ion i s observed to be highly depen-dent upon the specific locat ion of the new a c t i v i t y . For ins tance, the arrangement ( in equi l ibr ium) of the extra serv ic ing population would be remarkably d i f fe ren t i f the new a c t i v i t y were to locate in a small center near the system'sdominant (M th leve l ) center rather than in a center of comparable s ize c loser to the system's boundary (endpoints). In add i t i on , th is s t ructura l asymmetry i s viewed as a d i rec t i ve to any d i f fus ionary process which would unfold in that system: instead of 15 proceeding in a symmetric fash ion, central place adoption would now exh ib i t leading and lagging sectors rad ia t ing outwards from the system's dominant center . FOOTNOTES TO CHAPTER 1 Jef ferson (1939:231) evoked the p r inc ip le of the primate c i t y where: "A country 's leading c i t y i s always d isproport ionate ly large and except ional ly expressive of nat ional capaci ty and f e e l i n g . " Z ip f , l i k e the c i ted e a r l i e r observers, was more concerned with the s i ze re la t ions amongst many centers in a region: hence the evolut ion of the rank-s ize ru l e : p ( l ) = R b p ( R ) where R i s the rank of the c i t y , p(R) i s the population of the c i t y of rank R, p( l ) i s the population of the largest c i t y (of rank 1 ) , and b i s an empi r i ca l l y derived constant. I t has been widely stated that as a regional system becomes more complex and the communities more in te r re la ted — that i s , more integrated s o c i a l l y , p o l i t i c a l l y , and economically — there occurs a s h i f t from primate to rank-s ize features in the c i t y s ize d i s t r i b u t i o n . The author says aspatial because space i s only recognized as being a "container" of a set of c i t i e s in the s tochast ic approaches wh i le , on the other hand, in the central place perspective space i s seen as an element which organizes (constrains) the r e l a t i ve a t t r ibu tes of any community v is -c i -v is i t s neighboring communities. These theories have been only " p a r t i a l l y successfu l " because they are not (yet) we l l - a r t i cu la ted in terms of both t he i r in ternal s ta te -ments and the domain(s) of r e a l i t y which they purport to cover. ^Parr (1973) has recent ly suggested a means of modell ing the Loschian argument. Losch (1954) had an i n i t i a l model which was fundamentally d i f fe ren t from the models discussed here; in that o r ig ina l case, the var ie ty amongst central place populations rested upon the s ize of the nesting factor per se. 16 17 Mull igan (upcoming 1976) has demonstrated, however, that C h r i s t a l l e r i a n theory and the rank-s ize rule are indeed compatible under somewhat d i f f e ren t cond i t ions. I t i s only f a i r to note that Dacey, in f a c t , was r e a l l y the f i r s t to embrace a model wi th in the correct geometry. See Berry (1967:3). Dacey (1966) e x p l i c i t l y suggested th is d i s t i nc t i on but f a i l e d to u t i l i z e i t in his subsequent formulat ions. By independent growth the author means that the population expansion (contract ion) of the system occurs independently of the d i f f u -sionary process; the growth of any one center i s dependent, however, on the growth of the ent i re system. The composition of the service mu l t i p l i e rs might be under-stood by introducing the concept of threshold (as a surrogate for supply and demand analys is ) into the modell ing argument. Some knowledge of th is composition becomes c r i t i c a l for long run analys is because that i s the appropriate scale for tes t ing hypotheses of s t ructura l change in central place systems. As the fourth chapter demonstrates, the composition of each bundle i s less of a problem for comparative s ta t i c s ana lys i s . Chapter 2 THE GENERALIZED HIERARCHIAL MODEL 2.1 Introduction In th is chapter at tent ion i s devoted to the terms, notat ion and conceptual form of the t rad i t iona l modelling procedure. The argument i s i n i t i a l l y concerned with e l i c i t i n g a h ie rarch ia l inter-urban structure compatible with economic base theory. From that vantage, the urban employ-ment l inkages of the space-economy may be t yp i f i ed by a pa r t i cu la r sor t of input-output approach. The discussion then turns to the problem of g iv ing a sa t i s fac to ry evolut ionary in terpre ta t ion for the central place format and a naive s tage- l i ke development, i m p l i c i t to the c l a s s i c a l argument, i s proposed as a basis for future extensions. 2.2 Terms and Notation A center that provides the m th bundle {fm> of goods and serv ices i s said to have the composite funct ion m (1 < m < M) and i f that center does not provide l f m + - j ) i t i s said to have order m as w e l l . Since the center provides the m th basket for a complementary area, i t is said to m - dominate the ent i re population in that surrounding area. 18 19 The notat ional format adopted here is a synthesis of those other formats already ex is tent in the l i t e r a t u r e . Since several new concepts are to be u t i l i z e d (and hence symbolized), i t was f e l t that a l l concepts in the argument should be e x p l i c i t l y symbolized before proceeding: m: the h ie rarch ia l leve l w i th in the system (m = 1,2,**«,M); a lso refers to the bundle of goods and services associated with that level (see Figure 2 .1 ) ; M: the to ta l number of h ie ra rch ia l l eve ls wi th in the system; a lso refers to the to ta l number of bundles offered wi th in the system; e^: the to ta l employment in a center on the m th level ( i . e . in a center o f fer ing bundles l ,2 ,«»«,m); e ^ . : the sector of e^ engaged in o f fe r ing the i th bundle ( i . e . engaged in o f fer ing {f.} in a center which i s i t s e l f on the mth l e v e l ) ; note that 1 < i < m and that : T T T T e = e , + e \ + • • • + e m ml m2 mm g e m : the export-or iented (basic) body of employment in a center on the m th l e v e l ; B T e the sector of e . engaged in o f fer ing the i th bundle to households res id ing outside of the center on the m th l e v e l ; i . e . the basic sector providing the i th bundle; note that : e B = e B + e B + • • • + e B m ml m2 mm Pi e m : the non export-or iented (non-basic) body of employment in a center on the m th l e v e l ; note that : e T = e B + e N m m m N T e m i : the sector of e . engaged in o f fer ing the i th bundle to a l l households res id ing wi th in a center on the m th l e v e l ; i . e . the non-basic sector providing the i th bundle; note that: and: e = e 1 + e 0 + * * , + e m ml m2 mm e T . = e B . + e N . mi mi mi N T em i - j : the sector of e ^ engaged in o f fe r ing the j th bundle to the basic component e B . and to a l l m mi n sectors ( inc luding i t s e l f ) which service e . with a ' mi the bundles 1,2, • • • , i , j , • • • , m ; note that : e - = e , . + e „ . + * , , + e mi mil m2i mmi eo: the number of employees in each basic rural area; E m : the tota l employment on the m th l e v e l ; includes both the employment in the m th level center and the employment in the (complementary) area which that center serves; note that: c ~T , emm E = e + - j — m m k m d . : the number of dependents for each member (employee) 1 T of the sector e . ; mi d 0 : the number of dependents for each rural employee; P m : the to ta l population of a center on the m th l e v e l ; P m . j : the population sector of a center on the m th level engaged in o f fe r ing the i th bundle; note that : p T . = (1 + d.) e T . Hmi v i 1 mi and: pm ~ pml + pm2 + * " + pmm B T pm 1-: the export-or iented port ion of p ^ ; note that : pi, = (1 + d.) Kmi v i mi 21 p : the to ta l export-or iented population of a center on m the m th l e v e l ; note that: B = B + B + . . . + 'B pm pml pm2 *** pmm p^: the non export-or iented (non-basic) population of a center on the m th l e v e l ; note that: rm rm rm r m : the population of the complementary area of a center on the m th l e v e l ; the population of the center i t s e l f i s not inc luded; ri i s the population of each basic rural area; P m : the tota l (market area) population served by a center on the m th l e v e l ; note that : P = p + r m Hm m k : a service mu l t i p l i e r for the m th level ;^ note that: m M 0 < k < 1 and Y km < 1 m L , m m= 1 K -j: a nesting fac to r , representing the geometry of the c i t y system, which spec i f i es the (equivalent) number of market areas of level m-l which are contained in a market area of level m (m > 1 ) ; in a C h r i s t a l l e r -type central place system note that: K , = K for a l l m > 1 m-l K _i - 1: the (equivalent) number of s a t e l l i t e centers of level m-l which are served by a center on the m th level (see Figure 2 .1) . 2.3 Economic Base Considerations Suppose that employment in a, rura l complementary area is eo and that each employee has d 0 dependents. Then the tota l ( i . e . household) rura l population in each of these fundamental spat ia l units i s : SITE MARKET A R E A BOUNDARY O (m-l) st level centre o (m-2) nd level centre Figure 2 .1 . A central place system with nesting factor K , = K = 3; Source: Parr (1969: 241). m " ' 23 n = e Q + e 0 ( d 0 ) = (1 + do) e 0 (2.1) g Each f i r s t level center has a basic sector of e n employees so le l y con-cerned with o f fe r ing a commodity bundle {fi> to these rural households. I f , fo r a spec i f i c serv ice mix and production technology, kx f i r s t leve l employees are required for each rural employee (household), then: e? i = k i e 0 (2.2) g In add i t i on , however, these e n households must be serviced by a non-basic sector (which provides for i t s e l f as wel l ) of e^i employees where: e n = en ki + ( k j 2 + • k e B " 1 - k x The tota l employment e[ in a f i r s t level center i s then: T T B . N e i = e n = e n + e n B e n (2.3) 1 - k. (2.4) I f each employee has di dependents the export-or iented population g Pi of a f i r s t leve l center i s : P? = p? i = (1 + e? i (2.5) and the to ta l population p i s : P i = (1 + d J eL (2.6) 24 The exerc ise i s simply repeated for second level centers . In th is case the employment e 2 in the basic sector has two d i s t i n c t com-B B ponents: e 2 i concerned with the provis ion of (fi) and e 2 2 concerned with the provis ion of the new bundle { f 2 } . According to the geometrical constra ints of central place theory (see Figure 2 .1) : B B . e 2 1 = e n = ki e 0 e 2 2 = k : Ki e 0 + (Kx - 1) el (2.7) M M Now e 2 n and e 2 i 2 employees are required to provide {fi> and {f2} respect ive ly to the component e 2 i (and to one another); i . e . : where: and e S u + e 2 1 2 = e? i i ( k x + k 2 ) + (k i + k 2 ) 2 + P N _ k i efx e 2 1 1 1 - ki - k : P N _ k 2 e ^ e 2 12 1 - k i - k2 (2.8) (2.9) N N Likewise e 2 2 i and e 2 2 2 employees are needed to serv ice the second component e 2 2 : e 2 2 i + e 2 2 2 - e 2 2 (k i + k 2 ) + (ki + k 2 ) 2 + (2.10) where: 25 and: „ B 1 - k x - k N _ k 2 e 2 2 , . e 2 2 2 _ r T T T T T : (2.11) Total employment e l i engaged in f i r s t level a c t i v i t i e s i s : e l i = e ^ i + e 2 1 = e 2 i + e 2 n + e 2 2 i = e B i + k ^ J e | j ^ e L l ( 2 . 1 2 ) and to ta l employment e l 2 devoted to providing second order goods i s : J J 4. J e 2 2 = e 2 2 + e 2 2 _ B , N N - e 2 2 + e 2 i 2 + e 2 2 2 I t fo l lows that the overa l l employment e l in a second level place i s T T ^ T e 2 = e 2 i + e 2 2 Q B A „B _ e 2 i + e 2 2 k 1 - k 2 I f the households of employees engaged in the two a c t i v i t i e s are of s i ze P 1 + d i and 1 + d 2 respec t i ve ly , then the export-or iented population p 2 of a second leve l center i s : 26 B P2 B , B P21 + P22 (1 + d x ) e?i + (1 + d 2 ) e L (2.15) In l i k e manner the to ta l population p 2 of th is same center i s : P2 = T T p 2 1 + p 2 2 (1 + d J e L + (1 + d 2 ) e l 2 (2.16) By the same procedure i t fol lows that the components of the basic sector e m of a center on the m th l e v e l , concerned with the m provis ion of the set f ^ --Cf-> I i = l , 2 , * " , m - - of commodity bundles, are: I 1 J eml = k l e ° em2 = k z { K i e ° + ( K l " 1 } 'm3 e ! . = k 3 KiKzeo + (KiK 2 - K 2) e j + ( K 2 - 1) e l ^ B rm-1 mm m f-m-1 e „ = k„ 4 n K.e0 + i = l m-1 n K. - n K. 1=1 1=2 m-1 m-1 n K . - n K. 1=2 1=3 K , - 1 m-1 J 1 'm-1 (2.17) where: 27 em ml em2 em3 + e1 mm m i - l m M o + I k. n K . e 0 + I k. i=2 1 j=l J i=2 ' K i - i - 1 m a -2 T + I J e i ' k « a=3 i=l a - 1 a - 1 n K. - n K. j = i J j = i+l J (2.18) Now em'j2'**"'emim ^ 5 1 5 m ) employees are needed to provide the elements of {f..} to the component e^. (and to one another); that i s : mil mi 2 , N B + e . = e . mim mi (kj + k 2 + + U nr where: + (k i + k 2 + • • • + k m ) 2 + r B k. e . _ j mi "mij 1 - I'. (2.19) (1 < j < m) (2.20) m However, tota l employment e .. engaged in i th level a c t i v i t i e s i s then: e T . = e B . + e N . mi mi mi B . N , N A e • + e , . + e m 9 . + mi ml i m2i + e mmi n k. (e B + e B + + e B ) B + i - v ml m2 mm mi 1 - k i - k 2 - • • • - k m (2.21) and overa l l employment e m becomes 28 T T T e = e , + e' + m ml m2 + e" mm m n m R m R k x Y e B . + k 2 I e B . + . . . + k I e B . m D 1 , L mi . ' - . m i m . L , mi B , i=i i=i i=l i = l m mi 1 - X k i I = I L , 1 l e B . + e B + . - . + e B ml m2 mm m 1 - ,h k< eB + e B + . . . + e B ml m2 mm m 1 " ,1 k i 'm 1 - 1 k i (2.22) where 1 m represents the export base m u l t i p l i e r . 1 - Y k. i i i 1 Again, i f the households of employees engaged in the m a c t i v i t i e s are of s i ze 1 + d^ ( i = 1 , 2 , . " ,m) respec t i ve ly , then the export-or iented popu-la t i on p of a m th level center i s : rm m B x B . Pml + pm2 + *' mm = (1 + d j e ^ + (1 + d 2 ) e B 9 + -m2 + (1 + d m ) e' m' mm (2.23) 29 and the to ta l population p m of t h i s same center is p = p \ + p^"0 + • • • + Km Kml Km2 Hmm These formulations are simply refinements of a recent statement in the l i t e ra tu re by Parr , Denike, and Mull igan (1975). The c r i t i c a l new assump-t ion i s that household s ize var ies according to commodity bundles (due, fo r instance, to d i f ferences in wage leve ls ) but independently of where those bundles are o f fered. The refinement is so le ly intended to i so la te the d i s t i n c t ef fects of serv ice mix and family s ize mu l t i p l i e r s in creat ing the s ize d i s t r i bu t i on of urban communities. In add i t i on , several in te res t ing propert ies of central place systems are revealed in the above arguments. Since the non-basic a c t i v i t y N e m of a m th level place i s : m r N N N N e = e , + e o + • • • + e m cml m2 mm m i=l 1 B < (2-25) m m 1 - I Ic, i = l 1 i t fo l lows that the basic/non-basic ra t i o ( for employment) b m i s : 30 m i = l m ,1 (2.26) * Besides, the ra t io b>m of the export-or iented population (supported by the basic employees) to the local serv ice population (supported by the non-basic employees) i s : The l as t two equations are s i g n i f i c a n t because they make i t apparent that : ( i ) the b a s i c / n o n - b a s i c r a t i o decreases as cen te rs inc rease in s i z e ; ' ( i i ) the b a s i c p o p u l a t i o n / n o n - b a s i c popu la t i on r a t i o decreases (assuming, of c o u r s e , t h a t a l l dj are s i m i l a r in s i z e ) as cen te rs i nc rease in s i z e ; and ( i i i ) both r a t i o s are independent of the sys tem 's t opo logy . 2.4 Hierarch ia l Input-Output Linkages Closer examination of the economic base in terpre ta t ion suggests that employment constrained in a h ierarch ia l manner permits the urban share of the space-economy to be amenable to a type of input-output ana l ys i s . The format used here to describe ind iv idual urban economies bears a generic resemblance to the t rad i t i ona l sectoral (serv ices , manufacturing, transpor-t a t i o n , e t c . ) input-output model of spaceless economics but, ins tead, de l imi ts employment l inkages between the d i f fe ren t h ie rarch ia l sectors of central p laces . (2.27) 31 Considering the service mix mu l t i p l i e r s as surrogates for technical c o e f f i c i e n t s , reca l l (2.4) where, rearranging terms: T T T B e i = e n = k x e n + e n (2.4) Now consider the second level case and create a d i s t i n c t i o n between vectors J D of gross employment e 2 and net employment e 2 : T e 2 i e l 2 k i k x k 2 k 2 T B e 2 i e 2 i + T B e 2 2 e 2 2 (2.28) where the matrix k of serv ice mix mu l t i p l i e r s spec i f i es direct labor re-3 quirements for each bundle. Solving by matrix a lgebra: 1 - k 2 k x T e 2 i T e 2 2 1 1 - k i - k2 1 - k x B e 2 i e 2 2 (2.29) _* The new coe f f i c i en t matrix k simply determines the to ta l (d i rec t + ind i rec t ) labor requirements for each a c t i v i t y . (2.29) may be s imp l i f i ed to : T B e 2 1 e 2 i T B e 2 2 e 2 2 _ _ 1 - k i - k; k i k : k 2 k 2 B e 2 i B e 2 2 (2.30) which i s iden t i ca l to (2.12) and (2.13) together. 32 In add i t i on , the population vector p 2 may be spec i f ied as: T P21 T P22 1 + d i 1 + dj e l . T e 22 Considering the general case for. e^ then: "ml J J "m3 J "mm ki k i k i • • • kj k 2 k 2 k 2 • • • k 2 k 3 k 3 k 3 • • • k 3 k k k m m m m T em2 T em3 • + m3 • • e T mm • emm (2.31) (2.32) I t may be demonstrated that th is impl ies: B eml m3 • = em3 • mm • e B mm 1 m ki k i k x • • • k i k 2 k 2 k 2 • • • k 2 k 3 k 3 k 3 • • • k 3 km km km m m m m 'ml 2m2| B "m31 'mm I which i s ident ica l to (2.21). Now the population vector p^ i s : (2.33) 33 T pml 1+dx T Pm2 0 T pm3 • = 0 • T pmm 0 l+d 2 0 0 l+d 3 0 0 0 0 1+d. m 'ml T J "m3 J 'mm (2.34) Readers fam i l i a r with comments on the re la t ion between the economic base and t rad i t i ona l input-output approaches w i l l not ice that the bas ic / non-basic dichotomy i s given a very spec i f i c in terpre ta t ion by central place theo r i s t s . In both empir ical and ana ly t i ca l studies controversy has always exis ted over de l im i t ing a basic industry in an open economy. Recent ly, fo r instance, Romanoff (1974) has directed considerable at tent ion to the problem of c los ing the input-output model for basic a c t i v i t i e s ( i . e . moving the productive requirements of basic indust r ies from f i na l to intermediate demand). His remarks deserve the concern of those interested in the short run est imat ing propert ies of the regional input-output format. However, in a pure centra l place system, goods and services may not be exported from the ent i re system (a large economic region) but they cer -t a i n l y are exported from centers of one level to surrounding rural areas and 4 smaller centers . I t i s not unreasonable, then, to consider an a c t i v i t y basic and non-basic simultaneously with reference to the spec i f i c market area of a central p lace. The two proport ions, basic and non-basic, may be spec i f ied as above. 34 2.5 The Problem of Evolution The discussion now turns to a p a r t i c u l a r l y d is turb ing aspect of h ie ra rch ia l model l ing: the problem of how to ra t i ona l i ze the evolution of a central place system with h ie ra rch ia l proper t ies. Parr (1970, 1973) has pointed out that in the seminal e f fo r ts by C h r i s t a l l e r (1966) and Losch (1954) i t was not en t i r e l y c lear whether the authors were concerned with descr ib ing economic transformations in an idea l i zed region or whether they were presenting competing theories for equ i l ib r ium patterns of supply points and market areas at a cer ta in moment in time. In any case, the derived models (see Parr (1970) for some a l t e r n a t i v e s ) , which are more explicit in the i r concern for income l e v e l s , population f i gu res , t ransportat ion cos ts , e t c . , are best interpreted in a s t a t i c sense. Recently there have been attempts to i so la te the d i s t i n c t e f fec ts of parametric changes in populat ion, technology, e tc . on the funct ional composition of the hierarchy as i t ex is ts at one point in time (see Parr and Denike (1970)). The analys is was pa r t i a l ( in that concern was di rected to the individual goods that make up commodity bundles) but provided i n te r -est ing p o s s i b i l i t i e s for future ana ly t i c extension. Nourse (1968), on the other hand, observed such parametric inf luence in a much more aggregate and long run fash ion. By focussing upon the bundles of goods and services themselves - as opposed to the i r ind iv idual components - he was able to phrase his argument in a system-wide manner and not jus t confine his d i s -cussion to change within d i s t i n c t central p laces. Unfortunately, Nourse was unable to e x p l i c i t l y define the system's service mu l t i p l i e r s at d i f f e ren t points in t ime. 35 Hopeful ly , addi t ional invest igat ions along these l i nes w i l l bring about a much improved understanding of how at t r ibu te (numbers of employees, establ ishments, e tc . ) changes within central places may be re lated to : ( i ) the somewhat permanent s t r u c t u r a l p r o p e r t i e s (nesting f a c t o r s , d i s t a n c e s between c e n t r a l p l a c e s , etc.) of the c e n t r a l place systems themselves; and ( i i ) the more ephemeral processes (the innovation and adoption of new techniques, a c t i v i t i e s , e t c . ) which occur amongst the s e t ( s ) of c e n t r a l pI aces. In f a c t , the remaining discussion of th is thesis i s l a rge ly devoted toward creat ing a set of ana ly t i ca l perspectives for deal ing with such temporal change wi th in central place systems. I t i s only f a i r to note, however, that nowhere i s a comprehensive evolut ionary argument being out l ined -i t i s th is author 's opinion that such an argument i s s t i l l qui te a distance of f - but rather a basis i s being a r t i cu la ted for jus t d iscussing systemic centra l place changes in a space-time framework. As a r e s u l t , i t might be best to begin with a s tage- l i ke i n te r -pretat ion of a developing system which has been already b u i l t into the s t a t i c model. While th is only serves as an idea l i zed explanation of how h ie rarch ia l propert ies could a r i s e , the del iberate modi f icat ion of i t s more i m p l i c i t assumptions could h igh l ight cer ta in factors in the theory of inter-urban systems that have been neglected to date. 2.6 A Proposal of Stage-Like Development Suppose that a geometr ical ly closed central place "system" matures ( i . e . central places take on addi t ional a t t r ibu tes in the form of a c t i v i t y 36 bundles, employees, e tc . and inter-urban re la t ions are redefined over time) 5 in a ser ies of stages as suggested in Section 2 .3 . By geometric c losu re , the author i s e f f ec t i ve l y avoiding the problem of determining the various central place s i tes v i s - S - v i s a rural population base: he is assuming that these s i tes have already been selected ( i . e . that a geometry already ex is t s ) but that the a c t i v i t i e s (concerned with providing goods and serv ices) which are f i r s t housed there are of a minimal ( f i r s t level or convenience) nature. Suppose, in add i t i on , that new bundles of a c t i v i t i e s are i n t r o -duced to th is s t ruc tu re , e i ther through innovation or importation of tech-nique; in any case, assume that such new a c t i v i t i e s : ( i ) are introduced one bundle a t a time; ( i i ) are introduced in accordance with t h r e s h o l d o r d e r i n g and locate according to c e n t r a l place p r i n c i p l e s ; ( i i i ) are c h a r a c t e r i z e d by s e r v i c e m u l t i p l i e r s which do not change a f t e r those f u n c t i o n s have been introduced.6 Then, i t may be demonstrated that a central place system, featur ing a wel l-ordered s ize d i s t r i bu t i on of communities ( i . e . arranged according to d i s t i n c t s ize c lasses ; see Parr (1969) for the rank ordering of centers according to these s ize c l a s s e s ) , would develop over time. Furthermore, th is process of systemic development would r e a l l y be character ized by the successive integrat ion of i den t i ca l subsystems into one large system. During th is s tage- l i ke maturation, h ie rarch ia l a t t r ibu tes of the subsystems would be modified - although h ie rarch ia l domination would be retained - while the rural population base would increase in density so as to accommodate 37 the increments of population in the urban centers themselves. The system, then, i s viewed as being the f i na l resu l t of processes of increasing complexity wi th in and integrat ion amongst a set of iden t i ca l subsystems. The postulate of a rural base ( s e l f - s u f f i c i e n t or trading with an outside region) having a constant density is retained from c l a s s i c a l theory. I t i s fur ther postulated that density increases would be spread evenly as development proceeded.^ For purposes of s i m p l i c i t y the argument deals with a standard Km -j = K C h r i s t a l l e r i a n geometry and u t i l i z e s the more t rad i t i ona l nota-t ion for h ie rarch ia l models ( i . e . the discussion is phrased in terms of populations and not employees and the i r dependents). These s imp l i f i ca t i ons are useful for the sake of notation although the more general cases could be discussed in the same manner. Before proceeding i t might well be best to introduce the new notat ion which i s necessary because of the temporal element in the argument: k 0 : the rural service m u l t i p l i e r , representing a g r i -m cu l tura l technology (k 0 < 1 - £ k-, where i=l k i , k 2 , * * * 5 k m are the usual urban serv ice mu l t i -p l i e r s of the s ta t i c argument); s t i pu l a t es , in ra t i o form, how many rural residents are required to serve a given body of urban residents (and other rural residents throughout subsequent rounds of product ion); t 0 : the i n i t i a l point of t ime; time at which the rural households are s e l f - s u f f i c i e n t and no central place s i tes e x i s t ; the f i r s t point in the m th time in terva l (m = 1 , 2 , « " , M ) ; the population engaged in providing the m th bundle of goods and services f i r s t appears at 38 time t while i t s supporting population i s subsequently formed during A t m ; the central place geometry is defined at time t i when the f i r s t leve l functions appear; P m L"t ] : the basic sector for the m th bundle; i t i s introduced at those s i t es which are on the m th , (m+1) s t ,««« , and M th leve ls when the system has c losed ; C D P m [A t m ] : the total supporting population needed for pm [ t m ] ; th is population i s d i s t r i bu ted throughout the rural areas and smaller centers dominated by the m th level place as well as being a l located to the center i t s e l f (note again that th is m th level center may or may not take on addi t ional higher order functions at l a te r times t n , ^ + 2 ' " * ' ^ ' S S pmi t h e P ° r t l o n s ° f P m t A t m-' e n 9 a 9 e d in providing a g r i c u l t u r a l , f i r s t l e v e l , second l e v e l , a n d m th leve l goods and 9 m _ serv ices ( i = 0 ,1,2, • • • ,m); pjj [ A t J = I p ^ [ A t J ; i =0 c f*i [A t m ] : the port ion of p m Q [At ] a l located to each basic rural area; S S p mi j L ' A t m- ' : t h e P ° r t i o n o f Pmn- [ A t m ] ( i = 1,2,• • • ,m-l) a l located to each central place on the i t h , (i+1) s t , * * * , and m th leve ls ( j = 1 , 2 , " « , m ) ; pm L ' t m + A t m - ' : t h e t o t a l direct a l l oca t ion to the m th level p lace ; P^ [ t m + A t ] = p B [ t ] + p S [At ] m^ L m mJ Km L mJ Kmm L mJ M-1 I f there are K points of supply introduced at a time t i , then M-1 i t may be argued that K rural market areas, each of population s ize r i [ t 0 ] , "ex is ted" at time t 0 ( t i > t 0 ) . The basic sector pT[ t i ] (which provides {fi> to a rural area) introduced at each of these supply points involves a population of : 39 Pi [ t l ] = kiTi [ to] (2.35) In the time in terva l [ A t J a supporting body of population p\ [A t x ] i s formed throughout each f i r s t level market area (both at the point of pro-duction and in the complementary area) where: P i [Atx] = p? [ t x ] +(k0 + k j + (k 0 + k x ) z + (ko + ki) B f -, 1 _ k o _ k l Pi L t d (2.36) P i [A t i ] has two components: P ?o [ A t j - 1 _ i: _ k i P ? [ t j (2.37) and: P u [ A t J k 1 B r . n 1 - ko - k! P l [ t l ] (2.38) which imply that the population in each rural complementary area at time t x + A t i i s : r i [ t i + A t i ] = r j [ t 0 ] + pfo [A t i ] r> l 4- kpk^ i [tp] (2.39). and the population in each f i r s t level place i s : 40 Pi [ t i + A t J = p? [ t j ] + P l l [ A t J = kiTi [ t 0 ] + k i 2 rl [ t 0 ] 1 - k 0 - k i (2.40) At time t 2 ( t 2 > t i + A t J basic sectors are introduced at M-2 K points in order to provide { f 2 } . Each of these second leve l s i t e s serv ices K rural areas and K f i r s t level places (one center i s " i t s e l f " : the f i r s t level population occupying the same poin t ; the geometric argu-ments by Dacey (1965, 1966) should e luc idate th is po in t ) . The number of persons comprising each of these basic sectors i s : Once again, now in the time in terva l A t 2 , a supporting population i s required: P2 [ t 2 ] - k 2 ^K P l [ t x + Atx] + Kri [ t : + A t J l (2.41) S B ' p 2 [A t 2 ] = p 2 [ t 2 ] j-(k 0 + k i + k z) + (k 0 + k x + k 2 ) 2 + • • • = (kQ + k i + k 2 ) P2 [ t 2 ] (2.42) 1 - k 0 - k i - k 2 which has three components: P20 [A t 2 ] = 1 - k 0 - k i - k 2 P2 [ t 2 ] (2.43) P « t A t 2 ] = ] . k Q \ . k z p? [ ta] (2.44) 41 It fol lows that each rural area has an incremental gain of: S r i [A t 2 ] = Vf- [A t 2 ] (2.46) Th is , of course, constrains the population growth in the K-l centers M-2 which do not take on second level a c t i v i t i e s (there are (K- l )K of these in the ent i re system). In each of these centers , which w i l l remain f i r s t leve l centers during the closure of the system, incremental growth i s : p i n [A t 2 ] = T - r ^ - r i [A t 2 ] (2.47) The remainder p 2 1 2 [A t 2 ] of incremental f i r s t level a c t i v i t y i s a l located to the second leve l center i t s e l f : P 2 1 2 [A t 2 ] = pf ! [A t 2 ] - (K- l ) p L i [A t 2 ] (2.48) In add i t i on , a l l persons d i r e c t l y providing {f 2} are residents of the second level center: P2 [ t 2 + A t 2 ] = p? [ t 2 ] + pf 2 [A t 2 ] (2.49) Hence, considering the system as a whole, there are: M-l ( i ) K rural areas, each of s i z e : Ti [ t 2 + A t 2 ] = rx [ t i + A t x ] + n [A t 2 ] (2.50) 42 M 2 ( i i ) (K- l )K " f i r s t level centers , each of s i ze : P i [ t 2 + A t 2 ] = p! [ t x + Atx] + p 2 1 1 [ A t 2 ] (2.51) and M-2 ( i i i ) K " second level centers , each of s i ze : Pa [ t 2 + A t 2 ] = px [tx + Atx] + p f 1 2 [A t 2 ] + p 2 [ t 2 + A t 2 ] (2.52) The argument i s repeated for subsequent time periods un t i l geo-metric closure at time t M + At^ . The a l l oca t i on process becomes a b i t more complicated but fol lows the above in a symmetric fash ion . The basic sectors introduced during subsequent time in te rva ls are of the form: P? [ t 3 ] = k 3 | K P 2 [ t 2 + A t 2 ] + K(K- l ) P l [ t 2 + A t 2 ] + K 2 r x [ t 2 + A t 2 ] } p? [ti»] = U -j Kp 3 [ t 3 + A t 3 ] + K(K- l ) p 2 [ t 3 + A t 3 ] + K 2 (K-1) P l [ t 3 + A t 3 ] + K 3rx [ t 3 + A t 3 ] PM [ t M ] = V ! < P M - 1 [ t M - l + A t M - l ] + ,M-2 + K" - ' (K-1) P l [ t M - 1 + A t M - ] ] + K M " ] r i [ t M - ] + A t M - 1 ] ' (2.53) 43 A careful scrut iny of the case leads to the conclusion that rura l population for the ent i re system i s : m ( l - k o ) O - I k.) Rm ^m + A t m ^ = ^ ^ ^ ^ (2-54) 1 " I k i i=0 at time t + At . In addi t ion to ta l population (rural and urban) for the ent i re system i s : Tm K + A tm] ' r . Ct«3 1 ' m k ° (2-55) 1 - I k i i=0 ' at time t m + A t m - This implies that the r u r a l / t o t a l population ra t i o R m =p- at time t + A t m i s simply 1 - £ k. (where the ra t ios decl ine m i=l according to the sequence; 1, l - k l 5 1 - k i - k 2 , e tc . as the system matures). U I t fol lows that the urban/rural population ra t io ^ at th is time i s : m m U T - R T J, k i , m m m m •, i = l to ac\ r = —R— = R~ ~ m — = B- ( 2 - 5 6 ) m m m 1 r , m where b m i s the basic/non-basic ra t i o defined in Section 2 .3 . This l as t resu l t serves to i l l u s t r a t e a very in te res t ing feature of the s tage- l i ke perspective on development in central place systems. As long as a l l serv ice mu l t i p l i e rs are assumed to remain constant over t ime, then the urban/rural population balance of the system (or any subsystem) i s 44 ( i ) independent of the rural serv ice m u l t i p l i e r and the system's (sub-/ system's) topology and ( i i ) equals the inverse of the basic/non-basic r a t i o of the largest center in the system (subsystem). This r e s u l t , of course, holds true for the s t a t i c model out l ined e a r l i e r in Sections 2.3 and 2.4 o ( i . e . the s t a t i c model represents the f i na l stage at time t + At ). 2.7 Concluding Remarks The in tent ion of th i s chapter was to introduce the reader to the elementary s t ructura l propert ies of the general h ie rarch ia l model. In doing so , several addit ions were made to the ex is t ing l i t e r a t u r e : the basic/non-basic ra t i o was given a simple i n te rp re ta t i on , the economic base method-ology was embraced in a h ie rarch ia l input-output scheme, and systemic evolu-t ion was given a naive s tage- l i ke perspect ive. The author advocates use of the h ie rarch ia l model because of i t s f l e x i b i l i t y - th is is i m p l i c i t to the arguments in Sections 2.3 and 2.4 and i s stressed throughout the remainder of the thesis - and i t s symmetry, both in a s t ruc tura l and funct ional sense (e .g . the economy of the largest center in the system was demonstrated to be c lose ly t ied to the urban/ rura l population balance of the ent i re system). FOOTNOTES TO CHAPTER 2 'The mu l t i p l i e r km may be interpreted as the number of house-holds wi th in a c i t y of level m (or higher) that are needed to provide m th leve l goods and services ( i . e . the m th level bundle), as a proport ion of the number of households served ( i . e . throughout the ent i re m th leve l market area of the c i t y ) . When a l l households are of the same s i z e , then "popula-t i on " may be subst i tuted for "households" in the above. Becker (1956:509) has stated that the: . . . traditional view, based usually on simple correlations, has been that an increase in in-come leads to a reduction in the number of children per family. If, however, birth-control knowledge and other variables were held constant economic theory suggests a positive relation between family size and income, and therefore that the traditional negative correlation resulted from positive cor-relations between income, knowledge, and some other variables. i den t i f i e s the to ta l employment involved in f i r s t and second level a c t i v i t i e s in a second level center . This i s analagous to the term of gross output used in t rad i t i ona l input-output analys is - that i s , the total output produced by an industry or a sec to r , inc luding the port ion consumed by the industry (sector) i t s e l f . i den t i f i e s the port ion of e 2 which i s not involved in serv ic ing itsel_f_ with goods and services - that i s , i t represents the basic sector of e l . The analogous term in the usual input-output ana lys is l e l i The vector T e 2 e 2 2 The vector e5 |e?i | e? 2 45 46 i s net output, which represents the port ion of gross output wi th in an industry (sector) not consumed by the various production indust r ies (sec to rs ) ; in other words, i t represents f i na l demand. The matrix k ind icates the production technology ava i lab le at the second level place for u t i l i z i n g employees (labor inpu ts ) ; the analagous term for sectoral input-output analys is i s the matrix of technical c o e f f i c i e n t s . By examining the general formulations of equations (2.32) and (2.33) the reader should see that th is technology i s invar iant despite changes in the s izes of central places ( i . e . there are no scale economies to be rea l i zed by performing the same a c t i v i t y in a larger center ) . By re lax ing assumptions, commodities may be exported from the system (as Parr (1970) has suggested), but the basic/non-basic ra t i o would not be conceptualized any d i f f e r e n t l y . The author says "system" because, to be en t i r e l y co r rec t , a l l the central place s i t es are not elements of a system un t i l a wel l -ordered s i z e d i s t r i bu t i on i s formed - that i s , not un t i l the various independent subsystems have merged together into one large system. I t might be argued that th is i s a determin is t ic counterpart to the ergodic p r inc ip le of s t a t i s t i c s . In s t a t i c central place modell ing i t i s assumed that service mu l t i p l i e rs are the same in d i f f e ren t - s i zed centers (perhaps th is i s a b i t of a sore po in t ) ; now, i t i s add i t i ona l l y assumed that these mu l t i p l i e rs remain constant over t ime. S t r i c t l y speaking, the argument assumes that there are no diminishing returns to agr i cu l tu ra l product iv i ty as the rural base increases. This simply complements the supposi t ion, mentioned above, that the urban serv ice mu l t i p l i e rs have f ixed labour-output r a t i o s . I t may be demonstrated that th is resu l t holds t rue , as w e l l , fo r the more general argument with a var iab le nesting fac to r . Chapter 3 CENTRAL PLACE DIFFUSION 3.1 Introduction Over the past fo r ty years a number of empir ical studies have focussed on d i f fus ion in an inter-urban context.^ From th is body of l i t e r a t u r e has emerged a corpus of induct ive laws re la t ing d i f fus ionary a t t r ibu tes for d iverse items ( innovat ions, opin ions, e t c . ) to var iables such as the ava i lab le transport technology, the soc ia l structure of com-municat ion, the distances from adopting centers to e a r l i e r adopters or nearest la rgest neighbors, e t c . However, i t i s only in the very recent past that cer ta in observers have attempted to phrase more systematic statements about the inter-urban adoptive process. Hudson (1969) was the f i r s t to advocate use of a postulat ional format - embracing the geometry of central place th inking with grav i ty -potent ia l p r inc ip les - which allowed fur ther s ta te -ments to be in ferred (and eventual ly tes ted) . Pederson (1970), in a con-temporary a r t i c l e , discussed a wide var ie ty of issues which should s t i l l be of in te res t to theoret ic ians and empi r ic is ts a l i k e ; unfortunately, his more ana ly t i ca l statements, based as they were on a misunderstanding of the C h r i s t a l l e r i a n topology, must be greeted with some skept ic ism. 47 48 The ideas expressed in these two papers served as invaluable methodological guidel ines for the present author when he f i r s t began to r e f l e c t about d i f f us ion in a central place context. While i t i s present ly argued that there are s i g n i f i c a n t inconsis tenc ies in e i ther of these con-t r ibu t ions - hopeful ly these are el iminated in the upcoming assert ions -the d i f fus ionary model which is out l ined in th is chapter i s r e a l l y fashioned from the more promising ( in th is author 's opinion of course) features of Hudson's and Pederson's proposals. More to the point , th is author f i rm ly advocates t he i r deductive approach as being the proper means of un i t i ng , in ana ly t i ca l form, structure and process in central place systems. Eichenbaum and Gale (1971:541) would perhaps agree that th is represents a spec i f i c attempt at the macro scale of making " . . . the t rans i t i on to form-funct ion-process. . . " from a s t r i c t l y form-function methodology. The i n i t i a l task of th is chapter i s to examine c lose l y the r e l e -vant propert ies of the two models. Since Hudson's proposal d id appear f i r s t and his s t ruc tura l ( i . e . geometrical) axioms were co r rec t l y stated ( reca l l that Pederson's were not ) , much of the upcoming discussion i s appropr iate ly d i rected towards " h i s " model. I t i s th i s author 's conten-t i o n , however, that these s t ructura l axioms require a second look when they are placed in a structure-process nexus. On the other hand, th is author advocates Pederson's conception of the time element ( i . e . continuous versus d iscrete) in the d i f fus ionary argument and also favors his preference towards parametric f l e x i b i l i t y in the grav i ty -potent ia l asser t ion . 49 Besides, th is author has several refinements of his own to add. Perhaps the most s i gn i f i can t stems from his ongoing experiences with h ie rarch ia l c i t y s ize modelling (as in the previous chapter) : he i s able to employ a much wider var ie ty of community s i ze d i s t r i bu t i ons in his argument than e i ther of his predecessors. In contrast to the previous chapter, though, the d iscussion proceeds in a very heur is t i c (more s p e c i f i c a l l y , numerical) fash ion. Even in the idea l i zed central place set t ing the reader should already appreciate that the in terp lay of such diverse factors might well create an assortment of acceptance patterns. I t was decided, therefore, that a non-analyt ica l approach, character ized by a ser ies of tables showing generated data , would be most helpful in suggesting how ind iv idual parametric values and s p e c i f i c d i f fus ionary schemes might be re la ted . The author had both phi losophical and prac t i ca l reasons for advocating the heur i s t i c approach. F i r s t l y , he shares the opinion with many others (see Bergmann (1957) and Rudner (1966)) that over - formal isa-t ion i s not necessar i ly useful in the ear ly stages of an exerc ise . Secondly, th is author i s not en t i r e l y cer ta in whether the argument may in fac t be stated in a su i tab le ana ly t i c fashion - perhaps an algor i thmic format i s the most rigorous poss ib le . 3.2 Cr i t i c i sms of the Ex is t ing Models Hudson and Pederson shared the common intent ion of attempting to re la te some rather disparate concepts in d i f fus ion theory. The "neigh-borhood e f fec t " of c lustered or contagious growth, innovation appearance 50 according to the s i ze d i s t r i bu t i on of c i t i e s , and the use of the l o g i s t i c curve in depict ing cumulative adoption were topics of special concern. They were each able to devise a Hagerstrand-1ike t e l l i n g process for i n i t i a l community adoption and lend some credence to the hypothesis that the above phenomena are not mutually exc lus ive , but systemat ica l ly r e l a ted , propert ies of d i f fus ionary processes at the macro l e v e l . The reader should soon see, however, that the degree to which th is hypothesis has been confirmed i s somewhat tenuous. To begin w i th , each of the i r models i s l im i ted due to the shared assumption that centra l place populations would increase according to the system's nesting factor for market areas ( i . e . in a K = 3 system com-munity populations might increase in the sequence 1000, 3000, 9000, 27000,•• • ) • In Parr (1970) may be found a comprehensive summary of com-peting h ie rarch ia l models where the s t ruc tura l equation: pm 1 - k i K ( 3 ' t yp i fy ing the i r assumption i s included. By a rather simple modi f icat ion of the statement in Mull igan (upcoming 1976), th is model - i n c i d e n t a l l y , f i r s t suggested by Losch - may be shown to be but a special case of the general model out l ined in the previous chapter of the t hes i s . Secondly, i t was an enl ightening idea for each to employ demo-graphic force as the impetus for inter-urban d i f fus ion but, unfortunately, Hudson res t r i c ted his argument to the c l a s s i c a l form: i . e . , where the distance exponent has a value of two (see Stewart and Warntz (1958)). Pederson, however, opted for the increased general i ty which could be achieved by applying the potent ia l formula: 51 P j P H !TH = G ~J~ ( 3 TH where i s the potent ia l expressed at a center of population p^ by a center of population p T located d . ^ units d i s tan t , the constant G i s simply a sca l ing fac to r , and the exponent B represents the f r i c t i o n of d is tance. At one time considerable debate revolved about the meaning, and therefore magnitude, of th is distance exponent (see Isard (I960)) but more recent ly theoret ic ians (see Niedercorn and Bechdolt (1969)) and empi r i -c i s t s a l i ke have shunned the s t r i c t Newtonian approach. Suf f i ce i t to say that in r e s t r i c t i n g the distance exponent to a value of two, an observer i s making an en t i r e l y a p r i o r i assumption - one which i s espec ia l l y un-tenable in the central place set t ing where in te rac t ion is so le l y d i rected by distr ibut ion-consumption motives ( i . e . the domain of t r i p purposes i s 2 l i m i t e d ) . With c loser scrut iny of Hudson's a r t i c l e , the reader w i l l doubtless agree that th is r e s t r i c t i o n was employed because i t had a "cancel 1ing-out e f fec t " when united with the h ie rarch ia l model mentioned above. More s i g n i f i c a n t l y , however, i t i s d i f f i c u l t to agree with various aspects of the author 's te l l i ng -hear ing procedures. For instance, one-way domination by c i t y s i z e , while being a s t ruc tura l tenet of C h r i s t a l l e r i a n theory, i s d i f f i c u l t to j u s t i f y as a necessary condi t ion for central place d i f f u s i o n . Since in theory two centers create the same potent ia l on one another (a lbe i t one place may be much larger) i t would be more natural for adoption to proceed in e i ther d i rec t ion although in accordance with the previous h is tory of the d i f fus ionary process. Perhaps 52 a v iab le way of looking at th is point i s to d is t ingu ish between the short run and the long run: from the fac t that a large center may have con-s iderably more secondary adoptions (due to second, t h i r d , e tc . hearings) than a small center , i t does not fo l low that the large center i n i t i a l l y adopted ( for the f i r s t time) p r io r to the small center . This seems an appropriate time to emphasize that the Hudson and Pederson contr ibut ions and the model to be out l ined below are a l l con-f ined to primary adoption; however, th is does not preclude the i r value as a basis for developing more sophist icated models which could incorporate secondary adopt ion. In add i t i on , the actual generative structure of Hudson's t e l l i n g procedure leaves something to be des i red. He has postulated that messages emanate from t e l l i n g centers p T to hearing centers p^ during d iscrete time in te rva ls and that these messages are only e f fec t i ve when the t e l l -ing center has i t s e l f already adopted. Unfortunately one i s l e f t with the impression that a l l e f fec t i ve t e l l i n g performed wi th in the same time in te rva l i s character ized by the same demographic force (po ten t ia l ) . As Appendix B demonstrates, th is i s not r ea l l y the case at a l l . But perhaps the real issue concerns the usefulness of d iscre te time in te rva ls for conceptual iz ing message generation at the macro level where the in te rac t ion between communities tends to be espec ia l l y continuous. Hagerstrand (1965) himself was even a b i t skept ica l about employing t h i s assumption at the micro ( ind iv idua l ) l e v e l . While the d isc re te in terpre ta t ion is espec ia l l y useful for s to -chast ic model l ing, the continuous perspective has other advantages. F i r s t l y , i t allows process time to be r e a l i z e d : i . e . an object ive measuring 53 of time in terms of a system's process (parameters). Spec i f ied periods of time may then be re lated on a ra t i o scale (perhaps, as Pederson suggested and i s the case below, with respect to the threshold time in te rva l between " invent ion" in a M th level place and i t s f i r s t appearance e l s e -where). In add i t i on , continuous time (a lbe i t condit ions are idea l ized) represents an improved basis for d iscussing predic t ion of process a t t r i -butes. The use of d iscre te time i n t e r v a l s , while being valuable fo r descr ip -t ion with h inds ight , i s a b i t tenuous for predic t ion when no ru les ex is t fo r s t i pu la t i ng the appropriate lengths of those i n t e r v a l s . F i n a l l y , in order to c l a r i f y what Hudson's cha in - l i ke adoptive procedure ac tua l l y e n t a i l s , i t becomes necessary to delve into a somewhat unique feature of central place th ink ing: the existence of equivalent centers. The discussion w i l l d isgress for a short time and then return to the Hudson model. As a consequence of t he i r symmetric conf igura t ions , a l l centra l place systems are character ized by an ordering p r i nc ip le whereby m th level places m - dominate the urban (as well as ru ra l ) population in the i r market areas. However, they directly dominate only shares of the sur-rounding places on the (m- l )s t , (m-2)nd, e tc . l e v e l s . In a K=3 system, for instance, an m th level place m - dominates two places of the (m-l)st order, s i x places of the (m-2)nd order, e tc . but i t d i r e c t l y dominates only two places (one-third of the s ix surrounding centers) of each lower order. I t i s in th is sense that centers dominate equivalent centers in the central place scheme. For cer ta in o p e r a t i o n s - for instance, simply counting up the number of places (points) belonging to a M level system - th is poses only a 54 minor problem. Since most of the centers are obviously i n t e r i o r to the system an assignment p r inc ip le need only be appl ied to those centers located along the boundary of the ent i re system (or , in the l i nea r case, a t the endpoints of the system). On the other hand, when at tent ion turns to domination as a d i r ec t i ve for a spec i f i c process (as for d i f fus ion in Hudson's argument), equivalence cannot j u s t i f i a b l y be overlooked. In the f i r s t p lace , i t s neglect gives a rather d is tor ted spat ia l perspective of a process which i s i d e a l l y symmetric (even in a s tochast ic sense; see Appendix B) . The tendency to ignore equivalence - even though th is may be an expedient approach for i l l u s t r a t i n g market nesting (e .g . see Berry and Pred (1961)) -is espec ia l l y dangerous, however, because i t s k i r t s the important issue 3 of closure for the ent i re system. The geometric argument for central place theory (character ized by an assumption of perfect adjustment) i s s u f f i c i e n t l y general that statements may re fer to : ( i ) one i s o l a t e d system of m l e v e l s ; o r to ( i i ) several adjacent systems of m l e v e l s . The f i r s t in te rpre ta t ion simply refers to a per fec t ly closed system. The second in te rp re ta t i on , to the extent that d i f f e ren t systems may share centers on the i r common boundaries (at the i r common endpoints) , refers to an open system.^ A system's state of openness (closure) is maximized (minimized) in the geometrical sense ( i . e . the system may be considered 5 per fec t ly open) when that system is completely enveloped by other systems. However, real world systems are a l l open: the property of c losure in the i r abstract representations i s simply the resu l t of an i n t e l l e c t u a l 55 operation - whether th is be due to the researcher 's judgement, convenience in gathering data, or some other reason. In any case, i t remains exceedingly d i f f i c u l t to speculate about how the actual degree of c losure ( i . e . the residual openness) in real world systems may inf luence the a t t r ibu tes of processes unfolding there in . At leas t by employing the concepts of perfect c losure and openness, the l im i t s of a t t r ibu te var ia t ion can be establ ished in the idea l i zed representations ( i . e . abstract models) of those real world systems. This i s hardly a problem which i s confined to central place systems and the interested reader might wish to consult Harvey (1969:419-420) for a few general comments on the issue. In a l l honesty, the author i s not at a l l suggesting a neat so l u -t ion to th is dilemma but only wishes to emphasize i t s ex is tence. By at leas t recognizing the problem and estab l ish ing bounds to i t s s i g n i f i c a n c e , the empir ical q u a l i f i c a t i o n of theoret ica l statements may proceed in an atmosphere of improved confidence. Returning to Hudson's con t r ibu t ion , then, i t appears that he has postulated the existence of a per fec t ly open system since a "demonstration" was given to i l l u s t r a t e that equal amounts of demographic force were being expressed at a l l centers d i r e c t l y dominated by the M th level p lace. However, according to the above argument, he has then neglected the potent ia l expressed on the (M- l )s t level places on (at) the system's boundary (endpoints) by M level centers in adjacent systems. The a l te rna -t i ve in terpre ta t ion - that of perfect closure - is precluded by the over-emphasis of the populations of these same (M-l )st level p l a c e s . 6 In the upcoming sect ion at tent ion i s di rected to th is s p e c i f i c problem as well as to the other qua l i f i ca t i ons mentioned before i t . When the reader becomes acquainted with the new model and sees that i t i s a modi f icat ion of Hudson's and Pederson's o r ig ina l endeavors, then, hope-f u l l y he should see th is lengthy c r i t i que in the r igh t l i g h t : not as an over fast id ious attack but rather as a c l a r i f i c a t i o n of methodology. 3.3 An A l te rna t ive Model of Central Place Di f fus ion I t i s now postulated that : ( i ) a c e n t r a l place system e x i s t s which may be area I or l i n e a r , c l o s e d or open, but whose urban popu-l a t i o n s may be expressed according t o the general h i e r a r c h i a l model of the second chapter; ( i i ) a d i f f u s i o n a r y process may be con c e p t u a l i z e d in a t e l l i n g - h e a r i n g manner; ( i i i ) the mechanism f o r t h i s t e l l i n g procedure i s i n t e r -c i t y p o t e n t i a l (as expressed in equation (3.2) above) which i s a r t i c u l a t e d according t o the p r i n c i p l e s of m - domination and which i s e f f e c -t i v e only when one of the c i t i e s has in f a c t adopted; t e l l i n g commences j u s t as soon as a center does adopt (knows) and i t may proceed in e i t h e r d i r e c t i o n between the two c e n t e r s ; ^ ( i v ) p o t e n t i a l i s expressed by the entire populations of r e l a t e d centers although, as was argued e a r l i e r , only shares of (m - l ) s t (nearest (m-2)nd, (m-3)rd, et c . ) lev e l places are d i r e c t l y dominated by each m th level p lace; f o r instance, in a K=3 system, an M th level place expresses p o t e n t i a l a t six surrounding centers housing ( M - I ) s t lev e l a c t i v i -t i e s but the populations of those ( M - I ) s t lev e l places are dependent on the system's c l o s u r e ; (v) complete hearing ( i . e . adoption) only occurs when a s u f f i c i e n t amount of r e s i s t a n c e has been overcome; f o r primary ( i n i t i a l ) adoption t h i s threshold i s the same f o r a l l centers in the system.8 If the modelling procedure i s s t i l l a b i t unclear the reader should turn to Appendix C where adoptive times are ca lcu lated in a 57 step-by-step manner for the open and closed cases, respec t i ve ly , of a K=3 four level central place system. This same procedure underl ies the gen-erated data given in the fo l lowing ser ies of tables - th i r ty- two in a l l were selected by the author (from a set of about f i f t y o r i g i n a l l y hand-computed) in order to i l l u s t r a t e jus t how parametric var ia t ion may drama-9 t i c a l l y modify adoptive patterns in the central place scheme. I f the notation in the table headings appears a b i t esoter ic ( i t i s common in c i t y s ize arguments), then the reader should again re fer to Appendix C where the spec i f i c population f igures are presented for each case. An acquaintance with the l i t e ra tu re on h ie rarch ia l modelling and grav i ty -po tent ia l theory should ind icate that the chosen magnitudes of the parameters are wi th in p laus ib le bounds. In add i t i on , Figures 3.1 and 3.2 are included to i l l u s t r a t e the adoptive times (for both the open and closed cases) in a K=2 four level context . 3.4 Proposi t ions Based on the A l te rna t i ve Model Scrut iny of the aforementioned tables should ind icate that adoptive patterns may indeed be permuted by a var ie ty of fac to rs . Unfortunately, i n the absence of a great number of such tab les , i t i s possible to specu-la te only loosely about the s ingu lar e f fec ts of parameters: the formulation of ce te r i s paribus statements i s a b i t r i sky when parametric values are so int imately re la ted . On the other hand, some general trends are apparent and these are considered in the upcoming d iscuss ion . I t should be emphasized again ( i f the reader missed the point in Appendix C) that the adoptive patterns presented in the tables are 58 Table 3.1 Adoptive Times for Centers in an Open Five Level Central Place System; K = 3, p! = 1000, p /p , = 4 ^ Km ^m-1 (1 < m < 5 ) , b = 1.5 Cumulative Adopti ng Center Equiv. No. Centers % Adopti ng Time % Total Time 5 1 1.23 0.00 0.00 4 3 3.70 1.00 1.93 3 9 11.11 3.84 7.45 2 1 15 18.52 8.64 16.76 2 3 21 25.92 11.87 23.02 2 2 27 33.33 13.68 26.53 1 i 33 40.74 15.98 30.99 , 2 39 48.15 37.37 72.48 1 6 51 62.96 43.10 83.59 1 3 63 77.77 43.88 85.10 1 5 69 85.18 50.57 98.08 1 k 81 100.00 51.56 100.00 59 Table 3.2 Adoptive Times for Centers in a.Closed Five Level Central Place System; K = 3, P l = 1000, p m / p m , = 4 (1 < m < 5) , b = 1.5 Cumulative Adopti ng Center Equiv. No. Centers % Adopting Time % Total Time 5 1 1.23 0.00 0.00 3 7 8.64 1.00 7.04 4 9 11.11 1 .55 10.92 2 1 15 18.52 1.67 11.76 I 1 21 25.92 3.04 21.38 2 2 27 33.33 3.57 25.14 I 2 33 40.74 7.14 50.27 I 3 45 55.55 9.42 66.31 2 3 51 62.96 9.79 68.94 1* 63 77.77 11.19 78.79 I s 69 85.18 13.87 97.65 I 6 81 100.00 14.20 100.00 60 Table 3.3 Adoptive Times for Centers in an Open Five Level Central Place System; K = 3, pi = 1000, P m /P m _- | = 4 (1 < m < 5 ) , b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 5 1 1.23 0.00 0.00 4 3 3.70 1.00 2.97 3 9 11 .11 3.00 8.93 2 1 15 18.52 5.07 15.08 I 1 .21 25.92 7.05 20.98 23 27 33.33 9.02 26.83 2 2 33 40.74 9.53 28.35 , 2 39 48.15 21.69 64.53 1 G 51 62.96 22.92 68.19 1 3 63 77.77 27.77 82.62 75 92.59 32.81 97.63 5 81 100.00 33.61 100.00 61 Table 3.4 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 3, p x = 1000, P m / P m _ 1 = 4 (1 < m < 5) , b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopti ng Time % Total Time 5 1 1.23 0.00 0.00 3 7 8.64 1.00 7.70 2 1 13 16.05 1.30 9.98 1 1 19 23.46 1.76 13.57 4 21 25.92 1.91 14.70 2 2 27 33.33 3.53 27.17 l 2 33 40.74 5.47 42.13 l 3 45 55.55 7.80 60.06 2 3 51 62.96 10.01 77.04 1" 63 77.77 10.96 84.39 1 5 69 85.18 12.42 95.62 l 6 81 100.00 12.99 100.00 62 Table 3.5 Adoptive Times for Centers in an Open Five Level Central Place System; K = 3, P l = 1000, p (R ) = p 5 (1 < m < 5 ) , b = 1.5 Cumul a t i ve Adopting Center Equiv. No. Centers % Adopting Time % Total Time 5 1 1.23 0.00 0.00 4 3 3.70 1.00 8.36 3 9 11.11 2.35 19.65 2 1 15 18.52 3.82 31.94 2 3 21 25.92 4.60 38.46 2 2 27 33.33 5.08 42.47 1 I 33 40.74 5.36 44.82 •I 2 39 48.15 10.23 85.53 1 6 51 62.96 10.43 87.21 1 3 63 77.77 11.00 91.97 1 5 69 85.18 11 .43 95.57 •, 4 81 100.00 11.96 100.00 63 Table 3.6 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 3, pi = 1000, p (R ) = p 5 (1 < m < 5 ) , b = 1.5 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 5 1 1.23 0.00 0.00 3 7 8.64 1.00 15.09 2 1 13 16.05 1.19 17.96 ' I 1 19 23.46 1.57 23.73 4 21 25.92 1.96 29.55 2 2 27 33.33 2.29 34.52 l 2 33 40.74 3.08 46.49 1 3 45 55.55 3.84 57.93 1" 57 70.37 4.95 74.64 I 5 63 77.77 5.54 83.67 69 85.18 5.56 83.85 l 6 81 100.00 6.63 100.00 Table 3.7 Adoptive Times for Centers in an Open Five Level Central Place System; K = 3, pi = 1000, p m . (R) = p 5 (1 < m < 5) , b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 5 1 1.23 0.00 0.00 4 3 3.70 1.00 13.70 3 9 11.11 1.89 25.89 2 1 15 18.52 2.33 31.92 l 1 21 25.92 2.41 33.01 2 3 27 33.33 3.47 46.71 2 2 33 40.74 3.60 49.31 I 6 45 55.55 5.71 78.22 1 2 51 62.96 5.79 79.31 I 3 63 77.77 6.84 93.70 . 1" 75 92.59 7.28 99.73 I s 81 100.00 7.30 100.00 65 Table 3.8 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 3, pi = 1000, p ( R ) = p 5 (1 < m < 5 ) , b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 5 1 1.23 0.00 0.00 I 1 7 8.64 1.00 16.58 2 1 13 16.05 1.02 16.85 3 19 23.46 1.06 17.56 2 2 25 30.86 2.41 39.96 4 27 33.33 2.43 40.36 1 2 33 • 40.74 2.48 41.12 I 3 45 55.55 3.29 54.55 1* 57 70.37 4.32 71 .63 I 5 63 77.77 4.93 81.83 2 3 69 85.18 5.76 95.56 1 6 81 100.00 6.03 100.00 66 Tables 3.9, 3.10 Adoptive Times for Centers in an Open Four Level Central Place System; K = 3, pi = 1000, P m /P m_-j = 4 (1 < m < 4) ; T 3.9: b = 1.5, T 3.10: b = 2 Cumulative Adopting Equiv. Centers Adopting % Total Center No. % Time Time 4 1 3.70 0.00 0.00 3 3 11.11 1.00 7.31 2 1 9 33.33 3.84 28.08 I 1 15 55.55 8.64 63.14 1 3 21 77.77 11.87 86.77 1 2 27 100.00 13.68 100.00 4 1 3.70 0.00 0.00 3 3 11.11 1.00 10.49 2 1 9 33.33 3.00 31.47 l 1 15 55.55 5.07 53.23 l 3 21 77.77 9.02 94.60 l 2 27 100.00 9.53 100.00 67 Table 3.11, 3.12 Adoptive Times for Centers in a Closed Four Level Central Place System; K = 3, pi = 1000, P m /P m _- , = 4 (1 < m < 4 ) ; T 3.11: b = 1.5, T 3.12: b = 2 Cumulative Adopting Equiv. Centers Adopting % Total Center No. % Time Time 4 1 3.70 0.00 0.00 2 1 7 25.93 1.00 10.22 3 9 33.33 1.55 15.85 l 1 15 55.55 1.67 17.07 1 2 21 77.77 3.58 36.53 I 3 27 100.00 9.79 100.00 4 1 3.70 0.00 0.00 2 1 7 25.93 1.00 10.00 I 1 13 48.15 1.30 12.95 3 15 55.55 1 .91 19.08 1 2 21 77.77 3.53 35.27 I 3 27 100.00 10.00 100.00 68 Table 3.13, 3.14 Adoptive Times for Centers in an Open Four Level Central Place System; K = 3, pi = 1000, p m (Rm) = p 4 (1 < m < 4 ) ; T 3.13: b = 15. T 3.14: b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 4 1 3.70 0.00 0.00 3 3 11.11 1.00 19.68 2 i 9 33.33 2.34 46.14 I 1 15 55.55 3.82 75.15 l 3 21 77.77 4.60 90.54 l 2 27 100.00 5.08 100.00 4 1 3.70 0.00 0.00 3 3 11 .11 1 .00 27.76 2 1 9 33.33 1.89 52.47 l 1 15 55.55 2.33 64.74 l 3 21 77.77 3.47 96.42 l 2 27 100.00 3.60 100.00 I 69 Tables 3.15, 3.16 Adoptive Times for Centers in a Closed Four Level Central Place System; K = 3, pi = 1000, p m (Rm) = p, (1 < m < 4 ) ; T 3.15: b = 1.5, T 3.16: b = 2 Cumulative Adopting Equiv. Centers Adopting % Total Center No. % Time Time 4 1 3.70 0.00 0.00 2 1 7 25.93 1.00 17.83 I 1 13 48.15 1.19 21.22 3 15 55.55 1.96 34.92 I 2 21 11.11 2.29 40.79 l 3 27 100.00 5.61 100.00 4 1 3.70 0.00 0.00 l 1 7 25.93 1.00 17.34 2 1 13 48.15 1.04 18.05 I 2 19 70.37 2.37 41.05 3 21 77.77 2.39 41.47 I 3 27 100.00 5.77 100.00 70 Tables 3.17, 3.18 Adoptive Times for Centers in an Open Four Level Central Place System; Ki = 3, K2 = 4, K3 = 3, P l = 1000, P m /P m _- | = 4; T 3.17: b = 1.5, T 3.18: b = 2 Cumulative Adopting Equiv. Centers Adopting % Total Center No. % Time Time 4 1 2.78 0.00 0.00 3 3 8.33 1.00 7.49 2 1 9 25.00 3.16 23.67 2 2 12 33.33 3.44 25.77 I 1 18 50.00 6.97 52.21 l 2 24 66.67 11 .47 85.92 l 3 36 100.00 13.35 100.00 4 1 2.78 0.00 0.00 3 3 8.33 1.00' 11.26 2 1 9 25.00 2.60 29.28 2 2 12 33.33 3.00 33.78 l 1 18 50.00 3.84 43.24 l 2 24 66.67 8.24 92.79 I 3 36 100.00 8.88 100.00 71 Tables 3.19, 3.20 Adoptive Times for Centers in a Closed Four Level Central Place System; K i = 3, K 2 = 4, K 3 = 3, P l = 1000, p ^ p ^ = 4; T 3.19: b = 1.5, T 3.20: b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 4 1 2.78 0.00 0.00 2 1 7 19.44 1.00 19.49 l 1 13 36.11 1.67 32.55 3 15 41.67 1.95 38.01 1 2 21 58.33 3.71 72.32 2 2 24 66.67 3.83 74.66 I 3 36 100.00 5.13 100.00 4 1 2.78 0.00 0.00 2 1 7 19.44 1.00 18.16 1 1 13 36.11 1.30 23.59 3 15 41.67 2.60 47.22 I 2 21 58.33 3.76 68.29 2 2 24 66.67 4.77 85.76 I 3 36 100.00 5.51 100.00 72 Tables 3.21, 3.22 Adoptive Times for Centers in an Open Four Level Central Place System; Kx = 3, K 2 = 4, K 3 = 3, P l = 1000, p m ( R j = p „ ; T 3.21: b = 1.5, T 3.22: b = 2 Cumulative Adopting Equiv. Centers Adopting % Total Center No. % Time Time 4 1 2.78 0.00 0.00 3 3 8.33 1.00 15.68 2 2 6 16.67 2.50 39.21 2 1 12 33.33 2.71 42.48 l 1 18 50.00 4.19 65.69 1 2 24 66.67 5.77 90.41 l 3 36 100.00 6.78 100.00 4 1 2.78 0.00 0.00 3 3 8.33 1.00 23.51 2 1 9 25.00 2.00 47.01 I 1 15 41.67 2.16 50.77 2 2 18 50.00 2.18 51.29 l 2 24 66.67 4.15 97.53 l 3 36 100.00 4.25 100.00 73 Tables 3.23, 3.24 Adoptive Times for Centers in a Closed Four Level Central Place System; K i = 3, K 2 = 4, K 3 = 3, P l = 1000, p m (Rm) = p , ; T 3.23: b = 1.5, T 3.24: b = 2 Cumulative Adopting Equiv. Centers Adopting % Total Center No. % Time Time 4 1 2.78 0.00 0.00 2 1 7 19.44 1.00 27.72 l 1 13 36.11 1.27 35.34 3 15 41.67 2.22 61.56 I 2 21 58.33 2.55 70.62 2 2 24 66.67 3.55 98.34 1 3 36 100.00 3.61 100.00 4 1 2.78 0.00 0.00 2 1 7 19.44 1.00 24.29 I 1 13 36.11 1.02 24.68 l 2 19 52.78 2.54 61.72 3 21 58.33 2.79 67.82 l 3 33 91.67 3.78 91.91 2 2 36 100.00 4.12 100.00 74 Table 3.25 Adoptive Times for Centers in an Open Four Level Central Place System; K = 4 , P l = 1000, pm(R ) = p„, b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Ti me % Total Time 4 1 1.56 0.00 0.00 3 4 6.25 1.00 17.90 2 1 10 15.62 1.56 27.98 l 1 16 25.00 1.68 30.02 2 2 22 34.37 2.42 43.27 I2 28 43.75 3.79 67.85 1" 34 53.12 4.02 71.96 I5 46 71.87 4.23 75.80 I3 58 90.62 4.90 87.72 l 6 64 100.00 5.59 100.00 75 Table 3.26 Adoptive Times for Centers in a Closed Four Level Central Place System; K = 4, pi = 1000, p m ( R j = p , , b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 4 1 1.56 0.00 0.00 I1 7 10.94 1.00 11.49 2 1 13 20.31 1.03 11.88 3 16 25.00 2.00 23.01 l 2 22 34.37 2.28 26.26 2 2 28 43.75 2.56 29.39 I3 40 62.50 3.74 42.95 r 46 71 .87 3.84 44.14 i 5 58 90.62 4.65 53.51 i 6 64 100.00 8.70 100.00 76 Table 3.27 Adoptive Times for Centers in an Open Five Level Central Place System; K = 2, P l = 1000, p m / p m _ 1 = 4 , b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 5 1 6.25 0.00 0.00 4 2 12.50 1 .00 6.55 3 4 25.00 1.80 11.80 2 1 6 37.50 1.99 13.02 I 1 8 50.00 2.00 13.10 2 2 10 62.50 5.58 36.53 V 12 75.00 8.17 53.52 l 2 14 87.50 11.33 74.20 I 3 16 100.00 15.27 100.00 77 Table 3.28 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 2, pi = 1000, p m / p m _ 1 = 4, b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 5 1 6.25 0.00 0.00 7 43.75 1.00 9.40 4 8 50.00 5.00 47.00 2 2 10 62.50 5.65 53.12 l 2 12 75.00 5.70 53.55 l 3 14 87.50 9.24 86.88 1- 16 100.00 10.64 100.00 7 8 Table 3 . 2 9 Adoptive Times for Centers in an Open Five Level Central Place System; K = 2 , P l = 1 0 0 0 , p ( R ) = p 5 , b = 2 Cumulative Adopting Center Equiv. No. Centers % Adopting Time % Total Time 5 1 6 . 2 5 0 . 0 0 0 . 0 0 l 1 3 1 8 . 7 5 1 . 0 0 1 6 . 1 6 2 1 5 3 1 . 2 5 1 . 8 2 2 9 . 3 8 3 7 4 3 . 7 5 3 . 4 7 5 6 . 0 2 I 2 9 5 6 . 2 5 4 . 1 2 6 6 . 6 2 4 1 0 6 2 . 5 0 4 . 5 2 7 3 . 0 4 l 3 1 2 7 5 . 0 0 5 . 8 8 9 5 . 0 2 2 2 1 4 8 7 . 5 0 5 . 9 3 9 5 . 7 8 1 " 1 6 1 0 0 . 0 0 6 . 1 9 1 0 0 . 0 0 79 Table 3.30 Adoptive Times for Centers in a Closed Five Level Central Place System; K = 2, P l = 1000, p m ( R j = p 5 , b = 2 Cumulative Adopting Center Equi v. No. Centers % Adopting Time % Total Time 5 1 6.25 0.00 0.00 l 1 3 18.75 1.00 9.95 2 1 5 31.25 1.82 18.09 3 7 43.75 3.47 34.50 1 2 9 56.25 4.12 41.03 I 3 11 68.75 6.11 60.82 2 2 13 81.25 7.01 69.78 4 14 87.50 8.73 86.94 1" 16 100.00 10.05 100.00 80 Tables 3.31, 3.32 Adoptive Times for Centers in a Four Level Central Place System; K = 2, P l = 1000, p m / p m _ 1 = 4, b = 2, T 3.31: open, T 3.32: c losed Cumulative Adopting Center Equi v. No. Centers % Adopting Time % Total Time 4 1 12.50 0.00 0.00 3 2 25.00 1.00 37.71 2 4 50.00 1.80 67.87 l 1 6 75.00 1.99 74.96 I 2 8 100.00 2.65 100.00 4 1 12.50 0.00 0.00 I 1 ,2 5 62.50 1.00 23.19 3 6 75.00 1.80 41.74 I 2 8 100.00 4.31 100.00 O o O o Q o O o O ( a ) 3 I2 2 I1 4 I1 2 I2 3 ( a ) ( b ) 16000 1000 4 0 0 0 1000 6 4 0 0 0 1000 4 0 0 0 1000 16000 ( b ) ( c ) 1.00 2.65 1.80 1.99 0 1.99 1.80 2.65 1.00 ( c ) ( d ) 2 5 3 4 I 4 3 5 2 ( d ) ( a ) type of centre (b ) population of centre ( c ) standardized adoptive times ( d ) order of adoption Figure 3.1. Adoptive times for centres in an open four level central place system; K = 2, {p |1 < m < 4} = {1000, 4000, 16000, 64000}; Source: Table 3.31. CO O o — O o Q o 0 o Q ( a ) 3 I2 2 I1 4 I1 2 I2 3 ( a ) (b ) 8 0 0 0 1000 4000 1000 64000 1000 4000 1000 8000 ( b ) ( c ) 1.80 4.31 1.00 1.00 0 1.00 1.00 4.31 1.80 ( c ) ( d ) 4 5 2 2 I 2 2 5 4 ( d ) ( a ) type of centre ( b ) population of centre ( c ) standardized adoptive times ( d ) order of adoption Figure 3.2. Adoptive times for centres in a closed four level centra l place system; K = 2, { p j l < m < 4} = {1000, 4000, 16000, 64000}; Source: Table 3.32. 00 ro 83 standardized to pa r t i cu la r thresholds. In order to compare "actual times" -say, for instance, the times that d i f fe ren t processes take to expire - i t is necessary that th is factor be accounted f o r . ( i ) Closure Since closure dramat ical ly inf luences the potent ia l expressed at centers on (at) a system's boundary (endpoints), i t i s a prime determi-nant of systemic adoptive patterns. When the marginal (M- l )s t level places adopt r e l a t i v e l y qu ick l y , as they tend to do with systemic openness, they may become secondary poles for the d i f fus ion process. This i s cons is tent , of course, with the idea that d i f fus ion i s & -state ordered process whose e a r l i e r a t t r ibu tes have a constra in ing e f fec t on l a te r a t t r i bu tes . Openness (espec ia l l y in the areal case) seems to have two d i s -t i n c t consequences. F i r s t l y , i t emphasizes community s ize as a fac tor promoting ear ly adoption. Secondly, i t creates an adoptive lag at the lowest level of the hierarchy: there i s a rather great discrepancy between the adoptive time for the f i r s t leve l places ( I 1 ) nearest the Mth level place and the adoptive times for a l l other f i r s t level places ( I 2 , ! 3 , * * * ) in the system (see Tables 3.1, 3.18, 3 .25). The s ign i f i cance of openness seems espec ia l l y re lated to the dimensioning of the system. In l i nea r systems - where the nesting factor i s minimized - there are more exceptions to the above statements. Closure, on the other hand, contr ibutes to a more wave-l ike adoptive pattern (Tables 3.4, 3.12, 3.20). 7 This general ly appl ies to the set of a l l centers in the system but pa r t i cu l a r l y appl ies wi th in each 84 s ize c l a s s . Closure, then, emphasizes the distance to the or ig ina l source of d i f fus ion ( i . e . the M th level center) as a prime factor in adoptive order ing. In add i t i on , openness (as one na tura l l y expects) i s conducive to a more rapid completion of the systemic process. ( i i ) Central Place Populations From one perspective i t i s the s ize d i s t r i bu t i on of centers that la rge ly determines the spat ia l cha rac te r i s t i cs of inter-urban d i f f u -s i on . Most s i g n i f i c a n t l y , when the mu l t i p l i e rs (see Chapter 2) decl ine rather rap id ly (as in the rank-s ize as opposed to progression component case) and the system is open, the adoptive lag ( in the set of f i r s t level places) tends to be diminished. However, under the same cond i t ions , a new lag may be introduced before the very f i r s t centers of the system adopt from the Mth level p lace. This l a t t e r tendency seems to occur, though, only when the urban system i s complex (compare Tables 3.3 and 3.7, 3.4 and 3 .8) . From another perspective (and a less r e l a t i v e one), process com-p le t ion times are reduced ce ter is paribus by large overa l l urban popula-t ions ( i . e . high dens i t i es ) : that i s , when k i i s large and the elements of the sequence k 2 , k 3 , • • • do not decl ine too rap id l y . One addi t ional point must be included. I t should be understood that the rank-s ize r u l e , even when res t r i c t ed to an exponent of one, i s consis tent with various d i f fe ren t sets of community s ize (see Appendix C ) . Hence i t would be a b i t tenuous to suggest that a spec i f i c d i f fus ionary pattern ( in space and time) might be t y p i c a l l y related to the rank-s ize 85 s t ruc ture . However, i t i s in te res t ing to note that the adoptive orderings per se seem to change very l i t t l e in disparate rank-s ize se t t ings : closed systems espec ia l l y re ta in orderings when l i nea r and areal systems of the same complexity (number of h ie rarch ia l l eve ls ) are being compared. ( i i i ) Distance Exponent The " f r i c t i o n of d is tance , " of course, expresses a s i g n i f i c a n t e f fec t on the overa l l rate of systemic adoption. On the other hand, i t apparently has a permuting inf luence on adoptive ordering only during the ear ly to middle stages of the process. The acceptance pattern amongst the f i r s t , second, e tc . order places ( I 1 , 2 1 , ' " ) nearest the o r ig ina l source may vary according to the value of the distance exponent but th is ear ly permuting v i r t u a l l y disappears as the smaller and more d is tan t places adopt (compare Tables 3.2 and 3.4, 3.15 and 3.16). In add i t i on , there seems to be a tendency for a larger exponent to reduce the adoptive lag in open systems and promote re la t i ve locat ion (with respect to the M th level center) as opposed to merely c i t y s i ze as an important determinant of adoptive ordering (compare Tables 3.1 and 3 .3) . In closed systems, increased f r i c t i o n of distance i s d e f i n i t e l y conducive to wave-l ike d i f fus ionary patterns. v ( iv) H ierarch ia l Levels The complexity of the system does not, ce te r i s par ibus, seem to be as strong a d i r ec t i ve factor as one might expect. Taking into account that d i f ferences in h ie rarch ia l s t ruc tur ing are accompanied by other 86 di f ferences (e .g . the absolute population of the M th level center , the distance to the boundary (endpoints) of the system) - a l be i t these are in t imate ly re la ted - d i f fus ion appears to proceed in much the same re l a t i ve fashion in systems which have disparate h ie ra rch ia l development (but which otherwise share the same parameters). Na tu ra l l y , though, the existence of a greater number of s ize c lasses in more complex systems tends to even out the percentage adoptive times ( re la t i ve to the to ta l time needed for process expi ra t ion) of ind iv idua l centers and must be considered i f one intends to re la te these adoptive times to other factors ( s p e c i f i c a l l y , to the cumulative number of adopt-ing centers as in the l o g i s t i c argument). (v) Geometry Geometric e f fec ts have already been noted insofar as closure and the s i ze of community populations have a topological bas i s . Dimensional i ty , however, should be emphasized. Linear systems seem to be, ce te r i s par ibus, much more conducive to wave-l ike d i f fus ion than the i r areal counterparts (compare Tables 3.7 and 3.29). Secondly, var ie ty in the nest ing factor may induce some change in adoptive pat terns. The p r inc ip les of organizat ion in areal systems do not, though, seem as c r i t i c a l as other factors in promoting spec i f i c acceptance pat terns. Th i rd l y , and perhaps most s i g n i f i c a n t l y , the geometry inf luences the cumulative numbers of equivalent centers that may adopt at cer ta in t imes. This again i s c r i t i c a l to remember when one wishes to hypothesize about the forms of l o g i s t i c curves. 87 (v i ) Cumulative Adoption Hudson's s tochast ic argument generated an S-shaped cumulative acceptance (by center) curve which varied in shape according to the nesting factor and the number of h ie rarch ia l leve ls in the central place system. : In the more heur i s t i c format here, nothing so e x p l i c i t may be s ta ted . Natura l l y , under cer ta in parametric cons t ra in ts , S-shaped cumula-t i ve adoption may be c h a r a c t e r i s t i c ; t h i s , however, cannot be supported as a general hypothesis. As mentioned above, the most c r i t i c a l determinant of the nature of cumulative adoption i s simply the number of h ie rarch ia l l e v e l s : fo r th is defines the number of centers and therefore the number of points (types of centers) that may be func t iona l l y re lated (percentage cumulative centers versus percentage cumulative t ime). Quite na tu ra l l y , a smoother curve, gradual ly r i s i n g with few plateaus ( i . e . l i t t l e tendency toward very high or very low s lopes) , is more cha rac te r i s t i c of complex systems simply because there are more points to r e l a t e . In add i t i on , closure and cumulative knowing appear to be i n t i -mately t i e d . The adoptive lag of open systems tends to produce an S-shape in cumulative acceptance curves: th is e f fec t seems espec ia l l y prevalent in those systems which are concomitantly areal and complex as well (Tables 3 .1 , 3 .3) . On the other hand, some closed systems - again when areal and complex - appear to be character ized by r e l a t i v e l y l i nea r cumulative adoptive curves (Tables 3.6, 3.19). F i n a l l y , on a s l i g h t l y d i f fe ren t note, i t should be emphasized that cumulative acceptance refers only to the number of equivalent centers 88 and that , therefore, the population adopting may vary considerably in accordance with the s ize d i s t r i bu t i on of the places in the system. ( v i i ) A Br ie f Synthesis Hierarch ia l d i f fus ion and the notion of a d i f fus ion wave (con-tagious growth) were phenomena more than s l i g h t l y at odds un t i l Hudson's con t r ibu t ion . I t i s now possib le to i so la te some factors that seem to promote e i ther of these p a t t e r n s : ^ ( i ) h i e r a r c h i a l d i f f u s i o n : (a) systemic openness; (b) r e l a t i v e l y slow d e c l i n e in the values of the elements of the s e t {k I I < m < M}; m 1 (c) low value of the f r i c t i o n of distance c o e f f i c i ent; (d) areal d i m e n s i o n a l i t y of the system; ( i i ) wave-like d i f f u s i o n : (a) systemic c l o s u r e ; (b) r e l a t i v e l y r a p i d d e c l i n e in the values of the elements of the set {k I I < m < M}; m ' (c) high values of the f r i c t i o n of di s t a n c e c o e f f i c i e n t ; (d) l i n e a r d i m e n s i o n a l i t y of the system. In the previous i l l u s t r a t i o n s , Table 3.1 perhaps best exempl i f ies ( i ) above while Table 3.8 (the areal case) and Table 3.30 (the l i nea r case) best represent ( i i ) . The argument lends support to some of the hypotheses which Hudson and Pederson suggested - pa r t i cu l a r l y to the assert ions that disparate 89 d i f fus ion patterns may be emobidied in the same spat ia l structure and that cumulative adoption by central places may be S-shaped in nature - but, of course, with the many stated reservat ions. 3.5 Addi t ional Comments on the A l te rna t ive Model The basic def ic iency of the newly out l ined model (and of the seminal models, for that matter) revolves about i t s ( the i r ) t e s t a b i l i t y . Since a l l these arguments are based on a s t a t i c spatial-economic theory which i s i t s e l f i dea l i sed and somewhat unamenable to t es t i ng , i s o l a t i n g domains of the real world which would accurately r e f l e c t the modell ing postulates becomes an extremely d i f f i c u l t task. Now some of the s p e c i f i c methodological issues are attended to . F i r s t of a l l , the present model, l i k e Pederson's, i s de te rmin is t i c . In contrast to the idea l i sed s t a t i c s e t t i n g , centers of the same type (here, too, might a r i se a c l a s s i f i c a t i o n problem) in the real world would not be expected to adopt a spec i f i ed item at exact ly the same time. Of course, the computed idea l i sed times could be construed as the means of adoptive p robab i l i t y d i s t r i bu t ions for d i f fe ren t types of central places but th is would necessar i ly ra ise cer ta in other problems of a more s t a t i s -t i c a l n a t u r e . ^ Even Hudson's s tochast ic argument, when phrased in the terms of equivalent centers (see Appendix B) , does not r ea l l y circumvent th is i ssue. 12 Secondly, the model i t s e l f i s s t a t i c . Real world centra l place a t t r ibu tes would l i k e l y be changing over time and th is might have a s i gn i f i can t e f fec t on the propert ies of an ongoing spat ia l process l i k e d i f f u s i o n . Therefore, i f a model i s to become a v iab le predic tor i t must 90 accommodate parametric change in some fashion (whether that change be exogenous or dependent on the adoptive scheme i t s e l f ) . The a l te rna t i ve model which has been discussed can successfu l ly represent such change -as i s shown in the upcoming chapter, by assuming that the system i s passing through successive stages of equi l ibr ium - and th is const i tu tes one advan-tage of making the threshold of adoption e x p l i c i t in the argument. Th i rd ly (and th is appl ies to the other models as w e l l ) , i t i s necessary to speci fy the domain of items to which the proposed model i s app l i cab le . The postulates have been stated in such genera l i ty that the model should be appropriate for representing the adoption of most items -such as organizat ions, consumer goods, d iseases, e t c . - which researchers have used in the past as ind icators of a d i f fus ionary process. In any case, the model would be most useful for typ i fy ing those processes where adoption, in the researcher 's op in ion, could be conceptualized according to a te l l i ng -hear ing dichotomy. I t might be added that adoption of cer ta in items might depend on pa r t i cu la r modes of the ent i re transportation-communi-cat ion matrix and th i s should be considered in any empir ical q u a l i f i c a t i o n of the proposed model. On the other hand, i t must once again be emphasized that a l l these models deal with primary ( f i r s t ) adoption. For a more general s ta te-ment on inter-urban d i f fus ion - one taking into account intra-urban t e l l i n g and inter-urban feedbacks for secondary adoption - an approach s im i l a r to that of Caset t i (1969) would have to be integrated into the present argument. F i n a l l y , th is a l te rna t i ve model assumes that a l l places in the system tend to contr ibute to the d i f fus ionary pat tern. By applying the threshold constra ints of the centra l place format, i t would be possible to 91 sa t i s f y the notion that cer ta in small places could not adopt higher order economic items. Besides, proposi t ions which were based on the idea of process completion (as in the previous sect ion of th is chapter) , might be given special at tent ion s ince , in the real wor ld, there are numerous s i tua t ions where a process may terminate prematurely because of a subst i tu te (economic i tem), remedy (disease i tem), or some other fac to r . 3.6 Concluding Remarks In th is chapter at tent ion has la rge ly been devoted to combining spa t i a l process and spa t ia l s t ructure - f>r the centra l place context - in a coherent and systematic fash ion. As much emphasis was placed on ( i ) de-l i m i t i n g the postulates needed for th is operation and ( i i ) qua l i fy ing the sparse l i t e ra tu re which deals with the top i c , the d iscussion has a decided "methodological" r ing to i t . Some spec i f i c consequences of a proposed a l te rna t i ve model (which i s capable of deal ing with much more parametric var ia t ion than ex is t i ng models) were then g iven: to the degree that these resu l ts were nei ther discordant with the scanty empir ical work ava i lab le nor c o n f l i c t i n g with i n t u i t i v e thought, they were construed as o f fe r ing support of th is a l te rna -t i ve model as being representat ive of central place d i f f u s i o n . In shor t , the model suggests that i t should not be surpr is ing to discover that : ( i ) i d e n t i c a l items may d i f f u s e q u i t e d i f f e r e n t l y in d i s p a r a t e c e n t r a l p lace s e t t i n g s ; and ( i i ) d i s p a r a t e items may d i f f u s e q u i t e d i f f e r e n t l y in i d e n t i c a l (or reasonably s i m i l a r ) c e n t r a l p lace s e t t i n g s . 92 The author has defended, as w e l l , an heur i s t i c approach to the argument at hand. I t was f e l t that s t r i c t formal isat ion would be s e l f -defeat ing at the present time and that ana ly t i ca l r igo r should be i n t r o -duced only when the confirmation status of the modell ing procedure has been subs tan t ia l l y enhanced. In the meantime, the author would welcome repet i t ions of the procedure (with d i f fe ren t parameters) in order to fur ther qua l i f y the proposit ions discussed above. FOOTNOTES TO CHAPTER 3 " See Bowers (1937), McVoy (1940), Crain (1966), Hagerstrand (1952), Hudson (1969), Pyle (1969), and Pederson (1970). "•Central place theory i s more than a b i t tenuous where i t concerns the issue of consumer in teract ion amongst d i f fe ren t communities. The com-p lex i t y of the topic has precluded, to date, the a r t i c u l a t i o n of any " l a w - l i k e " statements which would rest f i rm ly on the tenets of economic theory - a host of in te r re la ted f ac to r s , such as the state of t ransporta-t ion , technology , the frequency of multipurpose household t r i p s , the incomes of those households, e tc . would have to be l inked in a coherent and con-s i s t e n t manner. Fortunately, g rav i ty -potent ia l theory, when modified to the s p e c i f i c domination scheme of the central place s t ruc ture , does o f fe r a su i tab le surrogate methdology. Equation (3.2) expresses the idea that in terac t ion between centers of the same s ize - given that they are elements of d i s t i n c t systems or of the same system at d i f fe ren t points in time -may vary considerably fo r a number of unspecif ied reasons: for instance, d i f ferences i n t ransportat ion technology, in the p r o c l i v i t y of customers to t r a v e l , e t c . In add i t i on , i t should be emphasized that the potent ia l formula, while being most appropriate for a s ing le mode of t ransportat ion (com-municat ion), may be extended and made appl icable to multimodal in teract ion (by combining the various ind iv idual potent ia l funct ions) as w e l l . ^Closure i s now interpreted somewhat d i f f e ren t l y than i t was in the previous (second) chapter. Here closure refers to whether or not d i s t i n c t systems share boundary points (where potent ia l i s expressed) whi le e a r l i e r that term al luded to the existence of a wel l -ordered f r e -quency d i s t r i bu t i on (which, in tu rn , would be geometr ical ly determined) of centers in the centra l place s t ruc ture . Berry (1964, 1967), on the other hand, would perhaps have an en t i r e l y d i f fe ren t and more funct ional approach to the use of c losure . Unfortunately, no other expression seems so adaptable to the depict ion of such systemic proper t ies . 93 94 "'This author i s of the opinion that the re laxat ion of postulates -a va r ia t ion of Harvey's (1969) "as i f " methodology i f you l i k e - i s i n -dispensable to ana ly t i c advance in many spheres of the soc ia l sc iences. He a lso shares Par r ' s (1970) more spec i f i c views on the inherent f l e x i b i l i t y of the h ie ra rch ia l argument: espec ia l l y i t s a b i l i t y to accommodate various geometries, non- ter t ia ry a c t i v i t i e s , e tc . Therefore, i t should not be surpr is ing that the interconnection of independent systems i s present ly being advocated. While th is would be d i f f i c u l t to ra t i ona l i ze theo re t i ca l l y in so le l y economic terms (the s ing le system has enough of i t s own problems here) , i t i s a perspective with obvious conceptual and empir ical merit -Berry and Pred (1961:18), fo r instance, have mentioned C h r i s t a l l e r ' s own concern over " . . . the r e l a t i v e strength of any one system and adjacent systems. . . . " I t should be pointed out that with asymmetry in sets of systems -e i the r in terms of the geometry or central place populations of such systems - the s t ruc tura l symmetry of any one system would no longer be a s u f f i c i e n t condi t ion for that same system to be character ized by symmetric d i f fus ionary patterns. "'Openness i s maximized in a topological sense when a system i s completely enveloped by other systems (and th is is the only case considered in th is t h e s i s ) . However, openness must a lso depend on the other parameters ( i . e . the populations of centers , the f r i c t i o n of d is tance, e t c . ) of those adjacent systems. Besides, th is conception of openness i s a lso useful i f the researcher wishes to consider an adoptive process beginning at a smaller center ( i . e . a center of order m < M) in any centra l place system ( i t might be emphasized that Hudson's argument cannot accommodate th is poss i -b i l i t y e i t h e r ) . Of course, an asymmetric adoptive pattern would be char-a c t e r i s t i c here as w e l l . -' Pederson has d e f i n i t e l y deal t with a closed system (due, in large measure, to his er r ing in terpre ta t ion of the C h r i s t a l l e r i a n geometry). As implied e a r l i e r , th is simply means that potent ia l is an impetus to d i f fus ion between only those centers which are func t iona l l y re la ted . The assumption that the threshold i s equal for a l l centers i s , admit tedly, of an a p r i o r i nature. The form of the general argument would s t i l l be app l i cab le , however, as long as adoptive thresholds remained a funct ion of c i t y s i z e . This threshold was made e x p l i c i t in Pederson's case but was deal t with only i m p l i c i t l y by Hudson. 95 JThere are two points of concern here. F i r s t l y , the tables which were omitted d id not c o n f l i c t with any of the tables which are present ly inc luded: for instance, adoptive times for C h r i s t a l l e r ' s data (see Beckmann and McPherson (1970:31) were s im i l a r to those given in other tables of t h i s chapter - hence, they were not included. Secondly, some very s l i g h t discrepancies might ex i s t in the per-centage (adoptive) to ta l times because, in a few cases, these times were computed with an excessive number of s i g n i f i c a n t f igures (see the t e l l i n g process in Appendix C for a case in po in t ) . l uStatements (b) in ( i ) and ( i i ) deserve some addi t ional com-ment. To begin w i th , i t i s simply being asserted that , ce te r i s par ibus, overa l l large central place populations would tend to induce h ie rarch ia l d i f f us ion while overa l l small central place populations would tend to give r i s e to more wave-l ike patterns. Di rect ing at tent ion to the rate of decl ine in the service mu l t i p l i e r s is but a preferable way - in the ana ly t i ca l sense - of s ta t ing such an hypothesis. To take an example, the strong h ie rarch ia l e f fec t exhib i ted for the open case in Table 3.1 and the closed case in Table 3.2 (where r x = 2000 and {k } = {.3333, .1667, .1250, .0937, .0703}) i s s i g n i f i c a n t l y diminished foY the comparable (open and closed) cases in Tables 3.3 and 3 .4 , respect ive ly (where n = 2000 but {k = .3333, .1185, .0779, .0462, .0313}). m In addi t ion th is " d i s t r i b u t i o n a l " approach i s superior to s ta t ing that the d i f fus ion pattern depends on the sum of the mu l t i p l i e r s alone. Considerable var ia t ion in the rate of decl ine of the mu l t i p l i e r s - and, hence, considerable var ia t ion in d i f fus ionary patterns - i s en t i r e l y con-sonant with an overa l l f ixed sum for those m u l t i p l i e r s . In f a c t , t h i s poses the in te res t ing question of determining the d i s t r i bu t i on of serv ice mu l t i p l i e r s which would - given cer ta in other parameters of the system (the nesting f a c t o r ( s ) , the f r i c t i o n of d is tance, e t c . ) as well as the sum of those mu l t i p l i e r s - minimize the time of l as t adoption in a central place system. Pa r t i cu la r problems would a r i s e , for instance, over the s p e c i f i -cat ion of appropriate time in te rva ls ( for these would determine whether or not centers of the same type adopted at "exact ly " the same time) and the actual postulat ion of a relevant adoptive p robab i l i t y d i s t r i bu t i on (normal perhaps?) for each central place type. ""The property of s tas i s induces a theoret ica l problem of some s i gn i f i cance . For an economic innovation to occur - except f o r , perhaps, " fash ion" changes in new automobiles, c lo thes , appl iances, e tc . - there would l i k e l y be a concomitant disturbance or s h i f t in the system's e q u i l i -brium pos i t i on . While th is represents an inconsistency of sor ts in the proposed model, i t i s a shortcoming which cannot - in th is author 's opinion -96 be resolved with the theoret ica l tools which are presently at hand (see footnote 2 above). In any case the author does give some a t ten t ion , in the fo l lowing chapter, to d iscerning changes which might occur in d i f -fusionary patterns as the whole system sh i f t s i t s equi l ibr ium posi t ion (which i s re f lec ted in the changing s ize d i s t r i bu t i on of urban communities). A l s o , the author fee l s that th is inconsistency i s more of a theoret ica l rather than pragmatic issue and the model-tester should not view i t as being an important bar r ie r to his empir ical qua l i f i ca t i on procedures. Chapter 4 P A R A M E T R I C I N F L U E N C E S O N S T R U C T U R E A N D P R O C E S S I N T H E C E N T R A L P L A C E S Y S T E M 4.1 Introduct ion In the second chapter of th is thesis i t was stated that central place theory, as t r a d i t i o n a l l y rece ived, i s merely a static formulat ion: that i s , a depict ion (a lbe i t somewhat idea l ized) at one point i n time of a cer ta in domain of the real world. Perhaps, however, the reader has already been convinced that there is considerable merit in re lax ing th is narrow viewpoint. In f a c t , the author contends that the property of s tas i s per se represents only a prima fac ie - and not far- reaching - constra int to the advancement of sound ana ly t i ca l work in the subject area. For instance, i t has been twice demonstrated j us t how the s p a t i a l -economic p r inc ip les inherent to the h ie rarch ia l model of c i t y s ize provide a useful framework for d iscussing attribute changes wi th in central place systems over a period of t ime. More e x p l i c i t l y , the author was able to c l a r i f y (by making the relevant centra l place geometries invar iant over time) the manner in which: 97 98 ( i ) aggregate c h a r a c t e r i s t i c s of c e n t r a l places (numbers of f u n c t i o n s , employees, e t c . ) and t h e i r r e l a t e d systems (the urban/rural popula-t i o n r a t i o ) cou Id be s h i f t e d during a p o i n t -s p e c i f i c s t r u c t u r a l t r a n s f o r m a t i o n ; and ( i i ) p a r t i c u l a r c h a r a c t e r i s t i c s of c e n t r a l places (has a center adopted a new a c t i v i t y ? ) and t h e i r r e l a t e d systems (how many centers have accepted t h i s new a c t i v i t y ? ) could be s h i f t e d during a p o i n t - s p e c i f i c d i f f u s i o n a r y process. Usua l ly , however, we tend to perceive the transformation in ( i ) and the process in ( i i ) as being somehow interwoven as economic a c t i v i t y i s extended throughout a region. But th is over looks, of course, a very in te res t ing and s i g n i f i c a n t problem in i t s own r i gh t : that being howinnova-t i o n s , adoptions, and s t ructura l changes are continuously blended in w e l l -def ined spatial-economic systems as growth proceeds. Natura l ly i t remains a much eas ier task jus t speculat ing loose ly about the form of a modell ing scheme which might s a t i s f a c t o r i l y deal with th is complex problem rather than ac tua l l y devis ing a set of re levant , l o g i c a l l y connected, and empi r i ca l l y testable statements.^ Mind you, the author has no present intent ions of putt ing together jus t such a set of statements - th is would simply require too much novel ana ly t i ca l and empir ical work; on the other hand, he wishes to o f fe r some addi t ional ins ights into how change and transformation may be perceived (and u l t imate ly discussed) wi th in the abstract central place context. I t i s hoped that from a corpus of par t ia l ins ights a more general theoret ica l model - inc luding feedbacks, m u l t i p l i e r s , and the l i k e - may eventual ly a r i se and that th i s w i l l lead to a subs tan t ia l l y improved understanding of the spat ia l conse-quences of economic growth in the real wor ld . 99 To begin w i th , i t i s quite possible to focus on d i f fe ren t parameters (populat ion, technology, e t c . ) in order to speci fy how the i r ind iv idual var ia t ion may be re lated to s t ructura l changes in diverse centra l place systems over time. By u t i l i z i n g the equi l ibr ium condi t ions of the h ie rarch ia l model, the instantaneous impact of such parameters may be i n i t i a l l y es tab l ished; fo l lowing th i s i t becomes possible to move on to the case of the long run where d iscre te system-wide changes (due, fo r instance, to the migration of productive factors) may be incorporated into the central place format. The former technique i s most typ ica l of the comparative s t a t i c s approach in which a system is taken to be in a state of equi l ibr ium and i t i s then "shocked" and taken to proceed to a subsequent state of e q u i l i b r i u m . 3 To the author 's knowledge Nourse (1968) represents the sole attempt to introduce parametric impact with e x p l i c i t system-wide "before 4 and a f te r " cond i t ions . Unfortunately his argument was based on the obsolete Beckmann (1958) model - a model, i n c i d e n t a l l y , which remains i n f l e x i b l e even when properly reformulated - so that a l l of Nourse's comments must be greeted with some reservat ion . Hopeful ly , too, when the reader has compared the upcoming discussion v i s -& -v i s Nourse's c o n t r i -bution he may be able to expel some of Nourse's pessimism about the app l i c -a b i l i t y and usefulness of the h ie ra rch ia l perspective for deal ing with regional economic growth. Secondly, the author reviews the macro-dif fusionary argument of the previous chapter in a new l i g h t . By v i s u a l i z i n g independent parametric change as an important force reshaping spat ia l adoptive processes TOO i t becomes feas ib le to hypothesize regarding the pa r t i cu la r impacts that changes in central place populat ions, in leve ls of per capi ta income o r , perhaps, i n t ransportat ion technology might have upon the overa l l accep-tance pattern of a given item. The author suggests that th is new per-spect ive seems most appropriate when looking at passive items (e .g . a new arch i tec tu ra l form, a new household appliance) which would not be expected to engender s i g n i f i c a n t feedbacks in the central place structure as other much more active items (e .g . a new mass t rans i t scheme, a new community 5 hosp i ta l ) most ce r ta in l y would. One overa l l concluding remark i s deemed necessary. The tenor of argument i s somewhat uneven as i t var ies from the quite r igorous s ta te -ments of the comparative s ta t i c s approach to the more descr ip t i ve analys is used in the discussion of long run s t ructura l change and, f i n a l l y , to the heu r i s t i c treatment of central place d i f f u s i o n . As in the th i rd chapter, the author sees no pa r t i cu la r and immediate advantage in strengthening the ana ly t i ca l tone in cer ta in parts of his d iscuss ion . Once again, cer ta in hypothetical examples are given in order to i l l u s t r a t e the e f fec ts of temporal changes in the relevant central place se t t i ngs . 4.2 Population To begin with consider the case of population growth wi th in a centra l place system. Scrut iny of the argument in the second chapter should make i t apparent that one can i so la te the instantaneous ef fects of change -a change which i s said to be exogenously induced - in the employment (popu-la t ion ) cha rac te r i s t i cs of the basic sector of a central place upon the overa l l employment (population) cha rac te r i s t i cs of that same central p lace. 101 Note, however, that our ex i s t i ng framework seems en t i r e l y appl icable for d iscussing long run population changes as w e l l . Theory would lead one to suspect that ce te r i s paribus population growth over the long run could refashion systemic structure and process in but a few general ways; the precise impact, though, would appear to depend upon: ( i ) the actual magnitude of such growth (put d i f f e r e n t l y , i f population were t o expand throughout the system a t a constant r a t e , would the long run o r the very long run be the r e l e v a n t time s c a l e ? ) ; ( i i ) the t h r e s h p l d of emergence f o r new centers -tha t i s , one must con s i d e r " . . . the for c e s of i n e r t i a which may preserve the number of h i e r a r c h i a l l e v e l s and even the o v e r a l l number of c e n t e r s . . ." (P a r r and Den ike (1970:574)); and ( i i i ) the degree of m o b i l i t y in productive f a c t o r s -labor, of course, i s of paramount i n t e r e s t here. The reader should note that population decl ine may be conceptu-a l i zed in a symmetrical but opposite manner. Before proceeding to the analys is per se a few statements of a de f in i te methodological character , d i sc los ing the author 's personal views about the especial advantages of the ( s t r i c t ) systems approach, might be of some in te res t to the reader. The author pa r t i cu l a r l y favors the systemic model since i t guarantees a measure of consistency in the analys is as concern var ies over d i f f e ren t temporal and spat ia l sca les . More e x p l i c i t l y , the reader should recognize that change may be appropr iately examined w i th in : ( i ) a s p e c i f i c c e n t r a l place - which i s an individual i t s e l f as well as an element of the much wider system - at a p a r t i c u l a r i n s t a n t i n time; or ( i i ) a s p e c i f i c i nd i v i duaI c e n t r a l place system - i t s e l f an - over a p a r t i c u l a r i n t e r v a l of time. 102 I t i s perhaps a truism to remind the reader that the long run changes al luded to in ( i i ) above would be made apparent in the same centra l places which would exh ib i t the instantaneous changes of ( i ) . In the f i r s t of these cases an exogenous-endogenous dichotomy is rea l i zed wi th in the central place i t s e l f (since the change in external employment (population) necessi tates a d i rec t change in the serv ic ing or basic sector which, in tu rn , brings about successive rounds of change in the non-basic sec to r ) . In the second case, however, th is dichotomy - i f , in f a c t , i t ex is ts - must be somehow rea l i zed at the "boundary" of the en t i re system. In order to c l a r i f y th is l a s t point the reader should observe that th is d i s t i n c t i o n would be v a l i d , for instance, i f the ex i s t i ng centra l place populations were to remain innately stable while migration pers is ted in the long run. Then i t would be e n t i r e l y correct to assert that systemic population growth was brought about by an exogenous change. In cont ras t , the e f fec t would be iden t i ca l but the d i s t i nc t i on could not be s t r i c t l y upheld i f a l l growth were internal (caused by an increase in the b i r th rate perhaps). The author simply fee ls that the exogenous-endogenous dichotomy is a very useful one for ensuring precis ion or methodological r igo r in an argument but he wishes to emphasize that the meaning of that d i s t i n c t i o n must be considered when the temporal and/or spa t ia l scales become enlarged. Nevertheless, th is a b i l i t y to l i nk successive scales (see Harvey (1969:452) i s a major reason why the systemic approach has both conceptual and i n s t r u -mental merit when one i s undertaking the complex task of modell ing some port ion of the real wor ld. 103 In c los ing th is foreword i t should be noted that since popula-t ion change - both in the sense of growth and red is t r i bu t ion - i s a symptom of other forms of parametric change, th is i n i t i a l sect ion is considerably more deta i led than the subsequent two sect ions which focus upon the impact of changes in per capi ta income and d i f fe ren t types of technology on systemic equ i l i b r ium. ( i ) Structure Examination of the discussion in Section 2.3 should make i t apparent that an exogenous s h i f t in employment (population) for a pa r t i cu la r centra l place would be rea l i zed throughout the rural areas and the various smaller communities served by that central p lace. On the other hand, th is s h i f t i s u l t imate ly resolvable to a change in the rural densi ty alone, s ince a l l community populations are " b u i l t up" from th is rural base. The upcoming discussion i s i n i t i a l l y concerned with the former (and more general) case wherein a sh i f t in any of the components e - | , B B e m 2 ' * * * ' emm °^ a n m ^ level center has an impact upon the tota l number of employees (persons) res id ing there. Unfortunately, though, the recur-s ive format of the h ierarch ia l model makes the exercise of der iv ing these pa r t i cu la r impacts more than a b i t tedious. For th is reason the author includes an a l te rna t i ve method - one which gains in conceptual and compu-ta t iona l s imp l i c i t y what i t loses in ana ly t i ca l general i ty - wherein a s h i f t in the rural density of the system may be re la ted to a change in the employment (population) of any central p lace. The advantages of t h i s second approach become much c learer l a te r in the chapter when i t proves useful to B B B el iminate the sectors e ^ , e m 2»** *» e m m ^ r o m t n e argument. 104 To begin w i th , the "matrix" approach advocated in the second chapter seems espec ia l l y amenable for ind ica t ing the impact of sh i f t s B B in the components e . (1 < j < m) of the basic sector e of an m th mj - - m level place upon the to ta l employment (population) of that same p lace. Recal l ing (2.21) or (2.32), the reader sees that : e . = e .. + eB . + e B + + e B ml m2 mm mi mi A m (4.1) where: m A m - 1 - I k 1 (4.2) i = l This e f f ec t i ve l y div ides the tota l employment e ^ engaged in i th level a c t i v i t i e s in to i t s basic and non-basic por t ions. Assuming that ( i ) technology remains constant ( i . e . that there i s no var ia t ion in any of the service mu l t i p l i e r s ) and that ( i i ) a l l B T emi ' emi ^ - 1 - m ^ a r e d i f f e r e n t i a t e in a s u f f i c i e n t l y small neigh-borhood of the i r equi l ibr ium posi t ions as given in (4 .1) , then: 3e T . mi k. + A _j m m k. A 1 for i / j m for i=j (4.3) 3e T . The reader should note that — i n (4.3) represents, for any m th level center , the direct impact of a "smal l " exogenous change in employment 105 B T in the sector e m j . upon the total employment sector e mi However, i t must be stressed (perhaps the reader might review the second chapter) that any sector e , (2 < j < m) may be stated as a funct ion of e 0 and B ^emi I 1 5 1 < J } s o t n a t the total impact of a sh i f t in j th level a c t i v i t i e s becomes: 3e mi 3e l "mj 3e mi 3e m 3e B k Y m a a=j 3e . mj A. m m 3e B k T — ^ K i l . . B* a=j 3e A mj m for j < i fo r j > i (4.4) The discrepancy between the terms 3e' mi and 3e^ "mj 3e' mi 3e B* "mj stems from the recurs ive propert ies of domination which character ize the central place argument. The former term indicates the impact (exerted upon the body of a l l employees in an m th level place engaged in i th level a c t i v i t i e s ) of a s h i f t in employment in any of the j (1 < j < m) individual basic sectors del ineated by the central place topology (each of these i s i n d i -cated in equation (2.17) above). An a l te rna t i ve perspective i s to think of these sh i f t s as occurr ing in any one of the j complementary areas which the m th level center serves in the capacity of a 1st , 2 n d , " * , j t h , * * * , or m th level p lace. T 3e ' . j On the other hand, the term — ^ indicates the impact upon e^ mj of a s h i f t in all j th (1 < j < m) level basic sectors - not jus t the mi 106 s ing le sector which i s provided with the j th bundle - that are provided with the m th bundle by the m th level p lace. Natura l l y , most of these basic sectors (complementary areas) are provided with lower order goods and services by the various i th (1 < i < m) leve l places which are in turn m-dominated by the m th level p lace. Now a number of a t t r ibu tes of these impacts may be inferred from the two equations. I f , for instance, the serv ice mu l t i p l i e rs decl ine in a wel l -ordered fashion ( i . e . as in the usual case where ki > k 2 > • • • > k ), m then (4.3) suggests that : 9e ma 9e T mb 9e " m j 9e "mj 1 < a < b < m a,b f j (4.5) and: 9e' ma 9e mb 9e ma 9e mb 1 < a < b < m (4.6) while (4.4) suggests that : 9e 'ma 9e mb 9e l ' m j 9e B* "mj l < a < b < j < m (4.7) and: 9e m i ae' m i 9e B* "ma 9e B* mb 1 < a < b < m (4.8) with the pa r t i cu la r consequence that : 9e T 9e T. ma ^ mb > 3e m , 9e . ma mb 107 1 < a < b < m (4.9) Equations (4.7) and (4.8) are perhaps the most s i g n i f i c a n t and hence deserve some in terpreta t ion at the present time. The former states that , given the set of iden t i ca l basic sectors in which the exogenous s h i f t might occur and the condit ion that a < b < j , the impact would be greatest in the sector e ^ of a l l f i r s t level employment, second greatest in the sector e ^ of a l l second level employment and would progressively diminish for the subsequent j - 2 sectors of higher order a c t i v i t y . The l a t t e r equation complements th is in that i t s ta tes , given the endogenous sector in which the various exogenous s h i f t s might be expressed, the greatest impact would be induced by a change in f i r s t level a c t i v i t i e s , the second greatest by a change in second leve l a c t i v i t i e s , and that th is e f fec t would increas-ing ly decl ine for the M-2 higher order a c t i v i t i e s . In add i t i on , a f ter r e c a l l i n g (2.22) where tota l employment was shown to be: e m " e m l + e m 2 + < 4- , 0> I t fol lows (assuming now that e^ i s d i f f e r e n t i a t e in a s u f f i c i e n t l y small neighborhood of i t s equ i l ib r ium posi t ion) that: de T k! k 2 k. + A k m , " , , j m , , m T B - = 7 T + /T + + ^ A — + " A -de . m m m m mj m 108 where ^ - i s the export base mu l t i p l i e r fo r an m th level p lace. I t m i s nearly a truism to point out that : de 1 m de de 'mj deL 'n j j < m < n < M (4.12) On the other hand: T T de m 3e . m _ y mi B* ^ B* d e B . i=l 3e . mj mj (4.13) so that: de' m de' dec 'mj deL nj j < m < n < M (4.14) As implied above, however, the body of employment e^ can be resolved in to an expression that only involves the rura l employment densi ty eo ( i . e . the number of rural employees in each basic uni t a rea) , the set of nesting coe f f i c i en ts (K^  | 1 < i < m-l}, and the set of serv ice mu l t i p l i e r s {k^ | 1 < i < m}. This argument was f i r s t presented by Beckmann and McPherson (1970:27) in terms of the rural population densi ty ri although, i t should be added, those two authors apparently were not in terested in d iscerning the re l a t i ve e f fec ts of sh i f t s in d i f f e ren t parameters - more s p e c i f i c a l l y , sh i f t s in r x i t s e l f and in each of the mu l t i p l i e r s k. (1 < i < m) - upon the population p m of an m th leve l centra l p lace. F i r s t of a l l the reader might wish to return to the notat ional format out l ined in Section 2.2 of the second chapter. There the reader 109 should see that the tota l employment E m in an m th level market area ( i . e . the employment of the m th level place as wel l as that of the complementary area which i t m-dominates) i s : Em = e™ + i ( 4- 1 5 ) . m Reca l l ing next (2.22) makes i t apparent that for m > 2: e m A m " em-l Y l = i <4-16> Af ter def in ing D for m > 2 as: D = e' m m 'm-l (4.17) and u t i l i z i n g (4.15) and (4.16), i t fo l lows that : Dm • ^ (4.18) Now by observing that : mm = K 'mm-1 m-l k m-l V l - 1 'm-l (4.19) i t may be demonstrated that: no where, from (2.2) and (2.4) Ei e j ejj_ (4.21) But th is means that for m > 2: a m-l K. A. E = eo_ n i i m A> i=l A i + 1 (4.22) Therefore, a f te r redef in ing e as: 3 m m e' = e + n D. m . = 2 i (4.23) i t fo l lows from (4.18) and (4.19) that: m e 0 H k i + Mn. + k 3 K i K 2 Ai A XA 2 + A 2A 3 m-l k n K. m i=l 1 A , A m-l m (4.24) which proves to be an extremely useful statement in the upcoming discussion For present purposes, however, i t i s s u f f i c i e n t to note that: m-l de' m k n K. M 1=1 1 A m-l A (4.25) m where the as ter isk i s a simple reminder that th is change in rural employ-ment densi ty i s system-wide. The author now wishes to general ize some of the above statements fo r the case of population (as opposed to employment a lone) . To begin w i th , reca l l (2.24) where i t was stated that the total population p ^ i n of an m th level place which i s supported by i th (1 < i < m) level employees i s : " I I " " + d i ) e m i <4-26) Now assuming d i f f e r e n t i a b i l i t y of a l l p j . i t should become apparent that : 3p T . . 3e T . - J 1 - (1 < 4- 2 7> and that: 8e" . 1 3e l . mj mj 3p T . 3e T . m i = (1 + d.) m i B * \- u i ; B* 3e . 3e . mj mj 3e T . 3e T . where —pp- and — ^ have been derived above in (4.3) and (4.4) respec-3e D . 3e B . mj mj t i v e l y . Of course, statements s im i l a r to (4 .5 ) , (4 .6 ) , (4 .7 ) , (4.8) and (4.9) - only now for the e f fec t of an exogenous change in employment on the various population sectors - may be formulated by creat ing appropriate ordering cond i t ions . F i n a l l y , the reader should reca l l (2.24) where the argument demonstrated that: Pm = Pml + Pm2 + "* + PL <4-29) Then i t fo l l ows , assuming the d i f f e r e n t i a b i l i t y of p m , that : 112 dpm m 8e T l - H ' * ^ - f (4-30) and that: dp m 3e^. 7 T F = I (1 * d i ) ^ (4.31) which indicate that: dPm d P m n —g— < —g— for j < m < n < M (4.32) d e . d e . mj nj and that: dp dp —jpr < — ^ fo r j < m < n < M (4.33) de . de . mj. nj Unfortunately, however, (4.25) does not lend i t s e l f to a simple extension for the population case. The wary reader has perhaps noted that the s t a t i c format is useful in other ways for analysing the impacts of small exogenous changes in spec i f ied parameters upon other var iables of a more endogenous char-acter - es tab l ish ing some of these other p o s s i b i l i t i e s , in f a c t , comprises a good part of the remainder of t h i s chapter. But now in te res t turns to examining changes in central place populations over much greater periods of t ime. The fo l lowing discussion may be construed as both a c r i t i que and a rev is ion of the argument found in Nourse (1968:209-212). Consider, to begin w i th , the elementary case where a long run increment of population - say, for s i m p l i c i t y , that a l l rural (basic) 113 areas and centra l places grow by one percent - occurs in a central place system but there i s no tendency for any new centers to emerge. In other words, due to the r e l a t i v e l y small increase in populat ion, the system would show no i nc l i na t i on to e i ther exh ib i t areal expansion or fur ther h ie rarch ia l d i v i s i o n (which, of course, would be a rea l l y in ternal to the system). Quite obviously , then, consumer purchasing power must increase wi th in the market area of each central place in the system. This in turn ind icates that there would be a tendency for a s h i f t to occur - assuming, of course, fac tor mobi l i ty - i n the system's hierarchial marginal goods and services (to use a phrase coined by Berry and Garrison (1958) which re fers to the upper endpoints of a l l bundles in the system except the M th leve l one) due to a type of import subs t i tu t i on . Put d i f f e r e n t l y , a surplus of purchasing power would be created in the market areas of a l l centers on the m th (1 < m < M) level by th is population increase and that surplus might well be s u f f i c i e n t - here i t i s taken that a one percent increase is s u f f i c i e n t - to induce a c t i v i t i e s to migrate away from centers on the (m+l)st, (m+2)nd, "« , and M th leve ls in to those centers on the next lower l e v e l s . This sl ippage of a c t i v i t i e s may be demonstrated by using the t rad i t iona l cost and demand methodology of spaceless economics or by employing the threshold conception of C h r i s t a l l e r i a n theory: fo r a l uc i d synthesis the reader should re fer to Parr and Denike (1970). Unfortunately, outside of s ta t ing that k i increases and that k^ decreases in the long run, at th is time i t i s only possible to speculate about the nature of change in the other mu l t i p l i e r s of the set {k m |m=l,2, ' * ' ,M} cha rac te r i s t i c of the general h ie rarch ia l model. 114 Theoret ical improvements, focussed on the composition of each m u l t i p l i e r , are sorely needed before even qua l i t a t i ve changes can be spec i f ied for those remaining M-2 m u l t i p l i e r s . One perspective that appears useful for deal ing with such re -a l l oca t i on is to i so la te the changes, between the i n i t i a l and subsequent states of equ i l ib r ium, in the numbers of persons engaged in r u r a l , f i r s t order , second o r d e r , * " , and M th order a c t i v i t i e s respect ive ly throughout the ent i re system. The resu l tant changes in the central place populations fo l low accord ingly . These activity populations and re lated coe f f i c ien ts (which, s i g n i f i c a n t l y , are ident ica l to the service mu l t i p l i e r s ) express-ing those populations as ra t ios of the tota l systemic population may be computed as shown in Appendix E. Consider the fo l lowing hypothetical case designed to i l l u s t r a t e the above remarks. Suppose that a four level K=3 system ex is ts at time to with r-i = 2000 and {km [ t 0 ]> = {.3333, .1667, .1250, .0937}. I f the population of th i s system expands by one percent up to a time t i but there i s no red i s t r i bu t i on occurr ing in the time in terva l between to and t i , then r i would increase to 2020 where {k [ t i ] } = {k [ t 0 ] } . m m Eventual ly , however, the sl ippage of goods and services would take place as argued above so that the system might be character ized by a new set {km [ t 2 ] } of mu l t i p l i e rs at the s t i l l l a t e r time t 2 . From the second chapter i t should be apparent that a property of th is growth-real !ocat ion scheme would be that l km C t . ] - I km [ t l ] - I km [ t , ] (4.34) m=l m=l m=l 115 I f i t i s assumed that only k i and ki+ change during the red i s t r i bu t i on then {km [ t 2 ] } = {.3500, .1667, .1250, .0770} would represent a possible state of the h ie ra rch ia l system at time t 2 . The impl icat ions of th is pa r t i cu la r change, espec ia l l y as i t a f f ec t s : ( i ) the numbers of persons r e s i d i n g in centers of d i f f e r e n t s i z e s ; 9 and ( i i ) the numbers of persons engaged in s p e c i f i c a c t i v i t i e s ; are i l l u s t r a t e d in Tables 4.1 and 4.2 respec t i ve ly . The reader should note that in order to f a c i l i t a t e computations the t rad i t i ona l notat ion (with no account given of d i f f e r e n t i a l family s i ze ) of the l i t e ra tu re i s u t i l i z e d . I t i s in te res t ing to observe that , at leas t under these specia l r e s t r i c t i o n s , red i s t r i bu t i on appears to be a funct ion of c i t y s i z e : the percentage coe f f i c ien ts for the f i r s t , second, t h i r d , and fourth order places are ( looking at column 6 in Table 4.1 and column 8 in Table 4.2 as wel l as neglect ing rounding e r ro r s ) : 0 .73, 0.81 (0.60 + 0.22) , 1.31 (0.77 + 0.24 + 0.29) , and -2.85 (-0.43 - 0.46 - 0.29 - 1.67) respec t i ve ly . The intermediate to ta ls 1.67, 0.00, 0.00, and -1.67 in column 8 of Table 4.2 represent the changes in urban a c t i v i t y ra t ios and c l ea r l y r e f l e c t , in percentage form, the hypothesized changes in the o r ig ina l set £ . , . . , . 10 of serv ice m u l t i p l i e r s . The author must caut ion, however, that these stated propert ies of incremental growth and red is t r i bu t i on must in no way be taken as gen-e r a l i z a t i o n s : they simply resu l t from the specific assumptive framework. On the other hand, a technique has been created which al lows comparisons 116 Table 4.1 A Proposal of a One Percent Growth and Redis t r ibut ion Scheme in a K=3 Four Level Central .1250, . Place System; {k m 0937}; {km [ t 2 ] } = [to]} = {.3500, (k m [ t .J } = {. .1667, .1250, 3333, .1667, .0770} (1) (2) (3) (4) (5) (6) Time Population Population Sector Per Unit No. of Units Sectoral Population % Total Population FT 2000 27 54000 28.13 PT 1000 18 18000 9.37 to pT 4000 6 24000 12.50 pT 16000 2 32000 16.67 pT 64000 1 64000 33.33 192000 100.00 rT 2020 27 54540 28.13 pT 1010 18 18180 9.37 t i PT PT 4040 16160 6 2 24240 32320 12.50 16.67 pV 64640 1 64640 33.33 193920 100.00 rT 2020 27 54540 28.13 PT 1088 18 19584 10.10 t 2 PT 4304 6 25824 13.31 pT 17430 2 34860 17.98 pV 59112 1 59112 30.48 193920 100.00 Table 4.2 An Analysis by Ac t i v i t y Sectors of the Proposed One Percent Population Growth and Redis t r ibut ion Scheme; (k m [ t 0 ] } = {k [ t x ] } = {.3333, .1667, .1250, .0937}; {k [ t 2 ] } = {.3500, .1667, .1250, .0770} (1) (2) (3) (4) (5) (6) (7) (8) A c t i v i t y Sector Population Sector Time to Time t i % Total to , t l Time t 2 % Total t 2 - - ' * % Total t 2 -% Total t i r 1 "FT 54000 54540 28.13 54540 28.13 0.00 pT 18000 18180 9.37 19584 10.10 0.73 pT 12000 12120 6.25 13284 6.85 0.60 pT 12000 12120 6.25 13616 7.02 0.77 p i 22000 64000 22220 64640 11.46 33.33 21397 67881 11.03 35.00 -0.43 1.67 P 1 p i 12000 12120 6.25 12540 6.47 0.22 p i 8000 8080 4.17 8556 4.41 0.24 P 2 p i 12000 32000 12120 32320 6.25 16.67 11228 32324 5.79 16.67 -0.46 0.00 p i 12000 12120 6.25 12688 6.54 0.29 P 3 p i 12000 24000 12120 24240 6.25 12.50 11554 24242 5.96 12.50 -0.29 0.00 P* p i 18000 18180 9.37 . 14933 7.70 -1.67 TOTALS 192000 193920 100.00 193920 100.00 0.00 118 of rea l loca t ions to be made amongst diverse central place systems with d i f fe ren t m u l t i p l i e r s , growth increments, topologies, and the l i k e . This perspect ive, rest ing on the twin supposit ions that ( i ) systemic population would expand incremental ly and then s t a b i l i z e and that ( i i ) no new centers would emerge because th is expansion was so s l i g h t , must be considerably modified i f population growth i s taken to pe rs i s t i n the (very) long run. Theory suggests that the extension of i n t e r s t i t i a l purchasing power would, at some time, induce the emergence of new f i r s t order centers . Concomitantly, centers previously on the m t h , (m+l)st, (m+2)nd,*" , and (M- l )s t leve ls would be expected to acquire (m+l)st, (m+2)nd, (m+3) rd , ' " , and M th order a c t i v i t i e s respect ive ly as w e l l . Unfortunately, ex i s t i ng theory provides no ra t iona le in suggest-ing the nature of a threshold for such emergence - i t appears, once again, that considerable at tent ion must be di rected in the future to the a t t r ibu tes of the ind iv idual commodity bundles before even a tenuous so lut ion to the problem of emergence may be forwarded. General ly , however, i t seems reasonable to ant ic ipa te that extensive population growth would, ce te r i s par ibus, induce fur ther systemic d i v i s i o n according to the i n i t i a l organizat ional (geometrical) p r inc ip les of the central place s t ructure. To take an example, in a M level K=3 system, a two-fold increase in population over a lengthy period of time would lead to the a r t i cu l a t i on of three i den t i ca l M leve l systems wi th in the area o r i g i n a l l y housing the s ing le M level system. The i n i t i a l center on the M th level would serve ( i . e . M-dominate) one of these "new" systems and th is system, in tu rn , would be enveloped by s i x one-th i rd port ions of two "new" ident ica l (equivalent) systems. These l a t t e r two 119 equivalent systems would each be M-dominated by an equivalent center which was previously of the (M- l )s t order ( in the o r ig ina l system) before the long run population growth commenced. This systemic rest ructur ing would take place in a l l s ize c lasses so that i t might be s a i d , in summary, that there occurs a one bundle import subst i tu t ion in a l l ex i s t i ng centers except the o r ig ina l M th level p lace. This in terpreta t ion d i f f e r s s i g n i f i c a n t l y from that suggested by Nourse: he would argue that the o r ig ina l M th order place simply becomes an (M+l)st order place in a more complex s ing le system which i s confined to the o r ig ina l a r e a . ^ While th is appears perhaps more consonant with the real world experiences of economic growth in a reg ion, i t i s not -in th is author 's opinion - a va l i d ce te r i s paribus argument (and th is i s what Nourse desires) for the impact of population growth alone. For such h ie ra rch ia l extension to become evident , i t would be necessary to have new economic a c t i v i t i e s (spec ia l ized goods and services which could only be supported by the overa l l s h i f t in aggregate demand) accompany the growth in populat ion. On the other hand, even by "assuming away" the dilemma of an emergence threshold - suppose, for instance, that an x 0 - f o l d increase in systemic population is s u f f i c i e n t for a l l "new" centers to appear - i t s t i l l does not remain a simple task to speculate about the nature of systemic res t ruc tur ing due to an x - fo ld (xo < x < K - l , where K i s the nesting factor) increase in the long run. However, i f i t i s fur ther supposed that , a f ter such emergence, growth and red is t r i bu t ion occur so that ( i ) the o r ig ina l set of mu l t i p l i e r s i s retained and ( i i ) the population sectors a l l expand at the same ra te , 12 then i t may be demonstrated that : 120 ( i ) in each of the K "new" systems, the population of each u n i t i n the s e c t o r s r i , p i , p 2 , * " ,p^ | x+1 is -Tr- times the population of the same u n i t K before the population increase and concomitant h i e r a r c h i a l d i v i s i o n occur (note, of course, t h a t these same u n i t s do not occupy the same l o c a t i o n s as b e f o r e ) ; ( i i ) the K-l e q u i v a l e n t centers which acquire a c t i v i t i e s p r e v i o u s l y found in only the M th level place have populations consonant with the p r o p e r t i e s of the "new" e q u i v a l e n t systems which they p a r t i a l l y M-dominate; and ( i i i ) the o r i g i n a l M th le v e l place r e t a i n s i t s popula-t i o n but transforms i t s a c t i v i t y s e c t o r s in a manner whereby a c o n t i n u a l l y d i m i n i s h i n g " s u r p l u s " population - th a t i s , the amount over and above the populations of the other K-l emerging M th level places - provides the remaining goods and s e r v i c e s to the K "new" systems. What i s important to observe here is that the commodity set i s being con-t i nua l l y redef ined, although always with a tota l of M+l bundles, unt i l the time when x+1 = K and that during th is r ede f i n i t i on : 'PM * * kM + l = " ' ^ m ' J r (4.35) where p M and P M are the populations of the o r ig ina l M th level place and system respect ive ly . Note also that the mu l t i p l i e r -> 0 as x -* K - l , which simply indicates that with a ( K - l ) - f o l d increase in systemic populat ion, K "new" and ident ica l systems ex is t as was s t ipu la ted above. I t should be apparent that , with the rede f in i t i on of the com-modity set over t ime, th is argument has a s i m i l a r i t y to the proposal 121 out l ined by Nourse; however, by re ta in ing a s t r i c t e r conception of centra l place p r inc ip les (as well as having at hand a more powerful model), the present author fee ls he i s g iv ing a more accurate depict ion of the imp l i -cat ions of long run ce te r i s paribus population growth within a central place system. Quite obviously , the above formulation i s an i dea l i sa t i on but i t does represent an elementary ana ly t i ca l model of systemic transforma-t i o n . In add i t i on , i t could be ref ined (to accommodate sh i f t i ng m u l t i p l i e r s , var iab le topologies, e t c . ) but the present format seems en t i r e l y s u f f i c i e n t fo r the time being. Population decl ine may be examined in a symmetric fash ion. Depending on the postulated nature of such dec l ine , varying amounts of a c t i v i t y loss would be expected for the system as a whole and the threshold for a c t i v i t i e s would ce r ta in l y migrate up through the hierarchy ( i . e . an m th (1 < m < M) leve l place would tend to lose a c t i v i t i e s to places on the (m+l)st level and perhaps even higher leve ls i f the population decl ine pe rs i s ted ) . ( i i ) Process Gravi ty - potent ia l theory suggests that , under cer ta in circum-stances, population increases throughout a central place system would tend to accelerate a d i f fus ionary process which had already commenced. For many items (e .g . most consumer goods) the e f fec ts of t h i s growth might be considered neg l ig ib le as termination of primary adoption would be expected to occur somewhat qu ick ly ; nevertheless, for some items (frequently those 122 invo lv ing more cap i ta l r i s k : new productive techniques, in f ras t ruc ture improvements, e tc . ) d i f fus ion would l i k e l y proceed in a moderate fashion and concomitant systemic growth would doubtless have an inf luence on the nature of spat ia l adoption. To begin w i th , suppose a four leve l K=3 system ex is ts as in the previous sect ion where {pm> = {1000, 4000, 16000, 64000}. Suppose, for convenience, that ( i ) a l l central places grow at the same ra te , that ( i i ) th is rate may be standardized according to the threshold time of f i r s t adoption (see the previous chapter) , and that ( i i i ) e f fec t i ve potent ia l i s enhanced only a f te r an increment has been added to the t e l l i n g and hearing centers . These assumptions are simply useful fo r conceptual iz ing continuous growth in discrete terms and they could be modified according to the wishes of the researcher. Consider the fourth level center. I f , for instance, the adoptive threshold time i s t and growth i s one percent per uni t of t , then the po ten t i a l l y e f fec t i ve population of that center at each of the t imes: 0, t , 2 t , 3 t , " ' , nt i s , respect ive ly : 64000, 64640, 65286, 65939, (64000)(1.01 ) n . With growth conceptualized as such, the modell ing scheme of the previous chapter becomes amenable to speci fy ing divergences that would be expected between d i f fus ion which occurs in a s t a t i c mi l ieu and d i f fus ion which occurs in a more dynamic m i l i e u . Tables 4.3 and 4.4 i l l u s t r a t e such di f ferences for two growth rates in open and closed systems. I t should be apparent to the reader that population growth d i f f e r e n t i a l l y contracts adoptive times depending upon the nature of systemic c losure : ear ly adoptive times in the open case but l a te r adoptive times in the closed case seem espec ia l l y in f luenced. In terest ing ly enough, 123 Table 4.3 The Impact of Population Growth on Spat ia l Adoption in an Open K=3 Four Level Central Place System; {pm> = {1000, 4000, 16000, 64000}, b = 2 Adopting No 1% 2% Center Growth Growth Growth 4 0.00 0.00 0.00 3 1.00 1.00 1.00 2 3.00 2.94 2.87 1 1 5.07 4.88 4.70 l 3 9.02 8.37 7.89 1 2 9.53 8.81 8.37 124 Table 4.4 The Impact of Population Growth on Spat ia l Adoption in a Closed K=3 Four Level Central Place System; {pm> = {1000, 4000, 16000, 64000}, b = 2 Adopting No 1% 2% Center Growth Growth Growth 4 0.00 0.00 0.00 3 1.00 1.00 1.00 2 1.30 1.29 1.28 1 1 1.91 1.89 1.87 1 2 3.53 3.46 3.36 1 3 10.00 9.21 8.57 125 in both instances the adoptive orderings are retained from the s t a t i c argument of the previous chapter. I t should be a truism that ( i ) s im i la r growth in a d i f fe ren t centra l place system, or ( i i ) d i s s im i l a r growth in the same system, o r , f i n a l l y , ( i i i ) d i s s i m i l a r growth in a d i f f e ren t system each represents a var ie ty of circumstances in which adoptive divergences could be remarkably unl ike those indicated in Tables 4.3 and 4 .4 . But th is is a whole new Pandora's box - one, unfortunately, which must be opened i f proper empir ical q u a l i f i c a t i o n of the hear ing - te l l i ng model i s to be undertaken - and i t was decided by the author not to extend the discussion at the present time. 4.3 Technology The impact of a technological change - upon e i ther the a t t r ibu tes of central place structure or process - would be in t imate ly t ied to the very nature of that change. Perhaps two r e l a t i v e l y simple examples, exh ib i t -ing how long run sh i f t s in the general state of technology may have con-t rad ic tory e f fec ts on the structure of the space-economy, can help e luc idate th is point . F i r s t l y , a substant ia l ce te r i s paribus improvement in agr i cu l tu ra l product iv i ty would be expected to induce a large number of rural house-holds - and, i n d i r e c t l y , many households in the lower order central places as well - to migrate into the very largest central places in the system. On the other hand, the growth of the more intermediate s ized places might be d ispropor t ionate ly enhanced by improvements in communications tech-nology which would allow very spec ia l i zed services ( for instance, s c i e n t i f i c 126 research or higher education) to " f i l t e r down" from the very la rgest centers ( th is i s a type of urban decen t ra l i za t ion ) . This sect ion opens by considering the e f fec ts of instantaneous changes in the service mu l t i p l i e r s upon the equi l ibr ium states of the 14 endogenous employment and population sectors . Spec i f i c at tent ion i s given to the issue of breaking down these mu l t i p l i e r s in to t he i r ind iv idua l components so that i t becomes possible to d is t ingu ish between the impact of: ( i ) a s h i f t in the aggregate m u l t i p l i e r i t s e l f (which might be induced by the a d d i t i o n of a new. a c t i v i t y to an e x i s t i n g bundle ( p o s i t i v e s h i f t ) or by an improvement in the productive e f f i c i e n c y of the given a c t i v i t i e s i n t h a t same bundle (negative s h i f t ) ) ; and ( i i ) a s h i f t i n a p a r t i c u l a r component ( i . e . an i n d i v i d u a l a c t i v i t y ) of t h a t m u l t i p l i e r (negati ve sh i f t ) . This d i s t i n c t i o n i s useful because i t leads to the e x p l i c i t ana lys is -a l b e i t in a somewhat naive fashion - of supply and demand in the central place argument. Subsequently the longer run p o s s i b i l i t i e s of s t ruc tura l var ia t ion are considered. The above cases are mentioned again in th is new context and addi t ional concern i s d i rected to increases in agr i cu l tu ra l production and transportat ion improvements - both of which would be expected to accompany technological progress in a space-economy. F i n a l l y , the discussion b r i e f l y turns to analyse the s ign i f i cance of these induced s t ructura l changes in d is turb ing or a l t e r i ng adoptive schemes in the central place s e t t i n g . 127 ( i ) Structure The impact of a s h i f t in any element of the set {k. | 1 < j < m} i s f i r s t reso lved. This s h i f t should be construed as being aggregate in nature, since i t could be induced by a s h i f t in any of the s p e c i f i c a c t i v i t i e s which comprise the j th bundle. There are two approaches to th is problem. The f i r s t and more encompassing perspective i s to consider "shocking", the employment sectors emi ( I f i f m ) in (4.1) by any serv ice m u l t i p l i e r k.. The formulations which describe th is e f fec t are r e l a t i v e l y complex but they do, conveniently enough, col lapse somewhat when the overa l l e f fec t on e^ i s taken up. The a l te rna t ive approach i s to u t i l i z e (4.24) immediately although now, of course, only the overa l l e f fec t on e^ may be ascer ta ined. Since the arguments are presented in the usual m th level format, the reader may wish to re fer to Appendix F for a comparison of the two methods in the context of a second level central p lace. Turning now to (4.1) and assuming d i f f e r e n t i a b i l i t y of each of the mu l t i p l i e rs k. (1 5 j 5 m) and each of the employment sectors B T e • , e . (1 5 i < m), i t fol lows that : 128 3e mi 3k. J , 9 e B -(A ) m i + A * V 3k.. m J . B f m de i u V ma K i l . 3k. ^ a=j -V m R + k- I e B k  1 b-l m b (A ) 0 1 1 + A [m> 3k. Am .1 * V for f m 3e^ m D 1 m n for (4. . B f m oe^ m D k y m a + k y e B m K i I. ak.J K i . i emb A for j > i which means, r eca l l i ng (4.10) and assuming d i f f e r e n t i a b i l i t y of e T , that : m m de T I e° + A m I L - mi m L. m 3e B 'ma m i = l dk. J m 3k. ( A J ' m eT + y -ma-ni ^ . 3k. a=J J A m for 1 < j < m (4.37) This i nd i ca tes , for instance, that: 15 de T de T m < n dk. dk. J J j < m < n < M (4.35 In add i t i on , combining (4.26) and (4.29) with the assumption that each centra l place population p m i s d i f f e r e n t i a t e , suggests that : 129 dp m m 3e T . H k T - I ( 1 + d0.wr J '=1 j (4.39) so that: dp dp dk. dk. J J j < m < n .< M (4.40) On the other hand, d i rec t d i f f e ren t i a t i on of (4.24) makes the impact of a s h i f t in ki upon e T : r m de T dk, e 0 1 + k 2 K 1 (A 1 +A 2 ) + k 3 K 1 K 2 (A 2 +A, ) + (Ax) 2 ( A x ) 2 ( A 2 ) 2 ( A 2 ) 2 ( A 3 ) 2 m-l km X K i l A ~ ' + A -I = I ^ m-l "m ( Am-l) 2 ( V ! (4.41) The same operation may be performed for the remaining m-l mu l t i p l i e r s in order to reach the general resu l t that : de T 'm dki = e 0 i i . ? M V i + V ^ { ( A J * i=2 ( A . ^ ) 2 ( A . ) 2 c=l K° J de T 'm dk. 6Ji dk = e 0 -4 j-l n K c-l C m k (A + A ) i-1 1 J — i _ i !_ N K 2 / . x 2 , C [ (A d ) • ( A . , , ) " ( A ^ c=l (4.42) e 0 1 m-l n K c=l c v nr for 2 < j < m-l 130 This ind ica tes , for instance, that: a-1 n K c = i J ' C ( A b ) 2 i=a+l ( A ^ ) ( A . ) ' c=l de T ^ de T m > m dk, < dk. a b for 2 < a < b < m (4.43) As in the previous sec t ion , d i f f e ren t i a t i on of (4.24) does not provide a statement analagous to (4.39) except, of course, for the t r i v i a l case when d = constant = d^ (1 < i < m). Suppose now that each of the mu l t i p l i e r s k^  i s resolvable into a set of smaller m u l t i p l i e r s : these l a t t e r coe f f i c ien ts would represent the ind iv idual goods and services which make up the j th bundle. In par-t i c u l a r , consider the l i near case where: k i = k n + k i 2 + • • • + k k 2 = k 2 i + k 2 2 + k. = k., + k . 0 + . . J J l j2 + k Izi 2z 2 + k jz. k = k 1 + k 0 + ' « « + k m ml m2 mz (4.44) m If the reader wishes to think of these mu l t i p l i e rs as being ordered so as to r e f l e c t increasing threshold requirements ( i . e . the threshold requirement 131 for funct ion f . , with associated m u l t i p l i e r k. , i s less than that fo r Ja j a funct ion f \ b i f a < b) , then the set {k j z | 1 < j < m} would repre-sent the h ie rarch ia l marginal goods in the system. I t should immediately fo l l ow , because of the l i nea r re la t ionsh ips hypothesized in (4.44) , that : 3 e mi 3 e mi 1 < 1, j < m W~=WT , (4-45) jn j 1 < n < z . and that: d e I d e I 1 < j < m jn d k j 1 < n < z i f d i f f e r e n t i a b i l i t y of k^n for 1 < j < m and 1 < n < z^. i s assumed. (4.44) becomes espec ia l l y use fu l , however, when a d i s t i n c t i o n can be drawn between the demand per employee (household) for each ind iv idua l funct ion and the various outputs per employee (household) which character ize production of the d i f fe ren t ind iv idual goods and se rv i ces . I t i s conven-ien t , then, to define the mu l t i p l i e r k^n as : k i n = ^ (4.47) j n S j n where: ( i ) x > n i s the demand per employee (household) of f u n c t i o n f ^ ; i m p l i c i t t o t h i s p a r t i c u l a r argument i s t h a t x. n i s i d e n t i c a l f o r a l l employees (households) in the system; and ( i i ) s^.n i s the output per employee of those employees who are engaged in p r o v i d i n g f . 132 The imp l ica t ion of (4 .47) , a f te r assuming that x^ n and are both d i f f e ren t i ab le and that demand i s held constant ( i . e dx^ n = 0 ) , i s then d k i n ' " . in . - k, jn d s j n ( s . n ) 2 s j n so that in the aggregate case (see (4 .37)) , for instance: de T de T dk. _m _ m jn (4. ds . dk- ds . jn jn jn . B K • ( T m e J H _ l e T + y m < m L . k s . n A jn m (4. a=j JIV This l as t statement ind ica tes , for an m th level center , the extent of the rate of decl ine in to ta l employment e\ which would be prec ip i ta ted m by a "smal l " ce te r i s paribus improvement in the productive e f f i c i ency of funct ion f^. This formulation has only partial mer i t , however, since i t does not account for any subst i tu t ions in demand amongst the various goods - more sophist icated approaches in the future w i l l doubtless el iminate th is problem though. The author fee ls that the impact of rural p roduct iv i ty changes could be usefu l l y analysed wi th in a s im i l a r framework. In th is case i t would simply become necessary to d is t ingu ish ind iv idual mu l t i p l i e rs - one for each of the various agr i cu l tu ra l a c t i v i t i e s in the system - which, together, would comprise the rural serv ice mu l t i p l i e r k 0 (see the second chapter for some comments here). In c los ing the comparative s t a t i c s portion of th is sect ion i t should be pointed out that a "smal l " change in the ex is t i ng t ransportat ion 133 technology might be construed much as an income subsidy to the consumer and, therefore, analys is of that case is included in the upcoming sect ion which i s concerned with sh i f t s in per capi ta income. I t i s somewhat speculat ive to consider these technological changes in a ce te r i s paribus fashion over the long run. The author re f ra ins from making extremely spec i f i c statements, in a cause and e f fec t context , about the in te r re la t ionsh ips of technological changes ( inc luding t echn i ca l , marketing, and organizat ional innovations) and var ia t ions in the central place structure because: ( i ) most important innovations tend to c l u s t e r together in space and time ( i . e . they are r a r e l y independent phenomena); and ( i i ) these innovations tend t o be so enmeshed in the whole process of regional economic growth t h a t i t i s d i f f i c u l t to d i s c e r n what material improvements are a t t r i b u t a b l e t o a t e c h n o l o g i c a l change per se. However, i t may not be too hazardous to comment in a very general fashion on how systemic a t t r ibu tes might vary with cer ta in technological changes. Hopeful ly , the b r ie f upcoming discussion seems a b i t more p laus ib le than the argument given by Nourse (1968:215-218). Urban product iv i ty may be enhanced by new marketing schemes, cap i t a l - i n tens i ve techniques, agglomeration economies, and the l i k e . In any case, as Parr and Denike (1970) have i l l u s t r a t e d to some length, the cumulative e f fec t of such changes in technology i s a reinforcement of funct ional cen t ra l i za t i on in the intermediate- to- large s ized communities in central place systems. In other words, the a b i l i t y of large firms to employ economies of sca le , new adver t is ing schemes, e tc . leads, in the 134 most par t , to a t ransfer of goods and serv ices from lower to higher l eve ls of the h ierarchy. Rural p roduc t i v i t y , as pointed out near the beginning of th i s very sec t i on , may be enhanced by increased farm mechanization, new fe r -t i l i z e r s , i n s t i t u t i o n a l reform, and the l i k e . Ceter is par ibus, the improve-ment of ag r i cu l tu ra l output per uni t of labour input would be expected to induce a stream of rural - to-urban migrat ion. As a consequence, the coe f f i c i en t R° = A^ (see Appendix E ) , representing the ra t io of rural to tota l employment (populat ion), would tend to decl ine in the long run. A s i g n i f i c a n t t ransportat ion improvement - say, perhaps a new mode of t ravel or an upgrading, of some ex i s t i ng mode(s) - which i s system-wide in extent would al low benef i ts to accrue to households s ince : ( i ) a g r i c u l t u r a l products could be moved in a more e f f i c i e n t manner to c e n t r a l places (Beckmann (1968:77-83) gives a b r i e f but i n t e r e s t i n g account of such f l o w s ) ; and ( i i ) d e l i v e r e d p r i c e s of f i n i s h e d goods ( i n c l u d i n g processed a g r i c u l t u r a l commodities) would be reduced as we I I J ^ I t fo l lows , then, that a t ransportat ion improvement would be conducive to funct ional c e n t r a l i z a t i o n : that i s , a strengthening of the re l a t i ve importance of the larger centers in the system. F i r s t of a l l , the enhancement of customer mobi l i ty would permit households to trade o f f some of the i r shorter and more frequent t r i ps (to acquire convenience goods) for more multi-purpose t r i ps (e .g . consider the symptomatic decl ine of rura l serv ice centers on the Canadian P ra i r i es and in the American Midwest). Secondly, these same customers would enjoy an increase in real income (what Long (1971) has termed the transport income e f f e c t ; see Appendix G) 135 which would favor the purchasing of income-elast ic goods and se rv i ces . Since many low order commodities are of the income- inelast ic type - fo r ins tance, food, c l o t h i n g , and personal serv ices - i t seems reasonable to general ize that th is transport income e f fec t would st imulate addi t ional growth of the la rger places in a central place system. In general , then, technological improvements are c l e a r l y conducive to a progressive increase in the urban/rural population ra t io of the space-economy. More s i g n i f i c a n t l y , though, they tend to promote growth i n only the intermediate-sized and larger places in the system. In terms of the h ie ra rch ia l model, the higher order serv ice mu l t i p l i e r s {•••, k. ,»»*,k..} 3 ** would tend to increase at the expense of the lower order mu l t i p l i e r s { k i , # , , , k . ,} ( r eca l l that the serv ice mu l t i p l i e r s represent the a c t i v i t y ra t ios for the various goods and se rv i ces ) . ( i i ) Process Systemic adoption could seemingly be modified in d iverse ways by technological change. To begin w i th , a ce te r i s paribus t ransportat ion improvement could have a d e b i l i t a t i n g e f fec t on the " f r i c t i o n of d istance" ( i . e . reduce the distance exponent in the grav i ty -potent ia l formula) so that the comments in ( 3 . 4 : i i i ) above might be worth reviewing. On the other hand, th is improvement might be construed as being a surplus of income for central place residents so that the population terms in the potent ia l funct ion would have to be weighted accordingly (see Isard (1960: 507) for comments on th is weighting procedure). Unfortunately, at the present t ime, theory does not permit any estimations of the r e l a t i ve s ign i f i cance of these two e f f e c t s ; hopefu l ly , though, the considerable ta lents being di rected to the understanding of the demand fo r t ravel w i l l 136 soon o f fe r a so lut ion to th is problem (at least in the somewhat s t r i c t centra l place context ) . In add i t i on , there are a few other points - perhaps of a more speculat ive nature again - which should be touched upon. The s t ructura l changes in the space-economy (mentioned above) would be expected, ce te r i s par ibus, to st imulate in teract ion amongst the larger centers of the system at the expense of diminished in teract ion between those larger places and the various low order centers which they dominate. The consequences of t h i s would be that : ( i ) adoption ( i n a r e l a t i v e sense) would tend to be temporally r e s t r i c t e d to even a g r e a t e r extent in the d i s t a n t low level places in the system; and ( i i ) adoption might not even occur (due to a loss of population) in many of the low l e v e l places i n the system. 4.4 Per Capita Income Before examining the s ingular impact of a change in per capi ta income on the employment (population) and adoptive a t t r ibu tes of central p laces, a comment on the nature of personal income i s required. The par t ia l perspective of the central place argument has t r a d i t i o n a l l y allowed theor is ts to avoid making e x p l i c i t statements about the spat ia l d i s t r i bu t i on of income in the relevant systems: th is being a p rac t i ce , i n c i d e n t a l l y , which re f l ec t s the fac t that l i t t l e at tent ion has been devoted to the balance of trade (payments) in e i ther the C h r i s t a l l e r i a n or Loschian 1 g formulat ions. Parr (1970:228) has pointed out that an i m p l i c i t postulate of the h ie rarch ia l model i s : 137 . . . t h a t patterns of consumption w i l l be i d e n t i c a l between the r u r a l areas and the d i f f e r e n t l e v e l s of ce n t e r . In f a c t , Parr might have general ized th is statement even fur ther . I t can be argued that the relevant postulate i s that a l l house-holds in the system would earn the same' "apparent" real income - in other words, that each household would consume an ident ica l amount of each 19 commodity bundle. Th is , of course, i s the same assumption which was u t i l i z e d in the previous sect ion (on technology) of th is chapter. For present purposes i t i s not necessary to re la te real income and money income for a l l central place locat ions - though that might prove a useful exercise - but in Appendix 6 the author demonstrates those con-d i t i ons which are necessary and s u f f i c i e n t for real income to remain constant in the neighborhood of any given l oca t i on . The assumption of constant real income i s appropriate simply because e x p l i c i t supply and demand condit ions (which would ensure the spat ia l equ i l ib r ium of a l l households) for the central place scheme have not as yet been spec i f ied - in th is absence, having real income constant i s at least s u f f i c i e n t for re ta in ing the s t r i c t h ie rarch ia l propert ies which character ize that body of theory. ( i ) Structure To begin w i th , reca l l (4.47) and postulate that , with "sma l l " changes in the mu l t i p l i e r k. and household demand x . , there is no change in output s . ( i . e . d s i n = 0) . Then i t fo l lows that : 138 d k i n 1 k i n 1 - J - m jn jn x j n 1 < n < x. J Given any m th (1 < m < M) leve l place st ipuate next that the ( rea l ) per capi ta income of a l l households in that place i s Y. Now i f demand x . is taken to be a continuously d i f f e r e n t i a t e funct ion of income Y, dk. then the re l a t i ve e f fec t of a change in the mu l t i p l i e r k.n due to a sh i f t in every household's income Y, becomes: dk. dk. dx. jn = jn ,m dY dx. dY d e T More s i g n i f i c a n t l y , though, i t should be pointed out that the impact -jjy- of a s h i f t in income upon the tota l employment e^ of an m th level place i s (from (4.37) and (4.46)) : (4.51) k. dx. = j n jn x . dY jn In pa r t i cu la r note that th is en ta i l s that : Y d k j n Y d x . n e k . ,Y = k. dY = x . dY = e x . ,Y ( 4 ' 5 2 ) j n ' jn jn j n ' where e. v and e v a re , respec t i ve ly , the e l a s t i c i t i e s of the j n ' Y x j n ' Y mul t i p l i e r k^n and household demand X j n with respect to income Y. 139 m dY (4 This asserts (again, in a par t ia l sense only , because there i s no subs t i tu -t ion amongst productive inputs) jus t how great the rate of increase (decl ine) in the employment body would be, given an income-induced change in demand for the income-elast ic ( income- inelast ic) good or func-t ion f ^ . One property of t h i s - i n c rease (decl ine) to note i s i t s depen-dence, in an inverse manner, on the production coe f f i c i en t s.n (where income. This increase must r e f l e c t an outward s h i f t in the space-economy's p roduc t ion -poss ib i l i t y curve - a s h i f t which may be brought about by a var ie ty of f ac to rs , act ing alone or in combination with one another. Improvements in technology and organizat ion, better s k i l l s in labour, a p r o c l i v i t y on the part of f irms to subst i tu te cap i ta l for other inputs , population growth, and the emergence of external economies would be numbered amongst the most important of such fac to rs . In other words, a long run expansion in personal income cannot be postulated to be some sor t of independent phenomenon - as i t i s viewed in Nourse (1968:212-214), for instance - but must be seen as a symptom of economic growth. In the context of the h ie rarch ia l model there i s one sa l i en t point to consider (and perhaps the reader r eca l l s th is from the previous Consider now the case of a long run increase in per capi ta 140 sec t ion ) . As a per capi ta income increase took p lace, sh i f t s would necessar i ly fo l low in the elements of the set {km | 1 < m < M} due to di f ferences in the re la t i ve (income) e l a s t i c i t i e s of demand for the ind iv idua l goods and services (which comprise the system's bundles). That i s , as a consequence of sh i f t s in demand alone, changes would be induced in the numbers of employees (persons) who were involved in the provis ion of f i r s t order, second order, e tc . a c t i v i t i e s throughout the ent i re central place system. I t might prove use fu l , then, to construe the elements of the set {k^} as being appropriate demand ratios ( for the 20 various bundles) which cou ld , of course, be redefined over time. There-f o r e , the impact of a long run change in personal income could be con-ceptual ized much l i k e a long run change in population ( th is case was 21 analysed e a r l i e r and i l l u s t r a t e d in Tables 4.1 and 4 .2 ) . Besides, while commodity bundles would be comprised of a mixture of income- inelast ic ( i n f e r i o r , necessary) and income-elast ic (normal, luxury) goods and se rv i ces , i t could be expected that these bundles would be weighted in a progressively income-elast ic fashion as the i r orders 22 increased. The resu l t ing impl icat ion should be apparent. Regional growth contr ibut ing to income increases would tend to be l oca l i zed in the higher level places - whose primary purpose i t i s to o f fe r the more spec ia l i zed , cap i ta l i n tens ive , and income-elast ic goods and services - so that i n -creased funct ional cen t ra l i za t i on would occur in the c i t y system. This in terpretat ion d i f f e rs considerably from that suggested by Nourse. While he recognized the importance of considering income-e las t i c i t y , he was e i ther re luctant or unable to integrate i t s consequences into his 141 argument. In any case, his hypothesis that a system would exh ib i t h ie r -a rch ia l truncation due to ce te r i s paribus [ s i c ] income changes (over the long run) is more than a b i t d i f f i c u l t to accept (espec ia l l y when i t i s recognized, as argued above, that such increases can only be ind ica t i ve . of long run changes in the performance of the space-economy). ( i i ) Process The inf luence of a per capi ta income increase upon the a t t r ibu tes of a d i f fus ionary process may be conjectured to be s im i l a r to the case given for a population increase. By weighting (see Isard (1960:507)) the populations in the potent ia l funct ion according to such income increases, i t becomes possib le to represent - in an aggregate sense - such notions as consumers enhancing the i r demand for t ravel (or communication), pro-ductive innovators perceiving p r o f i t a b i l i t y in a d i f fe ren t fashion (due to market increases in purchasing power), and the l i k e . More s p e c i f i c a l l y , Tables 4.3 and 4.4 might j us t as well repre-sent standardized one percent and two percent growths in personal income (population held constant) as increases in populat ion. Of course, there i s again the concomitant problem of de l ineat ing how such income sh i f t s might a f fec t the value of the distance exponent in the potent ia l function but an answer here, unfortunately, must await the synthesis of theory re lated to the demand for t ravel to central place theory per se (see footnote 2 of the th i rd chapter) . 142 4.5 Concluding Remarks In th is chapter the s t a t i c modell ing argument out l ined e a r l i e r i n the thesis was re lated to parametric change. Instantaneous s h i f t s , as wel l as long run transformations, in central place a t t r ibu tes (employ-ment, populat ion, e tc . ) were seen to resu l t from system-wide changes in populat ion, technology, and per capi ta income. Some new theoret ica l f ind ings were engendered by introducing a type of comparative s ta t i c s analys is into the d iscourse; unfortunately, on the long run side the argument tended more to rea l i ze the paucity of s o l i d theoret ica l ground (upon which the h ie ra rch ia l model res ts) rather than make substantive ana ly t i ca l advances. Re la t i ve ly l i t t l e at tent ion was given to the e f fec ts of such changes on the d i f fus ionary patterns of central place systems - research in th is area might, in f ac t , be stymied unt i l fur ther a r t i cu la t i on of the central place and grav i ty -potent ia l theories can be fashioned. Perhaps, however, one point d id c r y s t a l l i z e from the various perspect ives. I t seems that changes in central place structure (as re f lec ted in the s ize d i s t r i bu t i on of a system's communities) are general ly dependent upon the balance between population growth (or decl ine) and economic growth (or dec l i ne ) . I f population increases were to ou ts t r ip product iv i ty in a dramatic way, for instance, then a long run trend towards systemic d i v i s i on ( into r e l a t i v e l y independent subsystems) would be expected to occur. On the other hand, i f innovations (or, perhaps, regional com-parat ive advantage) proved pers is ten t l y successful in enhancing produc-t i v i t y , then systemic propert ies would be reinforced but in such a way that favored the population growth (decl ine) of the intermediate-sized and larger (smaller) places in the system. FOOTNOTES TO CHAPTER 4 'This statement appl ies in pa r t i cu la r to those observers l i k e Maruyama (1963), Pred (1965, 1966), Lampard (1968), and Thompson (1968) who have advocated a cybernetics framework for d iscussing growth and transformation in an inter-urban context. In a l l honesty, though, i t should be stressed that those arguments are not t o t a l l y confined to the domain which central place theory purports to cover. ''In the case of an instantaneous s h i f t the researcher would seem to be on r e l a t i v e l y safe grounds for s t i pu la t i ng whether or not a s p e c i f i c systemic change exhib i ted ceteris paribus c h a r a c t e r i s t i c s . When the argument moves on to the long run case, however, i t becomes apparent that the ef fects of a l l parameters cannot be analysed holding "other things the same." Misconceptions of th is sort have occurred in the economic l i t e ra tu re in the past (see Friedman (1949) and h is treatment of the Marshal l ian demand curve) so i t might prove a useful exercise to remain aware of the nature and l im i ta t i ons of the ce te r i s paribus perspect ive. JThe equi l ibr ium states for the h ie rarch ia l model are given in equations (2.18) and (2.33). In comparative s ta t i c s ana l ys i s , at tent ion i s being devoted to the direction of any change from such an equi l ib r ium s ta te . With the paucity of ana ly t i ca l statements in central place theory which concern equi l ibr ium per se and the genuine d i f f i c u l t i e s which pe rs i s t in v i sua l i z i ng equi l ibr ium in real world systems, perhaps i t i s encouraging to reca l l Isard (1956: ix -x ) : Despite the consequent failure to attain equilibrium in the secular sense, there is s t i l l value in equilibrium analysis. It is thought pertinent and worthwhile by some who conceive of the socio-economic system as a body tending toward a moving equilibrium and by others who find in equilibrium analysis categories of reference with which the extent of disequilibrium can be measured. 143 144 Most important, equilibrium analysis is valuable because it enables one to grasp better the laws of change and the workings of a system. "'There are two points of in te res t here. F i r s t l y , the best demonstration of how systemic equi l ibr ium may be a l tered ( in a s p e c i f i c central place) is found in Parr and Denike (1970) but those authors were not so concerned with the system-wide impl icat ions of such a s h i f t . Secondly, long run change was examined by Nourse but i t remains d i f f i c u l t to see how such changes could r e a l i s t i c a l l y be fashioned by the individual var ia t ion of d i f fe ren t parameters (see footnote 3 above). ^An " innovation c l u s t e r , " as suggested by Berry and Horton (1970:87), would be an extreme example of an act ive i tem; such an item might well induce s u f f i c i e n t systemic change so as to red i rec t i t s own expected adoptive pattern. The temporal resolut ion scale used to examine the adoption of such an item would, of course, be much greater than the appropriate scale used for the adoption of a less s i g n i f i c a n t item. Perhaps a r e l a t i v e l y simple example would have considerable i l l u s t r a t i v e mer i t . The reader might wish to re fer to Appendix D where equation (4.4) i s out l ined for a second level p lace. The text includes only the simple example where 1 < a < b < j < m. For the case of 1 < j < a < b < m, note the fo l low ing : k. - kL m I 3e B 'mc B* B c=j 3e . ^ 3eD, mj > mb A_ < „_B* m 3e: mj 3eL ma 3e B* 'mj 3e T ^ 3e T, ma > mb . B* < _ B* 3e . 3e . mj mj Each mu l t i p l i e r k. (1 < j < m) is del ineated into smaller vJ component mu l t i p l i e rs for the purposes of comparative s ta t i c s analys is in Section 4.3 of th is chapter (where in te res t focuses on the impact of tech-nological change on the central place s t ruc tu re) . In the case of the long 145 run, however, e f fo r ts must f i r s t be taken to speci fy the threshold requi re-ments of the ind iv idual functions (goods and serv ices) before d isc re te sh i f t s in the aggregate mu l t i p l i e r s may be ascer ta ined. The population sectors r i and {pm | 1 < m < 4} speci fy the total populations res id ing in rural areas, f i r s t level p laces, second level p laces, th i rd level p laces, and the fourth level place respec t ive ly . 1 0 Note that {k [ t 2 ] - k [ t i ] | 1 < m < 4} = {.0167, .0000, .0000, -.0167}. m m 1 - -I t i s only f a i r to note that Nourse was deal ing with the ear ly Beckmann (1958) model which, even when properly reformulated, does not e l i c i t a set of re lated service m u l t i p l i e r s . ""Although th is argument may appear to c o n f l i c t with the inc re -mental perspective (used for the example of one percent population growth; see Tables 4.1 and 4.2) upon cursory examination, th is i s not r e a l l y the case. Here i t i s presently assumed that the mu l t i p l i e r s remain constant throughout systemic growth and red is t r i bu t ion because, for each emerging centra l p lace, the population served i s proport ionately less as w e l l . In the previous incremental argument, however, red is t r i bu t ion led to increases or decreases in the mu l t i p l i e rs due to r e l a t i ve shifts - from the "ground up" - of the populations served at each l e v e l . 13 The cases of "no growth" are those already shown in Tables 3.10 and 3.12 respec t i ve ly . ^Beckmann and Schramm (1972) a lso deal t with the e f fec ts of changes in the service mu l t i p l i e rs upon systemic propert ies although there impacts were not re lated to the central place populations per se; ra ther , in te res t was given to the e f fec ts of technical change upon ( i ) the c i t y / market area population ra t ios and ( i i ) the population di f ferences between centers on adjacent h ie rarch ia l l e v e l s . The inequa l i t i es expressed in (4.38), (4.40), and (4.43) depend upon a pos i t i ve incremental increase in the service mu l t i p l i e r kj ( i . e . dk. > 0) - a condit ion which would resu l t from a technical innovation 146 adding a new funct ion or a c t i v i t y to the j th level bundle (see footnote 16 below). These inequa l i t i es would necessar i ly be reversed i f the tech-n ica l innovation were considered to occur amongst the ex i s t i ng stock of j th level a c t i v i t i e s ( i . e . representing cap i ta l deepening or the sub-s t i t u t i o n of cap i ta l for labor in the production of a spec i f i c good or serv ice in the j th bundle). This l a t t e r case i s taken up in more de ta i l i n equations (4.44) through (4.49) where the appropriate service mu l t i p l i e rs are resolved into the i r more elementary (mu l t ip l ie r ) un i t s . The funct ion f (1 < n < z ) is a member of the composite funct ion {f^} or funct ion set {f | 1 < m < M} which was introduced in the second chapter. I t would be espec ia l l y useful to have, in the words of Henderson (1972:437), " . . . a set of pr ices and possib ly a set of demand and supply condi t ions that would y i e l d balanced trade. . . " amongst the rural and urban un i t s . See von Boventer (1963) and Henderson (1972). ' J T h i s point deserves some added cons iderat ion. F i r s t l y , the assumption of constant real income may be su i tab ly relaxed without destroy-ing the h ie rarch ia l propert ies of the model. I t could be postu lated, for instance, that there are z qua l i t a t i ve categories of labor (unsk i l l ed , p ro fess iona l , e t c . ) , each in a state of equ i l ib r ium, and that the employ-ment (population) body engaged in providing a cer ta in bundle may be weighted according to these z categor ies. This would simply create a space-economy with z d i f fe ren t leve ls of real income. Secondly, the usual u t i l i t y and constant money income argument of t rad i t i ona l economics may be extended from the intra-urban scale to the inter-urban scale by a mult icenter argument f i r s t suggested by Papageorgiou (1971) and then further embellished by Papageorgiou and Caset t i (1971) and Papageorgiou (1973) (see Appendix G). But here a s i g n i f i c a n t t rade-of f must be considered: i s i t worth f o r f e i t i n g geometrical elegance (and i t s very obvious conceptual advantages) for a somewhat t i gh te r ana ly t i ca l package? At issue here is the reason why real income i s most su i tab le v i s - c l - v i s the l a t t i c e approach ( i . e . where places are equal ly spaced from the i r surrounding neighbors). In that c l a s s i c a l approach i t i s assumed that intra-urban a c t i v i t i e s are dimensionless ( i . e . there i s a perfect scale dichotomy between the two reso lu t ion l eve ls mentioned above); hence, the consumption of res iden t ia l land (and i t s related component in the household's u t i l i t y funct ion) must be precluded for the urban case. Natura l l y , th is makes i t expedient to suppose that a l l households throughout the system would consume equal amounts of the "other" goods and serv ices . 147 In order to accommodate constant money income into the s t r i c t h ie rarch ia l format, a lengthy l i s t of qui te un rea l i s t i c assumptions - as in von Boventer (1963) - would be required (e lse households would migrate to the cheaper goods sources - the larger centers) . The spat ia l d i s t r i -bution of money income, given constant real income, may be establ ished in the neighborhood of a spec i f i c locat ion as shown in Appendix G. 20 Tinbergen (1967) has used demand ra t ios to out l ine the pro-pert ies of a s i m i l a r , but not i d e n t i c a l , inter-urban system. 21 S i m i l a r i t i e s at leas t ex i s t for the case of an incremental long run change. " S e e Parr (1970:235) for some relevant comments. The problem is s im i l a r (and i n t r i c a t e l y related) to the dilemma which faced Thompson (1965:146) when he s ta ted , while commenting on community economic s t a b i l i t y , that : . . . deduction fails us here. For every income-elastic import (e.g., automobiles and tourism) one can cite, one can come up with an income-inelastic import (e.g., food, cigarettes, and fuel). And for every income-elastic local good or service (e.g., entertainment, restaurant dining, and home repairs), one can cite a likely income-inelastic one (e.g., rent, u t i l i t i e s , and local transportation). Chapter 5 A MODIFICATION OF THE STRICT HIERARCHIAL FORMAT 5.1 Introduction In the previous chapter at tent ion was given to the temporal va r ia t i on of h ie ra rch ia l a t t r ibu tes in a central place system. However., i t i s quite possible to extend the mu l t i p l i e r argument in a nontemporal sense as w e l l . To the author 's knowledge, Parr (1970:228) has been the only observer to speculate on the addi t ion of localized a c t i v i t i e s of a non-centra l place type (e .g . spec ia l i zed manufacturing, resource exp lo i t a t i on , e t c . ) to the s t r i c t h ie rarch ia l fo rmatJ However, th is addi t ion was not given e x p l i c i t cons iderat ion: that i s , the tota l population re lated to such l oca l i zed increments was not a l located amongst the d i f fe ren t s ized centers of the c i t y system. The remainder of th is chapter deals with the impl icat ions - in regards to both systemic structure and process -of such an a l l oca t ion scheme. 5.2 Structure The case here concerns the in tegrat ion of but one new a c t i v i t y in to the h ie rarch ia l format, although numerous such a c t i v i t i e s may be 148 149 concurrent ly integrated by u t i l i z i n g the same procedure. The real concern, however, i s over the spat ia l a l l oca t i on of the accompanying serv ice sec tor , given the s ize and locat ion of the employment (population) body engaged in th is l oca l i zed a c t i v i t y . To keep the case r e l a t i v e l y simple consider an M leve l K=3 central place system and an increment p M of nonnodal a c t i v i t y . Suppose, a , n in add i t i on , that the rural population i s s u f f i c i e n t l y productive to provide 2 for p a ^ and the population which services i t . Then, i r respec t i ve of where p M i s located in the system, an addi t ional increment p M such a , IM a , n that: M Pa,M [ I k m ] p a ,M M 1 - I k m=l m i s needed to service p a ^ (and i t s e l f ) . The tota l population p j M to be a l located is simply: PI,M = p a,M + p a ,M P a,M M 1 " I km m=l m (5.2) Now i f p g M i s located at the M th level center , then the ent i re population p j M i s a l located to that center and th is a l l oca t i on may be added to the f igure generated by the h ie rarch ia l model (see Chapter 2 ) . Suppose th is a l l oca t i on may be expressed nota t iona l ly as : 150 a,M Mpa,M Mpa,M;M (5.3) whose meaning should become c lear as the argument proceeds. I f , however, pQ M is located at a (one-third) component of a (M- l )s t leve l equivalent center, then: M- l P a,M;M-l a,M M-l 1 " I k m=l m (5.4) i s a l located to that equivalent center and: M-l Pa,M;M M mil m M-l P a,M;M-l (5.5) i s a l located to the M th leve l center where: a,M ~ M-l a,M " M-"Ta,M;M-l M-l pa,M;M (5.6) These two a l loca t ions may be added to the ind iv idual central place popula-3 t ions generated by the h ierarch ia l model as w e l l . . To take the argument one step more, i f p a ^ i s located at one of the s i x (M-2)nd level p laces, then: M-2Pa,M;M-2 a,M M-2 1 - T k L. r (5.7) m=l 151 i s a l located to that (M-2)nd level p lace, and: M-2 Pa,M;M-l 'M-1 M-1 m=l M-2Pa,M;M-2 (5.8) m i s a l located to each of two surrounding (M- l )s t level components, 4 and: k. J M-1 1-2Ka,M;M M 1 " I km L m=1 m M-2pa,M;M-2 VM M 1 - L . m m= I M-2Pa,M;M-2 + 2 M-2 Pa,M;M-l (5.9) i s a l located to the M th level place where: T T T C j 1 Pa,M = M-2Pa,M = M-2Pa,M;M-2 + 2 lM-2Pa,M;M-lJ + M-2Pa,M;M (5.10) Once again these a l loca t ions may be appended to the relevant central place populations generated by the h ie rarch ia l model. The argument becomes progressively more d i f f i c u l t to express abs t rac t ly as p, M i s located in lower and lower leve ls of the hierarchy a , n but the log ic underlying the above i s simply continued. However, one sa l i en t d i f ference does a r i s e : d i f f e ren t i a t i on amongst the various types 152 (see Appendix B) of central places in the same s ize c lass becomes a p o s s i b i l i t y . The reader should re fer to Appendix H where two numerical examples may be found. In the fo l lowing tab le , th is a l l oca t i ve scheme i s i l l u s t r a t e d fo r a l oca l i zed increment of 1000 persons (that i s , a to ta l population change of 4742) in a f i ve level K=3 system where {pm> = {1000, 4000, 16000, 64000, 256000} according to the h ie ra rch ia l model. Seven of the twelve possib le typical locat ions for th is nonnodal a c t i v i t y are accounted for -a number which should be en t i r e l y s u f f i c i e n t to demonstrate v a r i a b i l i t y in the a l loca t ion of the concomitant serv ice sector , given the s ize and locat ion of the l oca l i zed a c t i v i t y . Two a t t r ibu tes deserve c loser a t ten t ion . F i r s t of a l l , the a l l oca t ion is r a d i a l l y confined according to the topology of the system. I f a ray, o r ig ina t ing at the f i f t h level center , i s swept through the area inf luenced by the a l l o c a t i o n , then ( i ) the minimum inf luence i s zero radians ( l oca l i zed a c t i v i t y at a fourth leve l component) and ( i i ) the maximum inf luence is IT radians ( loca l i zed a c t i v i t y at a f i r s t level center nearest the f i f t h level center ) . Secondly, a l l oca t ion i s dependent upon the type of center possessing the l oca l i zed a c t i v i t y and not jus t that center 's s i z e . This i s made most evident when the f i f t h level place is again con-s idered : as s ize c lass i s held constant, but the center of in te res t i s moved progressively more d is tan t from the f i f t h leve l place ( i . e . type of center v a r i e s ) , then the service population a l located to the f i f t h level place decl ines in l i ke fash ion. This fo l lows from the p r inc ip le of m-domination in the central place argument. 153 Table 5.1 Var ia t ion in Central Place Populations as a Consequence of a Local ized A c t i v i t y p a , = 1000; K=3, M=5, { k } = {.3333, .1667, .1250, .0937, .0704}, n = 2000 Populations of Centers Type of Equiv. No Loca l . Location of Loca l . A c t i v i t y Center No. A c t i v i t y 5 4 3 5 1 256000 260742 257187 257483 4 2 ' 64000 64000 67555(1 )* 64000(1) 64592(1) 64000(1) 3 6 16000 16000 16000 18667(1) 16000(1) 2 1 6 4000 4000 4000 4000 2 2 6 4000 4000 4000 4000 2 3 6 4000 4000 4000 4000 1 1 6 1000 1000 1000 1000 1 2 6 1000 1000 1000 1000 I 3 12 1000 1000 1000 1000 1* 12 1000 1000 1000 1000 l 5 6 1000 1000 1000 1000 I 6 12 1000 1000 1000 1000 CONTINUED 154 Table 5.1 (Continued) Populations of Centers Type of Equiv. Location of Local ized A c t i v i t y Center No. 2 1 l 1 1 3 5 1 258198 257236 258820 257602 4 2 64100(1) 64000(1) 65062(1) 64000(1) 64008 64509(1) 64000 3 6 16222(2) 16000(4) 16222(2) 16000(4) 16037(1) 16018(2) 16000(3) 16760(1 ) 16037(2) 16000(3) 2 1 6 6000(1) 4000(5) 4000 4167(2) 4000(4) 4167(1) 4000(5) 2 2 6 4000 6000(1) 4000(5) 4000 4167(1) 4000(5) 2 3 6 4000 4000 4000 4000 l 1 6 1000 1000 2500(1) 1000(5) 1000 1 2 6 1000 1000 1000 1000 I 3 12 1000 1000 1000 2500(1) 1000(11) 1* 12 1000 1000 1000 1000 l 5 6 1000 1000 1000 1000 l 6 12 1000 1000 1000 1000 Parenthesized f igures are the numbers of equivalent centers of the same type but with d i f fe ren t populat ions. 155 5.3 Process The asymmetric structure of the modified central place system would ce r ta in l y be expected to induce asymmetry in an adoptive process -and so i t does. The nature of th is process asymmetry, given the h ie ra rch ia l parameters of the underlying system, would be dependent upon the s ize and absolute locat ion of the l oca l i zed a c t i v i t y increment. This represents another advantage of the d i f fus ionary perspective given in the th i rd chapter: analys is i s permitted despite the nonexistence of s t r i c t h ie ra rch ia l proper t ies . However, with th is added v a r i a b i l i t y , i t becomes a somewhat tedious chore to generate adoptive times - there-fo re , only one example (see Table 5.2) i s given. The a l l oca t ion for th is example may be found in case ( i ) of Appendix H and i s i l l u s t r a t e d i n the f i f t h column of Table 5 .1 . By re fe r r ing to Table 5 .2 , i t may be ascertained that s t ruc tura l asymmetry has a rather s i g n i f i c a n t e f fec t on adoptive times. In f a c t , adoptive orderings may be so a l tered that some members of one type adopt before, and other members a f t e r , a l l the members of another type. The chain of (dominating) re la t ions with the fourth level component ( i . e . one-th i rd equivalent center) i s the stimulus of th is a l t e r a t i o n . In c l o s i n g , i t must be stressed that i t would be hazardous to speculate on the in te r re la t i ons between h ie rarch ia l parameters and the propert ies of the loca l i zed a c t i v i t y in generating asymmetry - an approach s im i l a r to that of the th i rd chapter would have to be u t i l i z e d . L ikewise, accommodating several l oca l i zed a c t i v i t i e s - d i f f i c u l t enough for the s t ruc tura l argument - would be an extremely d i f f i c u l t task. 156 Table 5.2 A Comparison of Adoptive Times (Standardized) Between ( i ) Centers in a Closed K=3 Five Level System and ( i i ) Centers in a Modif ied (Due to One Local ized A c t i v i t y ) Version of that System; b=2, {km} = {.3333, .1667, .1250, .0937, .0704}; n = 2000, p a , = 1000 a , o and Located at 4th Level Component Type of Center Adoptive Time in S t r i c t H ierarch ia l System Adoptive Time in Modif ied System 5 0.00 0.00 3 1.00 1.00 2 1 1.30 1.30 I 1 1.76 1.76 4 1.91 1.67 (1 /3)* 1.91 (5/3) 2 2 3.53 3.44 (1) 3.53 (5) l 2 5.47 5.48 I 3 7.80 7.75 (2) 7.81 (4) 2 3 10.01 9.58 (2) 10.02 (4) 1" 10.96 10.76 (2) 10.98 (4) I 5 12.42 12.28 (2) 12.45 (4) l 6 12.99 11 .97 (2) 13.02 (4) Parenthesized f igures refer to the numbers of equivalent centers adopting at that time. 157 5.4 Concluding Remarks This argument was so le ly intended to support Par r ' s (1970) con-tent ion that a l oca l i zed a c t i v i t y could be embraced in the s t r i c t h ie rarch ia l format. Extensive asymmetry in structure and process were stressed as concomitant features of th is modi f ica t ion. 158 FOOTNOTES TO CHAPTER 5 'There are two key supposit ion here: ( i ) that the population body engaged in th is new a c t i v i t y locates at an ex i s t i ng centra l place s i t e and the systemic topology is retained (perhaps a new technology j us t becomes ava i lab le for exp lo i t i ng a nonubiquitous resource), and ( i i ) that the addi t ional populations do not promote the sl ippage (import subs t i tu -t ion) of goods and serv ices as discussed in the previous chapter. Rural population increases may be considered by using the argu-ment in Chapter 2 but th is only complicates the issue at hand. Care must be taken with the (M- l )s t centers in terms of a l l o c a t i o n ; when d iscussing structure th is i s not c r i t i c a l but i t must be remembered that a component (and not a "whole" equivalent center) receives the a l l oca t ion when concern turns to d i f f u s i on . Chapter 6 SUMMARY AND CONCLUSIONS 6.1 Introduction In th is thes is the d iscuss ion was i n i t i a l l y concerned with estab-l i s h i n g various a t t r ibu tes of a general h ie rarch ia l model which f a i t h f u l l y r e f l ec t s the tenets of C h r i s t a l l e r i a n (central place) thought. From there in te res t passed to recognizing j us t how th is s t ruc tura l argument could provide a useful framework for modelling inter-urban d i f fus ionary (adoptive) processes. F i n a l l y , the discourse closed with the author considering the e f fec ts of d i f fe ren t parametric sh i f t s - both continuous and d iscre te in nature - upon the propert ies of the spec i f ied models. In other words the essence of the d iscussion has been ( i ) the a r t i c u l a t i o n of ex is t ing statements and ( i i ) the reso lu t ion of new s ta te -ments about inter-urban structure and process (at least wi thin the context of centra l place p r i n c i p l e s ) . While th i s exercise has been attempted - in a more i m p l i c i t fashion i t should be added - in the past, the author fee ls that his present contr ibut ion i s s i g n i f i c a n t l y more general and, most important of a l l , considerably more consistent than these past endeavors have been. The author, in f a c t , i s hopeful that his present work w i l l 159 160 contr ibute to the a r t i c u l a t i o n , at some future date, of a general s t ruc -ture-process theory of inter-urban growth and development. Now the chain of argument has a purposeful design in that the reader f i r s t became acquainted with the conventional s t a t i c cases (models) and then proceeded to see how ( i ) temporal (instantaneous and long run) change, with spat ia l r egu la r i t i es in the space-economy held constant, and ( i i ) spat ia l change, as re f lec ted in s t ruc tura l asymmetry with time held constant, could both be embraced wi th in these o r ig ina l models. As such i t was necessary for the author to cont inua l ly jump back and for th between the polar concepts of structure (form-function) and process when he wished to e l i c i t various par t i cu la r asser t ions . I t seems most appropr iate, then, that in th is f i na l summary chapter the more s i g n i f i c a n t of these assert ions should be grouped together under the two general thematic headings. In add i t i on , th is chapter contains a few b r ie f points in passing which confirm the author 's strong advocation of a systems methodology -for th is appropriate topic area, that i s - and then closes i t s e l f by suggest-ing a number of paths along which future research might fo l low. 6.2 Structure The second chapter demonstrated that the most general of the ex i s t i ng h ie ra rch ia l models of c i t y s i ze could be fur ther extended by incorporat ing a d i s t i nc t i on between employees and dependents for a l l the households in a central place se t t i ng . This led to the formulation of a r e l a t i v e l y simple statement for the basic/non-basic employment ra t i o of an m th (1 < m < M) level place as well as a s l i g h t l y more complicated expression for the basic/non-basic population r a t i o of that same p lace. 161 Both these expressions were seen to be independent of the topology of the central place system which was being considered and the employment ra t i o was also shown to decl ine for progressively larger centers ( th is would hold t rue, as w e l l , for the population ra t io i f there was l i t t l e or no var ia t ion in family s i z e ) . I t was then demonstrated ( e x p l i c i t l y for the case of a system with a constant nesting fac tor and no va r ia t ion in fami ly s ize) that the urban/rural population balance of any central place system (or subsystem for that matter) would equal the inverse of the basic/non-basic population ra t i o of the largest center in the system (subsystem). I t was a lso shown that the " c l a s s i c a l " argument, involv ing recur-s ive formulas to describe the employment and population in the market areas of centers of d iverse s i ze ( i . e . centers on the various leve ls of the h ierarchy) , could be interpreted in a somewhat d i f f e ren t manner so that i t bore a generic resemblance to the input-output scheme of spaceless economics. The essent ia l d i f ference between the two cases was seen to l i e in the categories of ana lys i s : in the former, l inkages are del imi ted between the various hierarchial sectors of central places wh i le , in the l a t t e r , l inkages are expressed between the various industrial sectors of the economy. Af ter es tab l i sh ing the equ i l ib r ium condi t ions for a centra l place system, instantaneous sh i f t s in the employment and population cha rac te r i s t i cs of the d i f fe ren t communities could be determined by a comparative s t a t i c s methodology. By c lose ly adhering to the systemic guidel ines of the h ie rarch ia l model, exogenous sh i f t s could f i r s t be expressed for employ-ment (populat ion) , technology (output), and income (consumption) and, 162 then, the impacts of such sh i f t s upon those equi l ibr ium condit ions could be discerned in turn. A considerable number of statements - which led to the inference of several qua l i t a t i ve hypotheses - were included in the d iscussion so as to demonstrate the h ierarch ia l d i f f e ren t i a t i on of these instantaneous impacts. Perhaps the most s i g n i f i c a n t of these hypotheses were that the greater the s ize of the central p lace, ( i ) the greater would be the impact of a s h i f t in any given basic employment (population) sector and ( i i ) the greater would be the pos i t i ve (negative) impact of a technical innovation br inging for th a new a c t i v i t y (capi ta l deepening in any e x i s t -ing a c t i v i t y ) . The long run impl icat ions of such changes were taken up as w e l l . However, th is part of the text not iceably suffered in terms of r igor because such a paucity of analytical e f f o r t has been devoted (to date, that i s ) to a r t i cu l a t i ng issues l i k e ( i ) the threshold requirements, the advantages found in spec ia l i za t i on and agglomeration, e t c . on the part of f irms with ( i i ) the incomes gained, the frequencies of multipurpose t r i p s , e t c . on the part of households in the central place context. As a consequence, i t i s presently impossible to s t ipu la te jus t how the serv ice mu l t i p l i e rs would discretely change in the long run as a r e f l e c -t ion of the sh i f t ( s ) in the relevant parameter(s). Out of these arguments, however, emerged a few in te res t ing points which had been hi ther to overlooked in the l i t e r a t u r e . I t was shown, for instance, that the serv ice mu l t i p l i e r s also represented the actual proportions of a system's to ta l population (with family s ize again held constant) which'were devoted to f i r s t l e v e l , second l e v e l , th i rd l e v e l , 163 e tc . a c t i v i t i e s respec t i ve ly . In add i t i on , i t became apparent that these same mu l t i p l i e rs also represented demand ra t ios for the d i f fe ren t bundles of goods and services provided throughout the system. F i n a l l y , the s t ructura l argument was modified in two spec i f i c ways. In the second chapter, the s tage- l i ke propert ies of the h ie rarch ia l model were incorporated into an elementary evolut ionary scheme which retained the various cha rac te r i s t i cs of the conventional s t a t i c case. The f i f t h chapter, on the other hand, introduced s t ructura l asymmetry -due to a noncentral place type of a c t i v i t y (manufacturing, mining, e t c . ) -into the pure C h r i s t a l l e r i a n model. In both these cases rather complicated procedures were out l ined in order to a l loca te the extra population ( t h i s , of course, would be d is t r ibu ted throughout the ex i s t i ng systems) which would be required to service or support these new increments. 6.3 Process The th i rd chapter was en t i re l y devoted to resolv ing some methodo-log ica l issues in the l i t e ra tu re regarding central place d i f fus ion and, then, a r t i c u l a t i n g a model which was (hopeful ly) superior to the ex is t i ng formulat ions. In that chapter the author f i r s t c l a r i f i e d , in a somewhat fas t id ious manner, h is reasons for u t i l i z i n g a strict systems methodology when modelling inter-urban d i f f u s i o n . I ts app l icat ion led him to recognize that the boundary condit ions of any central place system could dramat ical ly a f fec t the cha rac te r i s t i cs of any processes occuring within that system. I t was decided, as a consequence, that i t would be a useful exercise to 164 d is t ingu ish between ideal closed and open central place systems (a l l real world systems are at leas t p a r t i a l l y open) in the d iscuss ion . The h ie rarch ia l model was used in conjunction with grav i ty -potent ia l theory in order to generate appropriate (Hagerstrand-1ike) d i f fus ionary patterns for central place systems with d i f fe ren t propert ies (nesting f ac to rs , s ize d i s t r i bu t i on of communities, e t c . ) . I t was demon-st rated that h ie ra rch ia l and wave-l ike adoptive patterns seemed to be •polar concepts in that ( i ) systemic openness (c losure) , ( i i ) a r e l a t i v e l y slow (rapid) decl ine in the values of the service m u l t i p l i e r s , ( i i i ) a r e l a t i v e l y low (high) value of the f r i c t i o n of distance coe f f i c i en t in the potent ia l formula, and ( iv) areal ( l inear ) dimensional i ty appeared to induce the former ( l a t t e r ) type of d i f fus ionary pat tern. This argument was subsequently modified so that changes in such patterns - from the ideal s t a t i c case, that is - could be discerned for ( i ) independent parametric sh i f t s ( in populat ion, income, e tc . ) and ( i i ) the addi t ion of noncentral place type a c t i v i t i e s . The second case was perhaps the more in teres t ing in that rad ia l discrepancies emerged in the adoptive scheme, re f l ec t i ng the fac t that centers in t imate ly t ied to a locat ion of incremental a c t i v i t y would adopt sooner than the other centers of the same type throughout the system. 6.4 Di rect ives for Future Research I t i s only r ight that , before th is thesis c loses , a few comments should be advanced which might speci fy appropriate ( in the author 's opinion anyways) areas of concern for future research. 165 F i r s t of a l l , there are numerous ana ly t i ca l de f i c ienc ies on the s t ructura l side of the argument alone. As has been repeatedly pointed out in the tex t , the service mu l t i p l i e rs of the h ie rarch ia l model require a more precise in terpre ta t ion in terms of e x p l i c i t supply and demand cond i t ions. Only then w i l l i t become possible to i nd i ca te , in a su i tab ly rigorous manner, the equi l ib r ium state(s) through which a central place system would be expected to pass over the long run. Another useful exercise - and th is in pa r t i cu la r would ce r ta in l y enhance the a b i l i t y of the s t ructura l model to represent cer ta in domains of the real world - would be to introduce dimensions of purposeful and/or random var ia t ion into the determin is t ic and i dea l l y fashioned h ie rarch ia l model. This modi f icat ion might be i ns t i t u ted e i ther in terms of the areal s ize (depict ing random var ia t ion of community locat ions from the perfect l a t t i c e ) or the employment (population) density ( re f l ec t i ng di f ferences in the product iv i ty of ag r i cu l tu ra l land) of the basic rural areas or i t might be introduced into the service mu l t i p l i e rs themselves (so as to indicate random di f ferences in labor or management s k i l l s or , perhaps, even spat ia l var ia t ion in the d i s t r i bu t i on of amenities for an otherwise featureless and bland wor ld) . Another useful change would be to introduce d i s t i nc t i ons between productive and consumptive l inkages amongst central p laces. In that way, the expenditures leaking out of communities to higher level places could be considered in both those sectors of the space-economy ( i . e . fo r both ind iv idua l f irms and consumers in an m th level place purchasing goods and services from (m+l)st, (m+2)nd, e tc . level p laces) . This would a lso help to resolve ' the balance of payments problem in the t rad i t iona l argument 166 and would provide a framework for considering even other topics - such as the c y c l i c a l s t a b i l i t y of the ind iv idual urban economies or the i n te r -re la t ions of investment and long run change throughout the system - under the h ie rarch ia l format. Besides (and th is i s a topic of special in te res t to the author) , there are promising avenues along which to integrate the rent and popula-t ion d i s t r i bu t i on models of intra-urban economics with the inter-urban model of community s i z e . The d i s t i n c t i v e h ie rarch ia l t r a i t s of the overa l l system could then be discerned - in a theoret ica l sense at leas t -wi th in the ind iv idual centers of that system. On the s t r i c t l y empir ical s i de , the author would welcome test ing of his proposed d i f fus ionary model. With appropriate care taken in se lec t ing a su i tab le study area, a ser ies of d i f fe ren t items which have been adopted, e tc . some of the hypotheses of the th i rd chapter could perhaps be r e a l i s t i c a l l y q u a l i f i e d . In addi t ion i t would be useful to re la te the temporal changes in the service mu l t i p l i e rs of such a study area to changes in other parameters (such as average household income, travel times to nearest neighbor communities, and the l i ke ) so as to suggest - in the absence of an ex i s t i ng rigorous ana ly t i ca l argument - how long run changes in such mu l t i p l i e r s might be s p e c i f i c a l l y re lated to various ind icators of regional economic growth (or dec l i ne ) . Th i rd l y , i t would be in te res t ing to test the a l l oca t i ve scheme given in the f i f t h chapter for noncentral place type a c t i v i t i e s . This could be done, again, by se lec t ion of a su i tab le area with only a few such a c t i v i t i e s located at d i f fe ren t s i t e s . 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" Internal and external factors in the develop-ment of urban economics," in H.S. Perl o f f and L. Wingo, J r . (eds.) (1968). Issues in Urban Economics. Balt imore: John Hopkins Press, 43-80. Tinbergen, J . (1967). "The hierarchy model of the s ize d i s t r i bu t i on of cent res , " Papers of the Regional Science Association, Vo l . 20, Z ip f , G.K. (T949). Human Behaviour and the Principle of Least Effort: an Introduction to Human Ecology. Cambridge: Addison Wesley Press. APPENDIX A STAGE MATURATION OF A FOUR LEVEL K=3 CENTRAL PLACE SYSTEM Assumptions: k 0 = .05, k x = .30, k 2 = .20, k 3 = .15, ku = .10 rx [ to] = 750 t i : P® [ t j = .3 (750) = 225 p? [ A t l ] = ^ _ 1 2 2 5 1 = 1 2 1 Pio [A t i ] = 17 P i i [A t i ] = 104 A l l oca t ion in A t x (numbers of dominated units in parentheses) r i = 17 (1) r i [ t i + A t i ] = 750 + 17 = 767 Pi [ t i + A t i ] = 225 + 104 = 329 174 t 2 : Pz [ t 2 ] = .2 (3(329) + 3(767)} = 658 p S [ A t 2 ] = ^ 6 5 8 ) = 8 0 4 P20 [A t 2 ] = 73 p l i [A t 2 ] = 439 pi2 [A t 2 ] = 292 A l loca t ion in A t 2 : r i = 24 (3) p i n = 10 (2) p i 1 2 = 419 (1) r i [ t 2 + A t 2 ] = 767 + 24 = 791 Pi [ t 2 + A t 2 ] = 329 + 10 = 339 Pa [ t 2 + A t 2 ] = 329 + 419 + 658 + 292 = 1698 t 3 : p? [ t 3 ] = .15 (3(1698) + 6(339) + 9(791)} = 2137 p S [A t ] = - 7 ( 2 1 „ 3 7 ) = 4986 3 3 -3 P30 [A t 3 ] = 356 P?i [A t 3 ] .= 2137 pL [A t 3 ] = 1425 P? 3 [A t 3 ] = 1068 A l l oca t ion in A t 3 : r x = 40 (9) P 3 i i = 17 (6) p f 1 2 = 24 (2) p ? 1 3 = 1987 (1) P 3 2 2 = 62 (2) pSi23 = 1 301 (1 ) r i [ t 3 + A t 3 ] = 791 + 40 = 831 Pi [ t 3 + A t 3 ] = 339 + 17 = 356 p2 [ t 3 + A t 3 ] = 1698 + 24 + 62 = 1784 P 3 [ t 3 + A t 3 ] = 1698 + 1987 + 1301 + 2137 + 1068 = 8191 U: p? [ U ] = .1 (3(8191 ) + 6(1784) + 18(356) + 27(831)} = 6412 p s [ A U ] = M w z i = 25648 177 p?o [ A U ] = 1603 p5i [ A U ] = 9618 p?2 [AU ] = 6 4 1 2 p,3 [Ati,] = 4809 p5- [ A U ] = 3206 Al loca t ion in A t , : ri = 59 (27) p5n = 25 (18) p5i2 = 36 (6) p5i3 = 51 (2) p?m = 8850 (1 ) p522 = 91 (6) P523 = 1 30 (2) pL- = 5606 (1 ) pl33 = 401 (2) P53 l | = 4007 (1 ) r i [ U + A U ] = 831 + 59 = 890 P i [ U + A U ] = 356 + 25 = 381 p2 [ U + A U ] = 1784 + 36 + 91 = 1911 p3 [ U + A U ] = 8191 + 51 + 130 + 401 = 8773 p„ [ U + A t , ] =8191 + 8850 + 5606 + 4007 + 6412 + 3206 = 36272 Table A . l Populations of Rural Areas and Central Matures Places in a K= in a Stage-Like 3 Four Level Central Fashion* Place System Which Time Pi P2 P3 P* Total to 750 (27)** 20250 - - - -20250 t i + A t i 767 (27) 20709 329 (27) 8883 - - -29592 t 2 + A t 2 791 (27) 21357 339 (18) 6102 1698 (9) 15282 - -42741 t 3 + At 3 831 (27) 22437 356 (18) 6408 1784 (6) 10704 8191 (3) 24573 -64122 t , = A t , 890 (27) 24030 381 (18) 6858 1911 (6) 11466 8773 (2) 17546 36272 (1) 36272 96172 Figures are subject to rounding er rors . Figures in ;the parentheses re fer to the equivalent number of rural areas, f i r s t level p laces, e tc . CD \ 179 Note that while incremental growth in rura l population remains a constant f rac t ion (0.05) of incremental growth in to ta l population (rural plus urban), the r u r a l / t o t a l r a t i o decl ines in the sequence 1.00, 0.70, 0.50, 0.35, 0.25 and the urban/rural ra t i o increases in the sequence 0.00, 0.43, 1.00, 1.86, 3.00. APPENDIX B T H E N O T I O N O F E Q U I V A L E N T C E N T E R S I N H U D S O N ' S S T O C H A S T I C T E L L I N G P R O C E S S In order to i l l u s t r a t e Hudson's t e l l i n g process i t is useful to del ineate the actual t e l l i n g chains. While th is reduces the elegance of the o r ig ina l mathematical argument, a much truer spatial perspective of a d i f fus ionary process i s a t ta ined. In shor t , i t i s possible to demonstrate p robab i l i s t i c symmetry in spa t ia l adoption: a property that should be expected in the i d e a l i s t i c central place s e t t i n g . Consider, for instance, a f i ve level K=3 system with the sequence: 1, 2, 6, 18, 54 representing the number of centers by s ize c l a s s . Table B.l c l a s s i f i e s these centers into types according to the i r distances from the s ingle f i f t h level placeJ In Hudson's proposal , only th i s f i f t h level place would know (have adopted) at time t x . In fac t , i t would know with complete cer-t a i n t y ; i . e . with a p robab i l i t y of one. However, s i x surrounding f i f t h leve l p laces, each the geometric center of an ident ica l system, would a lso know at the same time. At time t 2 , then, the f i f t h level place 180 Table B.l Types of Equivalent Centers in a K=3 Five Level Central Place System Order Type Equivalent Distance from 5th Number Order. Center 5 5 1 0 4 4 2 3/3 3 3 6 3 2 18 2 1 6 /3 2 2 6 2^3 2 3 6 1 54 l 1 6 l l 2 6 2 I 3 12 n 1" 12 l 5 6 4 I 6 12 182 of i n te res t i s surrounded by s i x fourth leve l places - each knowing with p robab i l i t y 1 - and s i x t h i r d , second, and f i r s t level places -each knowing with probab i l i t y 0.333. In equivalent terms, two places of each level have adopted with complete ce r ta in t y . The t e l l i n g process for th is pa r t i cu la r system may be represented by the statement: t i : g/hj e(b)(c) = f. (B . l ) which means that at time t . ( i > 1 ) , centers of type h. (j = 1 ,2 , • • • )» ' each dominated by e center(s) of type g, increase the i r adoptive p robab i l i t y by f^  (0 < f. < 1) . The increment of p r o b a b i l i s t i c adoption in each type g place at the previous time t. i s represented by b and c is simply the rec iprocal of the nesting fac tor K. The t e l l i n g process a t e a c h time t. (1 < i < 5) in the f i ve leve l K=3 system i s then: 183 t i t 2 5/4 5/3,2M 1 4/3 4 / 2 2 , 2 3 , l 6 3 / 2 \ 2 2 , 2 3 3 / l 2 , l 3 , l \ l 5 2 V 1 M 2 3 / 2 x , 2 2 , 2 3 3 / l 2 , l 3 , l \ l 5 2 V 1 M 2 2 7 1 3 2 2 / l 3 , l \ l 6 2 3 / l \ l 6 2 3 / l 5 2 7 1 M 2 2 7 1 3 2 2 / l 3 , l \ l 6 2 3 / l \ l 6 2 3 / l 5 3(1) [.333) = 1 (1) [.333) .333 2(1)1 .333) = .667 ( I X .333) = .333 2( .333)1 .333) = .222 .333)1 .333) = .222 21 .333)< .333) = .222 .333)( .333) = .111 21 .667)( .333) z: .444 .667)( .333) = .222 21 .222)( .333) = .148 .222)( .333) = .074 .556) ( .333) = .185 .556)( .333) = .185 2( .556)( .333) — .370 2( .444)( .333) = .296 .444)( .333) - .148 .444) ( .333) - .148 2( .444)( .333) = .296 .444)( .333) = .148 which spec i f ies the fo l lowing incremental states of p r o b a b i l i s t i c adoption for a l l members of each type: 5 1, 0, 0, o, 0 4 : 0, 1, 0, 0, 0 3 0, .333, .667, 0, 0 21 0, .333, .222, .444, 0 22 0, 0, .556, .444, 0 23 0, 0, .556, .444, 0 l 1 0, .333, .222, .148, .296 l 2 0, 0, .333, .370, .296 I 3 0, o, .222, .481, .296 1- 0, 0, .111, .592, .296 1 5 0, o, .111, .592, .296 l 6 0, 0, .333, .370, .296 184 Therefore, the sequential states sums of incremental s ta tes , are: 5 : 1, 1, 4 : 0, 1, 3 : 0, .333, 2 1 : 0, 0, 2 2 : o, .333, 2 3 : o, 0, l 1 : o, .333, l z : o, 0, l 3 : o, 0, 1": o, 0, l 5 : o, 0, l 6 : o, 0, p r o b a b i l i s t i c adoption, representing 1, 1, 1 1, 1, 1 1, 1, 1 .556, 1, 1 .556, 1 , 1 .556, 1, 1 .556, .704, 1 .333, .704, 1 .222, .704, 1 .111, .704, 1 .111, .704, 1 .333, .704, 1 These two l i s t s are useful fo r iden t i f y ing the d i f fe ren t adoptive patterns of centers that are of the same s ize but which have d i f fe ren t r e l a t i ve loca t ions . Di f fus ion i s symmetric in the sense that a l l centers of the same type have iden t i ca l adoptive pat terns. The incremental l i s t i s the counterpart to Hudson's f i r s t - h e a r i n g matrix [A]. Note, for instance, that at time t 3 , the equivalent number of f i r s t level places hearing a message (adopting) for the f i r s t time i s : a 3 5 = .222(6) + .333(6) + .222(12) + .111(12) + .111(6) + .333(12) = 12 185 which i s an element of the matr ix: 0 0 0 0 0 2 2 2 2 [A] = 0 0 4 8 12 0 0 0 8 24 0 0 0 0 16 for a l l f i r s t - hea r i ng centers. The above argument i s so le ly intended to be a spat ia l q u a l i f i -cat ion of Hudson's proposal . I t i s th is author 's contention that i t i s simply more accurate to state that centers adopt with cer ta in p robab i l i t i es rather than to state that equivalent enters adopt with ce r ta in ty . On the other hand, the same empir ical test i s used to ver i f y e i ther in te rp re ta t ion . FOOTNOTES TO APPENDIX B ^On a hexagonal l a t t i c e the distance from one point to each surrounding point can be calcu lated by applying the formula: d i d = ( i 2 + i j + j 2 ) * 1 < i < j where i and j are integers representing distances on a s i x - f o l d ax is with unit t rans la t ion periods and a per iodic ro ta t ion angle of T T / 3 . See Dacey (1965). 2 The inconsistency of Hudson's argument may be found at time t 2 . In addi t ion minor rounding errors ex i s t because the p r o b a b i l i s t i c values are expressed in decimal form. 186 APPENDIX C THE ALTERNATIVE DIFFUSIONARY MODEL: PROPERTIES OF THE TABLES AND AN OUTLINE OF THE TELLING PROCESS In th is appendix at tent ion is f i r s t given to the th i r ty- two tables of generated data i n Chapter 3. In order to complement the table headings, the fo l lowing l i s t provides addi t ional information about the centra l place populations in open (closed) systems: Tables 3 .1 , 3.2, 3 .3 , 3 .4 ; 1 Populat ions: 1000, 4000 (2000), 16000, 64000 (21333), 256000 2 Tables 3.5, 3.6, 3.7, 3.8; rank-s ize re lated Populat ions: 1000, 2946 (1473), 8385, 21800 (7267), 54500 Tables 3.9, 3.10, 3.11, 3.12; Populat ions: 1000 (500), 4000, 16000 (5333), 64000 187 188 Tables 3.13, 3.14, 3.15, 3.16; rank-s ize re lated Populat ions: 1000 (500), 2846, 7400 (2467), 18500 Tables 3.17, 3.18, 3.19, 3.20; Populat ions: 1000, 4000 (2000), 16000 (5333), 64000 Table 3.21, 3.22, 3.23, 3.24; rank-s ize re la ted Populat ions: 1000, 3062 (1531), 9800 (3267) 24500 Tables 3.25, 3.26; rank-s ize re lated Populat ions: 1000 (500), 3857, 13500 (6750), 40500 Tables 3.27, 3.28; Populat ions: 1000, 4000, 16000, 64000 (32000), 256000 Tables 3.29, 3.30: rank-s ize re lated Populat ions: 1000, 1923, 3571, 6250 (3125), 12500 Tables 3.31, 3.32; Populat ions: 1000, 4000, 16000 (8000), 64000 The t e l l i n g process postulated in Chapter 3 i s now i l l u s t r a t e d for a four level system (open and closed) with a K=3 geometry. The equivalent central place populations of that system d i f f e r by a mul t ip le 189 of four and the distance exponent of the potent ia l funct ion i s two. The computed adoptive times are those given in Tables 3.10 and 3.12 (with a s l i gh t d i f ference due to var ia t ion in s i g n i f i c a n t f i gu res ) . The centers may be c l a s s i f i e d according to types as they were in Appendix B: i t should not be necessary to restate the appropriate distances from the M th level p lace. The. populat ions, then, according to types a r e : ^ type population 4 64000 3 16000 open 2 4000 system 1 1 1000 1 2 1000 1 3 1000 4 64000 3 5333 closed 2 4000 system 1 1 1000 1 2 1000 l 3 500 Since the t e l l i n g process is now continuous in nature, i t s in ten-s i t y (with regard to a spec i f i c hearing or adopting place) jumps in a step-wise fashion as new t e l l i n g centers (which have themselves jus t 190 adopted) j o in in according to the p r inc ip les of central place domination. Adoptive times may be standardized to the time of f i r s t adoption (for a pa r t i cu la r process in a pa r t i cu la r system) outside the M th leve l center (source): for these r e l a t i v e t imes, the sca l ing factor G of equation (3.2) may be el iminated and populations may be expressed in simpler terms ( i . e . not i n thousands). An increment of e f f ec t i ve po ten t i a l , expressed continuously at a type h center (hearer) by e type g center(s) ( t e l l e r s ) a f te r some time t^, may be represented by the statement: e ( p a p h ) V 9/h: 3__JL_ ( C l ) d gh where p^, p^ are populations of type g and h centers respect ive ly and d ^ i s the distance separating them. The threshold for adoption i s simply determined by the greatest potent ia l expressed outside the M th level place in any one system. (I) The Te l l i ng Process in an Open System 341.333 = threshold 85.333 64 16 18.286 4 /3 : 4 /2 : 4 / 1 x : 4 / 1 2 : 4 / 1 3 : 3 (64 x 16) 3 2 64 x 4 64 x 1 l 2 64 x 1 l 2 2 (64 x 1) ( / 7 ) 2 191 (1) 3 adoptive time = 1 remaining resistance 341.333 ( i i ) 2 time 1 : 85.333 . 2 (16 x 4) 2 (/J) incremental time = 3/2 = 42.667 256 85.333 + 42.667 = 2 adoptive time = 3 341.333 256 ( i i i ) l 1 time 3 2 / 1 1 3 (64) = 192 2 ( 4 X 1 ) = g 1 incremental time = 149.333 64 + 8 adoptive time = 5.074 2.074 341.333 149.333 ( iv) l 3 time 1 4 / 1 3 time 3 2 / 1 3 18.286 16 x 1 1 2 = 16 2 (18.286 + 16) = 68.572 2 (4 x 1) 1 incremental time 8 254.475 34.286 + 8 adoptive time = 9.018 = 6.018 341.333 254.475 (v) I 2 time 1 3 / 1 2 time 3 16 16 x 1 l 2 = 16 2 (16 + 16) = 64 O C1 O Q O incremental time = 3 2 + 3 = 6.533 adoptive time = 9.533 341.333 261.333 192 (II) The Te l l i ng Process in a Closed System 4/3 : 6 4 X 3 ^ 3 3 3 = 37.926 4/2 : (64.x 4) = 85.333 = threshold 4 /11 : 64^x2 = 6 4 4 / 1 2 : = 16 4 / 1 3 : 6 4 x Q - 5 = 4.571 ( /7 ) 2 remaining resistance . ( i ) 2 - 85.333 adoptive time = 1 ( i i ) i 1 85.333 time 1 : 64 21.333 2/1 > : • l _ ( 4 _ x j l a 8 21 333 incremental time = Q = .296 64 + 8 adoptive time = 1.296 ( i i i ) 3 85.333 time 1 : 37.926 47.407 2/3 • (4 x 5.333) _ 14.222 (^3) 47 407 incremental time = 37 926'+ 14 222 = " 9 0 9 adoptive time = 1.909 ( iv) l 2 85.333 time 1 : 16 2 : 1 2 2 = 8 time 1.909: .909 (.16 + 8) = 21.816 3:1* 5 1 3 ^ 3 _ x j . = 5 > 3 3 3 47 51 7 incremental time = 24 + 5 333 = 1 , 6 2 0 adoptive time = 3.529 l 3 time 1: 4.571 2 / l 3 : 4 - ^ 5 = 2 time 1.909: .909 (4.571 + 2) = 5.973 3 / 1 3 : 5 ' 3 3 3 1 x ° - 5 = 2.667 74 789 incremental time = 5 5 7 1 + 2 667 = 8 - 0 9 6 adoptive time = 10.005 FOOTNOTES TO APPENDIX C 'This i s the basic progression component model f i r s t suggested by Beckmann (1958) and then revised by Beckmann (1968) and Parr (1969). In th is case central place populations d i f f e r by a mul t ip le that i s constant for a l l l e v e l s : in th is example i t i s four . The bracketed f igures represent the effective populations of places on the boundary or at the endpoints of a closed system. ^The rank-s ize rule is indicated in several tables of Chapter 3 by the statement p m (R ) = pM -. the population of a place midway on the m th h ie rarch ia l level times i t s rank (see Parr (1969) for a good d i s -cussion) equals the population of the M th level p lace. Even in th is non-exponential case, the s ize d i s t r i b u t i o n var ies dramat ical ly with other factors (geometry, s ize of smal lest p laces, e t c . ) . Note that both systems contain the same central place popula-t ions but that the populations effective for potent ia l (and, hence, d i f fus ion) are greater for boundary points in the second and fourth s ize c l asses .o f the open system. 194 APPENDIX D THE IMPACT OF EXOGENOUS EMPLOYMENT SHIFTS IN A SECOND LEVEL CENTRAL PLACE The purpose of th is appendix i s simply to e luc idate (4.4) for the second level case. In add i t i on , statements are given to i l l u s t r a t e (4.13) and (4.31) , each of which depends on (4 .4 ) . Recal l ing (2.4) and (2.7) i t should be apparent that : e 2 i = k i e 0 where, given that: Ai = 1 - k x A 2 = 1 - k i - k 2 (D.2) i t fol lows that : 3ejj2 _ k 2 (K t - 1) (D.3) 195 196 The as ter isk is a reminder that the second level p lace, while i t provides the second bundle to these Kx - 1 f i r s t level p laces, does not provide the same places with the f i r s t bundle as w e l l . Now by using (2.12) and (2.13) , i t should be c lear that : 3e l i 3 e 2 i 1 + , . k 2 (K : - ! ) I n = ( 1 - k i ) ( 1 - k 2 ) + k i k 2 ( K i - 1 ) A i A 2 3eL 3 e 2 1 k 2 ( K 1 - l ) k^ A i A , 1 + k 2 ( K A - 1 ) A i k 2 ( l - k 1 ) K 1 A i A 2 3e l i 3 e 2 2 3eL 3eL M l ) A 2 k i d - k j A i A 2 3 e l 2 B* 3ef 2 9 e l 2 , k 2 ( l ) 3eL " A> _ ( 1 - k i ) ' A i A, (D. so that: 197 de l d e 2 i A, Ai A 2 del A B * d e 2 2 del d e 2 2 J _ A 2 (D.5) dp 2 • B* d e 2 i (1 + d j (1 + d 2 ) B* 3 d e 2 l 3 e 2 i dp 2 d e 2 2 (1 + d i ) 3e l i 3 e 2 2 + (1 + d 2 ) 3 e l 2 « B* 3 e 2 2 (D.6) APPENDIX E THE DIVISION OF CENTRAL PLACE POPULATIONS ACCORDING TO ACTIVITY SECTORS The fo l lowing argument - as i t precludes v a r i a b i l i t y in the family s ize mu l t i p l i e r and the central place topology ( i . e . the nesting factor) - does not have the same degree of general i ty as many of the previous assert ions in th is thes i s . The author simply fee ls tha t , in th i s pa r t i cu la r case, considerable i l l u s t r a t i v e merit might be l os t by making the notation excessively complex. As a r e s u l t , the present argu-ment i s phrased in terms of sectors of population - th is being, of course, the t rad i t i ona l uni t for h ierarch ia l modell ing - rather than sectors of employment. Consider now an M level central place system with nesting fac to r K. The tota l population P^ of th is syrtem may be divided into M+1 parts or general a c t i v i t y sec tors , designated by the members of the set { r 1 , p 1 , p 2 , " - , p m , * * « , p M}, where: ( i ) r 1 i s the population engaged in rural (ag r i cu l tu ra l ) a c t i v i t i e s ; and ( i i ) p m i s the population engaged in m th (1 5 m 5 M) level a c t i v i t i e s ( i . e . the sum of the populations 198 199 in each of the m th , ( m + 1 ) s t , • , and M th order places engaged in .providing the m th bundle of goods and se rv i ces ) . Values of these elements may be computed as: r» = K ^ r , p 1 = k 1 { ( K - l ) K M " 2 (pi+r i ) + (K -1 )K M " 3 (pa+r j + + (K- l ) ( p ^ + r j + (pM+r x)} p 2 = k 2 { ( K - l ) K M - 3 (p 2 +r 2 ) + ( K - 1 ) M " 4 (p 3 +r 2 ) + . . . + (K- l ) (pM_-,+r2) + (pM+r 2)} • + - + ( K - l )<P M . ,+ r m ) • (p M + r m ) } M P = k M { P M + r M } (E . l ) However, each of the a c t i v i t y sectors p m (1 < m < H) may be reformulated so that : m + i l l + J<2_ + J<JL_ + Ai AiA 2 A 2A 3 AM-1V (E.2) where: V . 1 - J, k i i=l (E.3) 200 But (E.2) may be reformulated i t s e l f by adding each of the terms one at a time: m m K r i , x p = ( E > 4 Now the tota l population P M of the system is (according to (4.22) in the main t ex t ) : .M- l " rtM so that the proportion R° of to ta l population engaged in rural a c t i v i t i e s i s : while the proportion Rm of to ta l population engaged in m th (1 < m < M) leve l a c t i v i t i e s i s : In other words the service mu l t i p l i e r s alone define the proportion of a central place system's to ta l population which i s devoted ta rural pro-duction or i s engaged in providing the 1 s t , 2 nd, or M th bundle of goods and serv ices . Inc iden ta l l y , the argument can be extended for a var iab le topology or be restructured to concern employment (as opposed to population) but (E.6) and (E.7) would remain true mutatis mutandis. APPENDIX F THE IMPACT OF AN EXOGENOUS SHIFT IN A SERVICE MULTIPLIER IN A SECOND LEVEL CENTRAL PLACE The purpose of th is appendix is to i l l u s t r a t e (4.37) and (4.42) and compare the resu l ts of each approach. The-effect of a s h i f t in the m u l t i p l i e r k 2 upon the tota l employment in a second leve l place i s considered. I f the reader r eca l l s (2.12) and (2.13) then i t should fo l low that : so that: 3 e T i _ Meg, + e? 2 ) + | f f - A 2 k l  8 k 2 ~ ( A 2 ) 2 g 3eL. A ^ + e L ) + | ^ A 2 A 1  d k 2 " ( A 2 ) 2 del I f f d k 2 ( A 2 ) 2 ( F . l ) (F.2) 201 202 where: Ai = 1 - k i A 2 = 1 - k i (F.3) I f i t is reca l led that: 3^22 9k2 (Ki-1) k i e 0 „ Ai + K l e ° (F.4) then i t can be shown that: de 2 _ Ki e 0 dk : (A 2 ) ' (F.5) But from (4.41): e l e 0 K L . kaKi Ai A iA 2 (F.6) which implies that: de l dk 2 Ki e ( ( A 2 ) : (F.7) as in (F.5) above. APPENDIX G A NOTE ON THE RELATION BETWEEN MONEY INCOME AND REAL INCOME IN THE HIERARCHIAL FORMAT The fo l lowing demonstration i s an extension of the neoclass ica l (Hicks-Samuelson) argument on consumer behavior found in spaceless economics. The proposals of Long (1971) are embraced in the "mult icenter d i s t r i b u t i o n " framework of Papageorgiou (1971). I t must be pointed out, however, that the argument represents a specia l in terpre ta t ion of the s t r i c t h ie rarch ia l format: that being, where agr i cu l tu ra l goods are not immediately consumed but are sent on to the urban centers fo r processing. In other words at tent ion i s devoted to only the M bundles of goods and serv i ces . Needless to say, analys is i s only par t ia l in nature. Suppose that: ( i ) M composite goods and serv ices are offered accord-to s t r i c t h ie rarch ia l p r i n c i p l e s ; ( i i ) the locat ion of any household under equi l ib r ium condit ions may be spec i f ied by a set {s |m=l,2,*••,M} of distances to the nearest points of supply fo r goods of order 1 , 2 , • , M respect ive ly ; {s } may be 1 interpreted as a M-dimensional vector as w e l l ; 203 204 ( i i i ) the pr ice p m of the m th bundled good may be e x p l i c i t l y re la ted to the m i l l p r ice f , the vector component s , and the transport rate III r\ (per unit distance) t ; that i s : ( i v ) the household, given i t s locat ion {s^}, maximizes a u t i l i t y funct ion u = u ( x i , x 2 , • • • . x ^ ) according to a budget const ra in t Y = Y({sm}) in money income terms. Then, the usual statement of spaceless economics (see a good text l i k e Samuel son (1947) or Henderson and Quandt (1958)): d x i = - ^ l i d p 1 - XD 2 .dp 2 - AD M i dp M + V l , i ( d Y - x 1 d P l - x 2 d p 2 - — - x M d p M } t r 0 , with Lagrangian \ , determinant D, and cofactors ( j = 1,2,•••,M+1), may be general ized to : 205 d x 1 " " £ '11 + D Mi 3 p M . f . 3 p M QTM + "^ n d t „ + -Kz- ds. 9 f M 9 t M 9 s M D 9Y 8s i ds.! - X] + l i ^ d S M - x m 9 P M . 9 p M + 9 p M , a ? : d f M + 3E7 d t M + 9 iT d S M 'M M (G.3) which implies that: 9x i - A 9s. J D D j i 3s, ap• Du . , . J D 8Y 9s ap. x • j as fa> 9Y aY 9sT * J 3p 9s 9x. 9Y 9S . * * * J (G.4) where * indicates that locat ion is held constant, * * indicates that u t i l i t y and locat ion are held constant, and * * * indicates that m i l l p r i ces , transport ra tes , and locat ion are held constant. Th i s , of course, i s the same as: 9x. i 9s . ax. 9Y 3Y 9sT * J 9x. ap ap. * 3 s j (G.5) a X i However, i f i s set equal to zero, then: 206 3Y 9s . J f!ii 3P. 9p. V 9s 9x. i |9Y (G.6) 9 Y where represents the sh i f t in money income which would be required by the household to compensate for a locat ional change away from the supply point of the j th composite good and s t i l l go on consuming the same amount of the i th composite good. The necessary and s u f f i c i e n t condit ions for real income to remain constant in the neighborhood of the 9x. household's locat ion are, then, determined by set t inq •—• n r , , J 3 9s. = 0 for a l l J i , j = 1 , 2 , - - . , M . The above argument i s useful for ind ica t ing the change in money income for incremental locat ional s h i f t s . The spec i f i ca t i on of real income at any point in the system may be estab l ished v is-et-v is the M th leve l center by an indexing procedure. In add i t i on , i t should be pointed out that : 3 t j fax,! 9t. - X. f a x ^ 9Y 9t . * * * J (G.7) which i s ident ica l to the statement in Long. He has ca l led the f i r s t term on the r ight hand side the transport subst i tu t ion e f fec t and the second term the transport income e f fec t . FOOTNOTES TO APPENDIX G In th is argument i t i s assumed that a household i s in spat ia l equ i l ib r ium i f i t has the same real income as a l l other households: put d i f f e r e n t l y , i t consumes the same amounts of goods and services as other households and i t , therefore, has no incent ive to change l oca t i on . This i s a picky point but {p } i s used to indicate pr ices only remainder of the thesis i t refers to i s d i rected to notat ion enthus iasts , in th is appendix - throughout the urban populat ions. 207 APPENDIX H THE ALLOCATION OF NONNODAL ACTIVITIES Two numerical examples of the argument in Chapter 5. are pre-sented below. Suppose a f i ve level K=3 system ex is ts with {k f f l} = {.3333, .1667, .1250, .0937, .0704} and {pm} = {1000, 4000, 16000, 64000, 256000}. The a l l oca t ion of : p T c = Pa>5 = J 0 0 0 = P a , 5 5 .2109 4 / 4 < i W-*> i - y k L* n m=l i s of i n te res t . Case ( i ) : Suppose p 3 r = 1000 locates at a (one-third) component of a a 5 D type 4 center; then: 2Q°3 = 3555 i s a l located to that type 4 component .0704 (3555) 1 1 Q _ . ^ . ^ 4_. . , 2 i Q g = 1187 i s a l located to the s ing le type 5 center 208 Suppose P a 5 = 1000 locates at one type 2 1 center; then • ^3^= 2000 i s a l located to that type 2 1 center .1250(2000/3) „ , . . , . . , ^ o 7 [- n = 222 i s a l located to each of two type 3 centers .0937(222/3) 0 K . . . . , ' - = 25 i s a l located to each of two 2813 type 4 components 2( .0937) (222/3) _ 5 Q i g a n o c a t e c | to one type 4 component-. • 1 2 5 ° ^ ° 9 0 / 3 ) + ^ ( 2 0 0 0 + 2 ( 2 2 2 / 3 ) } + (2000 + 2(222) + 100} = 2198 is a l located to the type 5 center 

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