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Linear programming model for land resource allocation in the Lower Mainland of British Columbia. Gardner, Andrew George 1971

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A LINEAR PROGRAMMING MODEL FOR LAND RESOURCE ALLOCATION IN THE LOWER MAINLAND OF BRITISH COLUMBIA. , by Andrew George Gardner B.Sc. Honours, U n i v e r s i t y of Edinburgh, 1968. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of A g r i c u l t u r a l Economics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL 1971 : "In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representative. It i s understood that copying or publication of this thesis for the financial gain sh a l l not be allowed without my written permission." Department of Agricultural Economics University of British Columbia .30th., April, 1971. ABSTRACT The expanding population i n the Lower Mainland of B r i t i s h Columbia, has been accompanied by a commensurate increase i n the pressure of demand for land. In recent years attention has been focused on the competition for the regional land resources by a g r i c u l t u r a l and urban i n t e r e s t s . Coincident with this c o n f l i c t , public concern has been expressed over market a l l o c a t i o n of a g r i c u l t u r a l land to urban uses. In this respect reference has often been made by the public, to the benefits of planning regional land a l l o c a t i o n . To date no quantitative economic examination of this trend i n land resource a l l o c a t i o n has been undertaken. This thesis i s an attempt to show how a mathematical technique - l i n e a r programming, - can be used to analyse and evaluate such land a l l o c a t i o n problems. The l i n e a r programming model optimizes a system based on a conceptual framework i n which the stated objective i s the a l l o c a t i o n of land to i t s "highest and best use". Economic, highest and best use of land i s shown to exi s t when the a c t i v i t y bidding the highest price for the resource i s allocated the s i t e . i i U t i l i z i n g Lower Mainland Regional Planning Board land inventories and population projections, and Vancouver Real Estate Board p r i c e data, the s p e c i f i c a t i o n s of the l i n e a r programming model were formulated such that the a l l o c a t i o n of land uses r e s u l t s i n the maximization of the aggregate value of the regional land base. The model was used to analyse the a l l o c a t i o n of land i n the Lower Mainland of B r i t i s h Columbia for four time periods: 1966-1971, 1966-1976, 1966-1981 and 1966-1986. In each case, the pattern of land use associated with the maximum aggregate land value as determined by the model, shows from an economic standpoint^the most e f f i c i e n t possible use • i . of. land i n the whole region. The plans derived are normative i n that they show the pattern of land use development which should be followed to achieve optimum,land a l l o c a t i o n . The model, apart from being normative, also appears p o s i t i v e i n that i t shows the actual present day trend of continuing urban expansion onto r u r a l land. The p o s i t i v e aspects of the model were a t t r i b u t e d to the mechanism by which the r e a l estate market operates, and i t was hypothesised that by being to some extent p o s i t i v e , the model could be used f o r p r e d i c t i v e purposes. This s p e c i f i c model appears l i m i t e d i n i t s a p p l i c a b i l i t y on account of c e r t a i n problems of s p e c i f i c a t i o n and data a v a i l a b i l i t y , but the methodology i s considered to be a s i g n i f i c a n t advance on present land use planning concepts which lack e x p l i c i t economic c r i t e r i a and objectives applicable to land resource development. i i i TABLE OF CONTENTS Page LIST OF TABLES v i LIST OF FIGURES v i i Chapter 1. INTRODUCTION.... 1 i • • -i 2. PATTERNS OF LAND USE IN THE LOWER MAINLAND OF BRITISH COLUMBIA ••• 3 The Lower Mainland of British Columbia.. 3 The Development of Agriculture i n the Lower Mainland 4 The Urbanization of the Lower Mainland Region. 8 3. ' LAND USE PLANNING 13 The Objectives of Land Use Planning........ 13 Economics and Land Use Planning «, 14 Land Use Control 16 Values and Assumptions i n the Planning Process 20 The Effect of Land Use Controls on Land Values......................... 24 Discussion.. 25 i v Page 4. LINEAR PROGRAMMING AS AN ALLOCATIVE DEVICE AND ITS APPLICATION IN LAND RESOURCE ALLOCATION .28 The Linear Programming Model 28 Assumptions of the Linear Programming Formulation '. . 30 Solution of the Linear Programming Formulation 32 Post Optimal Analysis 34 Applications of Li.near Programming i n Land Use- Planning 35 Evaluation of the Technique 39 5. A LINEAR PROGRAMMING.MODEL FOR LAND -USE PLANNING... • 42 The Objective Function.... 43 " The Constraints 50 The Input Output C o e f f i c i e n t s . . . . 53 The Mathematical Model 54 De t a i l s of the Model................... . .. 55 6. THE DATA. . . 66 The Components of the Objective Function... 67 The Right Hand Side Elements..... 69 The Input-Output C o e f f i c i e n t s 70 7. THE RESULTS 73 The Optimum Solution 73 The Shadow Prices 79 ] V Page 8. DISCUSSION 08 The Optimum Land Use Plans 88 The Derived Shadow Prices of the Constraints 90 General Considerations 97 9. SUMMARY AND CONCLUSIONS 103 BIBLIOGRAPHY 1° 6 i .-APPENDIX....... 109 v i LIST OF TABLES Table Page 1.1 Classification of Farm Land for Census Division 4, B.C 7 1.2 Distribution of Population in the Lower Mainland of British Columbia 9 7.1 Cumulative Increase in Urbanized Area - Projected as Occuring Between 1966 and the Dates Shown, 1971-1986.... 74 7.2 Area of Land in Agricultural Uses Projected for 1971-1986 and Compared to 1966. 75 7.3 Increase i n Urbanized Area by Municipality or Area, 1966-1971 80 7.4 Increase i n Urbanized Area by Municipality or Area, 1966-1976 81 7.5 Increase i n Urbanized Area by Municipality or Area, 1966-1981 82 7.6 Increase in Urbanized Area by Municipality or Area, 1966-1986 83 7.7 Derived Shadow Prices of Established Urban Lots i n Designated Years, 1971-1986 84 7.8 Derived Shadow Prices of Developing Urban Lots i n Designated Years, 19 71-1986 85 7.9 Derived Shadow Prices of Acreage Rural Areas i n Designated Years, 1971-1986 86 7.10 Derived Shadow Prices of Lowland Rural Areas in Designated Years, 1971-1986 87 7.11 Derived Shadow Prices of Population i n Designated Years, 1971-1986 87 v i i LIST OF FIGURES Figure Page 1 I l l u s t r a t i o n of the Pattern of Change i n Total and Improved Farm Area i n Census D i v i s i o n 4, B.C., 19 21-1966 5 2 I l l u s t r a t i o n of the Relationship between Economic Rent and Land Use-Capacity. 45 3 I l l u s t r a t i o n of the Relationship between Economic Rent and the A l l o c a t i o n of Land to Four Competing Uses 47 4 Projected Increase i n Total Urbanized Area, 1966-1986....... 77 5 Projected Increase i n Urbanized Area by Zoning. C l a s s i f i c a t i o n , 1966-1986 78 v i i i Acknowled gement s This study was made possible by assistantships awarded by the Department of A g r i c u l t u r a l Economics of the Univ e r s i t y of B r i t i s h Columbia. P a r t i c u l a r acknowledgement i s offered to Dr.' P.L. Arcus, Assistant Professor, Department of A g r i c u l t u r a l Economics, f o r suggesting the nature of the research, f o r assistance, encouragement and supervision during i t s progress. The author wishes to express gratitude to the Committee and the members and s t a f f of the Department of A g r i c u l t u r a l Economics. Thanks are extended to members of the Lower Mainland Regional Planning Board, New Westminster, who provided, much of the data. 1 Chapter 1 INTRODUCTION The uses to which a region a l l o c a t e s i t s land, i s an important factor • i n determining the well-being of the inhabitants of the region. The P h y s i c o c r a t i c school of economic thought, r e f l e c t i n g the peasant economies of Europe i n the l a t e seventeenth and early eighteenth centuries, took the p o s i t i o n that a g r i c u l t u r e alone produced the wealth of the community, and therefore there was no better use to which the land resources of a country could be a l l o c a t e d -than t o , a g r i c u l t u r a l purposes. i The vast increases i n population growth and the associated i n d u s t r i a l revolutions i n many countries of the world, have, however, brought rapid changes i n the uses to which land resources can be a l l o c a t e d . Land i s now used for i n d u s t r i a l , commercial, and r e s i d e n t i a l uses as w e l l as for a g r i c u l t u r a l purposes. This increase i n the number of p o t e n t i a l uses to which land may be allocated.has resulted i n discussion and i n some cases, argument, over the competition between the d i f f e r e n t uses f o r the land resources, and the manner i n which the land resources should be employed. Following from t h i s discussion, i t has often been suggested that i t would be i n the best i n t e r e s t s of society as a whole to plan the use and development of i t s land resources i n a r a t i o n a l manner. use and This thesis examines the concepts and development of a regional land resource r a t i o n a l e of the planning of the base. A r a t i o n a l economic land 2 use planning model i s derived f o r land resource a l l o c a t i o n purposes. The t h e o r e t i c a l model i s applied to a land resource a l l o c a t i o n problem i n the Lower Mainland of B r i t i s h Columbia, Canada. Solution of the model, to provide normative land use plans f o r t h i s region, i s by way of a maximizing l i n e a r programming technique. Evaluation of the u t i l i t y of the model, and the output i n planning resource use programmes, follows. The evaluation examines the extent to which the concepts and the model overcome the apparent weaknesses of other land use planning techniques by providing a more consistent basis for programming.„resource a l l o c a t i o n . 3 Chapter 2 PATTERNS OF LAND USE IN THE LOWER MAINLAND OF BRITISH COLUMBIA Introduction The Lower Mainland of British Columbia has developed rapidly since the mid 1800's. From a sparsely populated undeveloped area, i t has evolved into a complex economic region with a population in excess of a million persons in 1970. Associated with this rapid development there has been a vast reorganization of the land base of the region. This chapter examines: the pattern of change in land allocation in the Lower Mainland region, which has taken place and s t i l l is taking place. The format for the remainder of this chapter involves a brief examination of f i r s t l y , the development of agricultural land use in the Lower Mainland region, and secondly, the growth of the population of the region and the associated urbanization processes. Before examination of these topics, however, a geographical » description of the region i s appropriate. 1. The Lower Mainland of British Columbia The Lower Mainland of British Columbia consists of a portion of the south-west mainland of British Columbia. The area covers some 1790 square miles and is bounded by the international boundary with the United States on the south-west, and the North Cascade mountain range on the south-east. On it s northern side, the area is bounded by the Coastal Moutains which run East to West. The area consists l a r g e l y of a p l a i n which widens from about a mile or two at Hope at i t s eastern end, to some 16 - 20 miles wide at the western end - some 90 miles down the Fraser River. The p l a i n i s subject to periodic flooding i n many areas, and this r e s t r i c t s i t s p o t e n t i a l uses. The climate i s mild with a mean annual temperature of 50°F and a heavy annual r a i n f a l l ranging from 35" to 100". 2.- The Development of A g r i c u l t u r e i n the Lower Mainland Region The f i r s t attempt at systematic farming i n the Fraser Valley began i n the early 1860's soon a f t e r the discovery of gold i n the Fraser River. In 1862 and 1863 the Chilliwack and Sumas Valley areas attracted many farmers. Shortly afterwards, a g r i c u l t u r a l settlements began at Mud Bay i n Surrey, at Mission, Maple Ridge, P i t t Meadows, and the Nicoma Island."*" A g r i c u l t u r e developed r a p i d l y i n the region from that time, with the impetus being created by i n d u s t r i a l i z a t i o n and high rates of migration into the region providing a ready market f o r a g r i c u l t u r a l products. This phase continued u n t i l 1941. Figure 1 and Table 1.1 i l l u s t r a t e the trend. Study of the Figure indicates that the t o t a l area of occupied farm land increased between 1921 and 1941 by 14 per cent. An increase i n the area of improved land of 37.6 per cent took place over the same period. G.B. White, "The Development of the Eastern Fraser V a l l e y " , B r i t i s h Columbia  H i s t o r i c a l Quarterly, Vol. XII, 1948. - 5 -Figure 1 Pattern of Change i n T o t a l and Improved Farm Area, Census D i v i s i o n 4, B.C. 1921-1966 Farm Area (Acres) 1951 1961 1966 Source: Dominion Bureau of S t a t i s t i c s , Census of Canada f o r the years shown. 140,000 1921 6 However, 1941 marked the peak of a g r i c u l t u r a l land use i n the Fraser V a l l e y . This i s shown i n Figure 1 and elucidated i n the f i r s t row of Table 1.1. From i t s 1941 high of 330,259 acres, t o t a l farm area dropped to 256,235 acres i n 1966. The d i f f e r e n c e i n area of occupied farm land between the years 1941 and 1966 i s accounted for by land which has been put to other uses. These other uses are: i n d u s t r i a l uses, r e s i d e n t i a l development, and i n s t i t u t i o n a l uses. Furthermore, some has regressed to bush. Whereas i t has been pointed out that the t o t a l farm area has s u b s t a n t i a l l y decreased since 1941, i t can s i m i l a r i l y be observed i n Table 1.1 that the Improved Land area has not s u b s t a n t i a l l y changed. Furthermore, xv'hile i t i s true to say that a g r i c u l t u r a l land i n the Valley has been " l o s t " to a g r i c u l t u r e , i t i s also appropriate to point out that the remaining land has been used more i n t e n s i v e l y . There has, not unnaturally, been strong p u b l i c opinion voiced against the conversion of a g r i c u l t u r a l land to other uses. I t i s claimed that the region r e l i e s upon i t s a g r i c u l t u r a l sector to provide a s u b s t a n t i a l amount of employment and wealth i n the regional economy. Phrases such as "respect for a g r i c u l t u r a l land" or, "the bona f i d e farmer knows that land i s a tru s t from one generation 2 to another" have been used i n expressing disapproval of such land use changes. V.J. Parker, "Problems and Progress i n R a t i o n a l i z i n g the Use of Resources of the Fraser V a l l e y , " The Lower Fraser V a l l e y : Evolution of a C u l t u r a l Landscape, ed." A.H. Siemens (Vancouver: Tantalus Research, 1966), pp." 163-171. 7 TABLE 1.1 C l a s s i f i c a t i o n of Farm Land for Census D i v i s i o n 4, B.C.* 1921 1931. 1941 1951 1956 1961 1966 Total Farm Area 289,687 316,472 Improved Land Area 142,583 172,813 Acres 330,259 304,291 196,195 202,089 294,033 204,016 274,588 198,458 256,235 202,096 * Note: Census Divsion 4, B.C., includes a small area of land not included i n the Lower Mainland Regional Planning Board area. Source: Dominion Bureau of S t a t i s t i c s , Census of Canada, for the years shown. 3. The Urbanization of the Lower Mainland Region Associated with the economic development of the region, there has been a rapid increase i n the population of the area. Table 1.2 shows that the population of the Lower Mainland increased from 249,331 i n 1921 to 1,005,657 i n 1966, - an increase of over 400 per cent.. This increase has created a heavy demand on land for i n d u s t r i a l , i n s t i t u t i o n a l , r e s i d e n t i a l and transportation purposes. Accurate estimates of land area which has been urbanized are unavailable. However, the Lower Mainland Regional Planning Board estimated that 3 by 1966, 106.5 square miles of the land area had been urbanized. In i t s e l f , however, t h i s f i g u r e gives no i n s i g h t i n t o the patterns of urbanization, the reasons for the s p a t i a l form and i n t e r a c t i o n of the present urban uses, or the past or present trends i n urban land use i n the Fraser Valley. A more p o s i t i v e exposition of the trends and patterns of the urbanization of the region i s given i n Howell-Jones' work. Using p o s t a l data for the area from Confederation u n t i l 1964, he demonstrates the evolution of s p a t i a l i n t e r a c t i o n 4 patterns associated with urbanization. The r a t i o n a l e underlying t h i s technique i s that p o s t a l revenues can be used to determine the c e n t r a l i t y (and thus, i n e f f e c t , the f u n c t i o n a l importance as a s e r v i c e centre,) of a settlement. Further-more, th i s enables a ranking of service centres i n order of importance, i n the area. Thus a h i s t o r i c a l study of t h i s type i s a most l u c i d and convenient way i n which to demonstrate the dynamic nature of settlement processes over time. Lower Mainland Regional Planning Board, Personal Communication (1970). G.I. Howell-Jones, "The Urbanization of the Fraser V a l l e y , " The Lower Fraser V a l l e y : Evolution of a C u l t u r a l Landscape, ed. A.11. Siemens (Vancouver: Tantalus Research, 1966), pp. 139-161. 9 TABLE 1.2 D i s t r i b u t i o n of Population i n the Lower Mainland of B r i t i s h Columbia 1921 - 1966 1921 1931 1941 1951 1956 1961 1966 Metropolitan Area-*- 222 ,294 337,218 394,588 562,048 655,110 790,259 892,384 Valley Area2 27 ,037 34,101 45,464 74,500 87,873 103,885 113,273 Total Lower Mainland 249 ,331 371,319 440,052 636,548 752,983 894,144 1,005,657 Vancouver City and environs. 2 Fraser V a l l e y excluding Metropolitan area. For d e t a i l s of the m u n i c i p a l i t i e s included i n each area, see p. 51 Source: Lower Mainland Regional Planning Board, "Population Trends i n the Lower Mainland" (New Westminster, B.C. 1966). 10 The e a r l i e s t business centres i n the region were a g r i c u l t u r a l and other primary product processing centres, but were extremely small. With the westward expansion of the Canadian P a c i f i c railway, economic expansion began i n the Lower Mainland. The population of the Vancouver - New Westminster sub area began to grow r a p i d l y , and t h i s i n i t i a t e d an expansion of a g r i c u l t u r a l production i n the Fraser V a l l e y . In turn the a g r i c u l t u r a l industry became i n c r e a s i n g l y dependent upon Vancouver as a market. This phase, therefore, marked the beginning of the metropol i t a n i z a t i o n (defined as fu n c t i o n a l concentration of a c t i v i t y ) of the Vancouver area. Over the following f o r t y years t h i s f u n c t i o n a l concentration of the Vancouver area was expanded and consolidated. This was, apparently, the r e s u l t once again, of improved communications and modes of transport.^ This did not, however, have the same impact on the Valley service centres although there was some consolidation of service functions i n some centres which expanded at the expense of some neighbouring centres. Those which expanded were mainly on the f r i n g e of the Vancouver - New Westminster complex. The r e s i d e n t i a l development i n the Lower Mainland region during t h i s period was concentrated i n the already established centres with an " i n f i l l i n g " process taking place. Those f r i n g e centres which expanded also experienced increased r e s i d e n t i a l development, since the increased m o b i l i t y of a larger s e c t i o n of the population gave these centres another function - namely that of being -dormitory areas f o r the Vancouver area. Thus, by the end of the 1930's, the region exhibited a h i e r a r c h i c a l arrangement of service centres. Howell-Jones, o p . c i t . , p. 154. 11 From 1940 u n t i l the present day, a d i f f e r e n t phase i s discernable. I t began with an expansion of r e s i d e n t i a l population i n t o the Valley centres. The service functions of these dormitory areas did not keep pace with the r e s i d e n t i a l function and the dependence of these areas upon the metropolitan area became more marked over the early years of t h i s period. Simultaneously, atrophy of the small centres occurred.^ In e f f e c t , the economic, transport, and technological changes were responsible for the development of a dominant metropolitan area and an accompanying i n t e r urban r e s i d e n t i a l complex. Since 1950 the boundary of the metropolitan f r i n g e has moved from j u s t west of Cloverdale to east of Abbotsford - a distance of some twenty-five miles. However, during the decade of the 1960's, this a r e a l expansion of the metropolis r e s t r i c t e d as i t i s to a narrow band of eastwards development, has led to a resurgence of growth of the f u n c t i o n a l growth of l o c a l s e r v i c e centres to supply t h e i r expanded h i n t e r l a n d s . Consequently the dominance of the service functions of the metropolis i s being challenged. This i s not unexpected. In f a c t when viewed i n the l i g h t of the " g r a v i t y concept" of urban and r e g i o n a l growth i t i s e n t i r e l y p r e d i c t a b l e . This concept recognizes that future development of a region and i t s i n t e g r a l sectors and functions, i s c l o s e l y r e l a t e d to the p r e v a i l i n g g r a v i t a t i o n a l influence of sub areas i n a t t r a c t i n g migrants and industry. This i n . t u r n i s r e l a t e d to the present rank of the sub area as a service centre, and i t s distance from other sub areas.^ Acknowledgement of the gravity concept as a factor i n r e g i o n a l evolution reinforces the view that the system i s a dynamic one. Therefore, the future of the c e n t r a l metropolitan area, Howell-Jones i b i d . , p. 159° Klassens, E 0 , "Theories of Settlement Evolution", (unpublished paper, Dept. of Geography, U n i v e r s i t y of B r i t i s h Columbia, 1968). 12 and the whole Lower Mainland area i s rel a t e d to the a b i l i t y of the c e n t r a l core of the metropolitan area to compete for importance as a service centre as the population of the region grows. 13 Chapter 3 LAND USE PLANNING Introduction Land use i s the term commonly employed to refer to the s p a t i a l d i s t r i b u t i o n of land based functions within a region - i t s r e s i d e n t i a l communities or l i v i n g areas, i t s . a g r i c u l t u r a l , i n d u s t r i a l , commercial and r e t a i l business d i s t r i c t s or major work areas and i t s i n s t i t u t i o n a l and l e i s u r e time functions. Land use planning, therefore, can be interpreted as the design and c o n t r o l of the s p a t i a l d i s t r i b u t i o n of the land based functions i n a region or a country. In making an evaluation of land use planning, both g e n e r a l l y a n d i n p a r t i c u l a r i n the Lower Mainland region of B r i t i s h Columbia, Canada, i t i s imperative to thoroughly examine the r a t i o n a l e behind the concept. The objectives of land use planning, and the means by which i t i s hoped to achieve these objectives must be s i m i l a r i l y probed, enunciated, and evaluated. The examination and evaluation which follows w i l l be made i n an economic context and wherever possible an attempt w i l l be made to r e l a t e the planning' function to economic p r i n c i p l e s . 1. The Objectives of Land Use Planning Every plan must necessa r i l y have objectives.. If an idea or concept has no objective, then i t cannot be reasonably c a l l e d a plan. Therefore, i n t h i s i n v e s t i g a t i o n , i t i s reasonable to question the objective of land use planning and enunciate the r a t i o n a l e . B a s i c a l l y , the concept on which the whole notion 14 of Land Use Planning r e s t s , i s that land should be employed i n i t s "highest and best use". A c o r o l l o r y to this i s that when land resources are employed i n t h e i r highest and best use, then the owners of the land, or society i n general, w i l l be assured of optimum returns from these resources.^" At f i r s t sight these statements appear simple and easy to apply to any given s i t u a t i o n . However, an inherent d i f f i c u l t y i n a p p l i c a t i o n of th i s t h e o r e t i c a l base a r i s e s i n assessing the returns from the land resources. This r e s u l t s from the c r i t e r i a used during assessment of these returns. The c r i t e r i a may be s t r i c t l y monetary terms, 2 i n t a n g i b l e s o c i a l values, or a combination of both. The r o l e played by the values of the decision-makers ( i . e . , the body vested with the power to d i r e c t the course of land use change i n an area,) i n formulating such c r i t e r i a w i l l be examined l a t e r i n the se c t i o n . For the present, however, the factors involved i n the use of the monetary - economic - c r i t e r i a must receive an in t r o d u c t i o n . 2. Economics and Land Use Planning Economics has been aptly described by Reynolds, as being concerned with; "The objective expression of the subjective wants of i n d i v i d u a l s and the a l l o c a t i o n of scarce resources to meet them, using money mainly as a measuring rod."3 In planning, the economic system serves f i r s t l y as a means of expressing the wants of consumers and r e c o n c i l i n g them with the s c a r c i t y of resources a v a i l a b l e f o r t h e i r s a t i s f a c t i o n . Secondly, i t provides a l o g i c a l and quantitative ^ R. Barlowe, Land Resource Economics (Englewood C l i f f s , N.J.: Prentice H a l l , 1958). 2 " H. Ben-Shahar and others, "Town Planning and Welfare Maximization", Regional Studies, Vol. 3, (1969), pp. 105-113. 3 D.J. Reynolds, Economics, Planning and T r a f f i c (London: I n s t i t u t e of Economic A f f a i r s , 1966), pp. 15-16. 15 context with which to examine the concepts or evaluate the relevant v a r i a b l e s . Furthermore, i t i s claimed that by taking f u l l e r account of economic a n a l y s i s , planning ideas are more l i k e l y to be acceptable to those responsible f o r 4 a l l o c a t i n g resources to them. Thus, the foundations of the economic system provide those responsible for decision making with an i r r e p l a c e a b l e t o o l . In the c l a s s i c a l l a i s s e z - f a i r e economy, i . e . , an economy i n which there i s no governmental co n t r o l or r e s t r i c t i o n of the economic actions of i n d i v i d u a l s , i t i s assumed i n d i v i d u a l s w i l l attempt to use t h e i r land resources f o r the purposes which promise the highest monetary r e t u r n . T h e sum of the actions of the i n d i v i d u a l s , therefore, t h e o r e t i c a l l y r e s u l t s i n land being put to i t s highest and best use as measured i n terms of economic e f f i c i e n c y . R a t c l i f f e expounds t h i s theory, r e f e r r i n g to urban land, when he says:-"The structure of the c i t y i s determined by the d o l l a r evaluation of the importance of convenience. Each a c t i v i t y seeks to minimize the d i s u t i l i t i e s and costs of f r i c t i o n by l o c a t i n g where i t s transport costs are at a minimum. Each one must be w i l l i n g to pay.site rent up to an amount which added to the transportation cost i s j u s t less than the t o t a l of transport costs added to s i t e r e n t a l f o r a l t e r n a t i v e l o c a t i o n s . Assuming perfect competition, each s i t e thus becomes occupied by the a c t i v i t y which can use i t most e f f i c i e n t l y . 6 The highest and best use of any p a r t i c u l a r s i t e i s subject to change, and the dynamic pattern of land uses i s explained by R a t c l i f f e as follows:-"The a t t r i b u t e s of each l o c a t i o n , however, are dependent on the nature of the occupants of every other l o c a t i o n and bidding f o r a l l s i t e s i s not simultaneous. Hence, there i s a continuous 4 Reynolds, op. c i t . , pp. 108-109. ^ R.T. E l y , Outlines of Land Economics, II (Ann Arbor, Michigan: Edwards Bros, 1922), p. 15. ^ R.V. R a t c l i f f e , The Metropolis i n Modern L i f e , (New York: Fisher Co., 1955), p. 127. 16 s h i f t i n g of land uses as l o c a t i o n a l a t t r i b u t e s vary with changes i n the occupancy of other l o c a t i o n s . " 7 Furthermore, the pattern of land use w i l l be affected by the changing demand for land with d i f f e r i n g p o t e n t i a l i t i e s . For example, population growth i n an area w i l l " d i c t a t e " that extra land to be put to r e s i d e n t i a l use. Thus, t h e o r e t i c a l l y , i n such a l a i s s e z - f a i r e economy, the market would solve i t s own problems of resource a l l o c a t i o n and there would be no need f o r any i n t e r v e n t i o n i n order to secure the highest and best use of land, (viewed i t must again be remembered, from an economic standpoint). This echoes S i r Arnold Plant's argument that, "There i s no index of the r i g h t use of land which i s a s a f e r , le s s ambiguous guide to the community's wishes than the open market t e s t . " 8 However, there are two sides to every argument, and the contrary opinions which hold that some form of c o n t r o l and zoning i s necessary to achieve the highest and best use, must be heard and evaluated. 3. Land Use Control Under the system of property r i g h t s i n Canada, the owner of land has considerable freedom to decide how he w i l l employ his land resources. However, these r i g h t s although exclusive are not absolute, and public action to d i r e c t land use p r a c t i c e s on p r i v a t e property can be invoked by the sovereign power of 7 I b i d . , p. 128. S i r Arnold Plant, quoted i n N. L i c h f i e l d , The Economics of Planned Development, (London: Estates Gazette, 1958), p. 309. 17 governments, i n the i n t e r e s t s of public health, safety and general welfare. Land use c o n t r o l , therefore, i s the process whereby these p r i v a t e property r i g h t s r e l a t i n g to land ownership are constrained by governmental power. Several d i f f e r e n t methods of enforcement of t h i s governmental p o l i c e power are prevalent. In e f f e c t , however, these measures can a l l be c l a s s i f i e d under the general heading of zoning. Zoning i s the process whereby a land resource i s divided into sub sections i n which c e r t a i n s t i p u l a t e d standards p e r t a i n i n g to use of the resource must be met. Some zoning regulations are enacted to prevent use of the land i n a manner not i n keeping with i t s present, predominant use or uses. Other zoning regulations which are commonly used, enforce b u i l d i n g codes, s u b d i v i s i o n p o l i c i e s and health standards. In sum, the enforcement of zoning controls are designed to regulate - "performance standards 9 that protect the best i n t e r e s t s of both the property owner and the p u b l i c . " (a) Purpose of Land Use Control There are two views of what land use control should attempt to do. Th f i r s t states that control of urban space i s the prerogative of the r e a l estate market since i t r e f l e c t s the sum t o t a l of the whims of s o c i e t y , and that the powers of zoning vested i n the a u t h o r i t i e s are merely a regulatory t o o l with which to smooth out imperfections. Davis has commented i n t h i s d i r e c t i o n : -" I t i s abundantly clear the the purpose of zoning i s to 'adjust' or 'correct' the working of the p r i c e mechanism i n the urban property market." x u ^ Barlowe, op. c i t . , p. 465. O.A. Davis, "The Economics of Municipal Zoning" (unpublished Doctor's d i s s e r t a t i o n , U n i v e r s i t y of V i r g i n i a , 1964), p. 67. 13 It i s not, however, made e x p l i c i t by Davis or by R e i n e r ^ , (who also enunciates t h i s view) as to what i s to be adjusted or corrected. The opposing camp hold that c o n t r o l and planning must constitute a complete market replacement, and as such must function as an instrument f o r the scheduling of goals and the means to reach them. This view stems from the premise that the market i s an i r r e l e v a n t , inappropriate and imperfect method for arranging and c o n t r o l l i n g the use of space, since i t merely sums the actions of i n d i v i d u a l s acting i n s e l f i n t e r e s t . Therefore the i n t e r e s t s of the community as a whole may be ignored and go u n s a t i s f i e d . ^ 2 Thus, s u b s t i t u t i o n for the o r i g i n a l pattern of land market values, by a new pattern which evolves through the imposition of zoning regulations, i s the aim of th i s school of thought. (b) Evolution of Land Use Control The early h i s t o r y of North American land, use con t r o l regulations r e f l e c t s the adjustment process from r u r a l communities to enlarged urban communities. Thus, i t i s seen that the r u r a l a t t i t u d e of i n d i v i d u a l i s m was c a r r i e d over to the c i t i e s , and the view of zoning was based on the view of govern-ment as umpire. Hence the courts, r e f l e c t i n g the times, accepted zoning as i t could be j u s t i f i e d by the benefits to the i n d i v i d u a l members of the community. General welfare was interpreted to mean health and safety for the residents of a community. T.A. Reiner, Journal of the American I n s t i t u t e of Planners (March 1966), p. 115. 1 2 J.B. Ukeles, The Consequences of Municipal Zoning (New York: Urban Land Economics Research I n s t i t u t e , 1966), p. 23. Ukeles, op. c i t . , p. 23. 19 However, today the emphasis has moved i n two d i r e c t i o n s . F i r s t l y , the r a t i o n a l e has s h i f t e d to giving prime consideration to the community i n t e r e s t and secondly, land use c o n t r o l has been directed towards r a t i o n a l i z a t i o n of the use of land space i n an attempt to promote a pattern which i s best suited to 14 the t e r r a i n , the economic r e l a t i o n s h i p s and s o c i a l structures of the community. (c) The Goals arid Objectives Formulation of adequate and e f f e c t i v e goals for planning land use i s by no means an easy task. The need for goals of some sort i s agreed upon, but the precise nature of the formulation process - i . e . , what the goals should r e f l e c t and do, and how they are to be used, i s not c l e a r . This had led Willhelm to say: "Statements of goals i n plans and zoning ordinances are usually extremely general, and are derived from a vague sense of purpose rather than from c a r e f u l study and analysis."15 The goals of a plan r e f l e c t the desired future state of l i f e i n the region as envisaged by the c o n t r o l l i n g (planning) commission, and the regulations perpetrated thereby r e f l e c t the standards which are required i n order to reach t h i s end st a t e . Thus, as Ukeles points out, the fundamental objective, of zoning i s to: "...cause the pattern of structures, a c t i v i t i e s and persons d i s t r i b u t e d i n the area to approximate the pattern suggested or prescribed by the zoning ordinance. Therefore, the achievement of t h i s pattern requires that c e r t a i n processes of change be hal t e d , slowed down, or redire c t e d and that others be maintained.""^ 14 15 16 S.M. Willhelm, Urban Zoning and Land Use Theory (New York: Free Press of Glencoe, 1962), p. 165. Ib i d . , p. 87. Ukelesj op. c i t . , p. 26. 20. If the evaluation of a plan and an assessment of how far i t has met i t s goals and objectives i s to be of any worth, then the c r i t e r i a of assessment must be framed i n terms of these s p e c i f i c processes. This i s seldom the case at present. 4. Value Systems and Assumptions i i i the Planning Process The purpose and objectives of any plan, land-use or otherwise, are the product of the de c i s i o n makers, i . e . , planner's, value system. A value system i s an i n t e r - r e l a t e d set of concepts which i n d i v i d u a l s u t i l i z e to decide what i s to be desired, negated, nondesired, or what i s to be considered appropriate or inappropriate.''"^ The s i g n i f i c a n c e of t h i s statement would appear to require l i t t l e e laboration. However, i t i s of importance to c l a s s i f y the general o r i e n t a t i o n which planner's values assume. Expressions by decision makers i n d i c a t e the presence of numerous values. Kluckhohn has c l a s s i f i e d the s o c i a l value systems which appear to be 18 basic f o r the d e c i s i o n makers to a r r i v e at zoning decisions, i n t o three s e t s . These are: (a) The Economic - P r o t e c t i v e Value o r i e n t a t i o n (b) The I n d i v i d u a l - C o l l e c t i v e Value o r i e n t a t i o n . . -(c) The Present - Future Time Value o r i e n t a t i o n Willhelm, op. c i t , , p. 99. C. Kluckhohn, "Values and Value Ori e n t a t i o n i n the Theory of Action", Towards a General Theory Of Action, eds. T."Parsons and E. S h i l s (Cambridge: Harvard University Press, 1954), pp. 409 - 412. 21 (a) The Economic - Pr o t e c t i v e Value Orientation Of the three value sets t h i s i s the most important for the d e c i s i o n maker. I t i s alleged that the Economic - Pr o t e c t i v e value set dominates the zoning process. Certain d e c i s i o n makers espouse economic values while others resort to protective values. Those de c i s i o n makers holding to the economic - value o r i e n t a t i o n , approach zoning by arguing that land use must be dictated by the market s i t u a t i o n , on the basis of supply and demand, and that the market must be allowed to determine land values unhindered. The c o r o l l a r y to t h i s l i n e of thinking i s that the optimum use of land i s that which y i e l d s the highest investment returns. A contrary opinion i s held by those with a p r o t e c t i v e - value -o r i e n t a t i o n . Here the view i s that c o n t r o l of land use by non market forces i s necessary since land use patterns i n themselves are not a true r e f l e c t i o n , . . 19 of economic forces. (b) The I n d i v i d u a l - C o l l e c t i v e Value Orientation Those who hold d i s t i n c t l y . i n d i v i d u a l i s t i c values put forward the viewpoint that the p u b l i c welfare i s ultimately served as each i n d i v i d u a l forwards h i s own s e l f - i n t e r e s t s . C r u c i a l to t h i s i d e a i i s the notion that the summation of each pr i v a t e increment consequently y i e l d s the best r e s u l t s , and t h i s i s the welfare function that w i l l best serve the c o l l e c t i v e i n t e r e s t . The presence Willhelm, op. c i t . , pp. 95 - 96. 22 of a c o l l e c t i v e agent i s seen as unnatural. Since control over land use resides i n personal ownership of property, d i r e c t interference through c o l l e c t i v e representation challenges p r i v a t e welfare notions and eventually destroys c o l l e c t i v e i n t e r e s t s . On the other hand, de c i s i o n makers whose value o r i e n t a t i o n i s c o l l e c t i v e , advocate the existence of a normative system designed to promote the c o l l e c t i v e welfare apart from i n d i v i d u a l a s p i r a t i o n s . Therefore, i n t h i s view, land use must conform to the s t i p u l a t i o n s of a given c o l l e c t i v e welfare concept. Thus, zoning must accord with established planning standards that are thought to 20 promote these c o l l e c t i v e i n t e r e s t s . (c) The Present-Future Time Value Orientation This approach i s t i e d very c l o s e l y to the economic - p r o t e c t i v e -value system. Those arguing from the economic appraoch tend to regard time as simply a dynamic force which i n i t i a t e s change i n land u s i n g ' a c t i v i t i e s : i t i s an i n e v i t a b l e development d i c t a t i n g s h i f t i n land use patterns r e s u l t i n g from the operation of economic fo r c e s . Thus, s h i f t s i n the economic forces w i l l bring about an automatic, and hence predictable pattern of land use development. Future time viewed from such a stance then, i s that period which i n i t i a t e s a 21 c e r t a i n developmental pattern i n accordance with economic conditions. I b i d . , pp. 117 -I b i d . , pp. 149 -119 152. 23 The opposing p r o t e c t i o n i s t viewpoint of future time span i s conceived as being the time yet to come during which land use change w i l l be brought about through co n t r o l measures, ( i . e . zoning in,-a manner complementary to protective values). The pr o t e c t i v e o r i e n t a t i o n , therefore, denies that time brings changes i n land use, but instead holds to the value that time i s e s s e n t i a l for accomplishin changes i n agreement with preservation of present or proposed land uses and needs. Thus, i n e f f e c t , such an o r i e n t a t i o n endeavours only to perpetuate e x i s t i n g land use patterns i n accordance with p r o t e c t i v e values with the i m p l i c i t desire that the future s h a l l not y i e l d a land use contrary to the pr o t e c t i v e value o r i e n t a t i o n of the present time. Thus, i n sum, the view of time held by those espousing a generally economic o r i e n t a t i o n does not perceive zoning as a p r e d i c t i v e instrument: rather, zoning according to this perspective merely mirrors those economic factors that determine land use i n the time yet to come. On the other hand, because p r o t e c t i o n i s t s maintain as part of t h e i r value o r i e n t a t i o n that zoning must determine - not r e f l e c t - land values, they endeavour to make zoning i t s e l f a p r e d i c t i v e device to f o r e t e l l the course and pattern which land uses w i l l assume 22 over txme. With the i m p l i c i t incorporation of such values i n the statement of the objectives of land use planning ordinances, i t i s possible to i d e n t i f y a basic element of paradox involved i n the r a t i o n a l e of land use c o n t r o l . On the one hand, c o n t r o l i s aimed at s o c i a l or community c o n t r o l of land areas by some i n d i v i d u a l s . On the other, c o n t r o l i s envisaged as a pr o t e c t i v e mechanism Willhelm, op. c i t . , pp. 156 - 158 24 for preserving the i n d i v i d u a l ^ property and amenity r i g h t s . This basic paradox i s s u c c i n c t l y enunciated by Ukeles when he states that: "Zoning today i s an uneasy amalgam of strongly i n d i v i d u a l and b a s i c a l l y communal i n t e r e s t s . A large amount of the c o n t r o l of space has passed from the i n d i v i d u a l to the community. However, the i n t e r e s t s , values and goals to be served by t h i s p ublic c o n t r o l are s t i l l to a large extent i n d i v i d u a l and private."23 5. The E f f e c t Of Land Use Controls on Land Values The e f f e c t s of land use c o n t r o l measures on the value of land i s extremely d i f f i c u l t to test or demonstrate, since land values are dependent on a great many var i a b l e s ranging from n a t i o n a l economic trends to l o c a l supply-demand r e l a t i o n s h i p s i n the land market. Nevertheless, the argument concerning the e f f e c t s of zoning on property values has been p a r t i c u l a r l y f i e r c e . On the one hand, i t i s claimed that by 24 i n h i b i t i n g c e r t a i n uses,property values w i l l be suppressed. No exact empirical evidence i s a v a i l a b l e to elucidate t h i s contention, but t h e o r e t i c a l l y the degree to which t h i s would be so would be r e l a t e d to the d i f f e r e n t i a l i n p r o f i t a b i l i t y between the use y i e l d i n g the biggest net return allowed under the regulatory conditions, and the most p r o f i t a b l e use under free market conditions. However, the opposite view i f also convincingly argued and the point i s made that the l i m i t a t i o n on property owners choice has only adverse e f f e c t s Ukeles, op. c i t . , p. 29. P.A. Benson and others, Real Estate P r i n c i p l e s and P r a c t i s e (Englewood C l i f f s , N.J.: Prentice H a l l Inc., 1954), p. 410. 25 25 on speculators. The Uthwatt Committee i n 1942 i n the United Kingdom argued very sensibly however, by saying that: "...wisely imposed planning control does not diminish the t o t a l sum of land values, but merely r e d i s t r i b u t e s theni by increasing the value of some land and by decreasing the value of other land." Thus planning controls create, destroy and s h i f t land values, i n eff ect superseding or at le a s t modifying the supply and demand mechanism of the land market. This would i n some instances appear to be contradictory to the aims of land use c o n t r o l . 6. Discussion The r a t i o n a l e of land use planning, i t s concept, objectives, values and means have been examined. In perspective the whole concept at present appears lacking i n i t s a b i l i t y to be of any use as a t o o l to p o s i t i v e l y a f f e c t r e g i o n a l or urban a c t i v i t y unless the objectives are reinfo r c e d by strong p u b l i c commitment to such p o l i c i e s . Furthermore, Ukeles has demonstrated an inherent i n a b i l i t y of zoning p r a c t i c e s to succeed i n t h e i r s p e c i f i e d goals, by showing that there i s c o n t r a d i c t i o n between the means of con t r o l and the fundamental 27 s o c i a l and economic aspects of land use. The basic weaknesses of land use con t r o l measures appear to be c l o s e l y re l a t e d t o : -(a) The f a i l u r e of planners to appreciate a l l the factors involved i n the u t i l i z a t i o n of land. 25 Ukeles, op. c i t . , p. 43. 26 The Uthwatt Report, Report of the Expert Committee on Compensation and  Betterment, Cmnd. 6386 (London: H.M.S.O., 1942). 27 Ukeles, op. c i t . , pp. 44 - 51. 26 (b) The f a c t that there e x i s t s no s a t i s f a c t o r y framework of theory with which to approach the dynamic aspects of land use change. (c) A f a i l u r e to develop c r i t e r i a which are s u f f i c i e n t l y precise to be of value i n formulating and guiding land a l l o c a t i o n d e c i s i o n s . With regard to the f i r s t and second weaknesses - (a), - the lack of the planner's appreciation of the factors involved i n land u t i l i z a t i o n , and (b), - the lack of a s a t i s f a c t o r y dynamic land use theory, i t should be noted that a major f a c t o r i n t h i s f i e l d i s the widely held opinion of the planners that the present shape and form of urban North America i s i r r a t i o n a l , and therefore formulation of an appropriate theory of the s p a t i a l form of these areas i s i r r e l e v a n t . . However th i s view i s rejected by McConnel who describes the process of d e c e n t r a l i z a t i o n of land uses and land values i n North American urban areas as being the c l e a r r e s u l t of open market forces r e s u l t i n g at l e a s t i n part from the problems of obsolete urban centres that have been unable to provide s a t i s f a c t o r y services f o r mobile consumers. He concludes that the d e c e n t r a l i z a t i o n of a c t i v i t i e s may w e l l be the most economic and s o c i a l l y s a t i s f a c t o r y answer to the 28 problems r e s u l t i n g from the problems of contemporary personal m o b i l i t y . The view that urban North America i s i r r a t i o n a l i n i t s land using a c t i v i t i e s , t y p i f i e s the predominant o r i e n t a t i o n , i . e . , the p r o t e c t i v e value o r i e n t a t i o n , encountered i n the planning f i e l d . In part t h i s may be due to an attempt on the part of the planners to compare urbanization i n North America S. McConnel, "Economic Aspects of Land Use Planning", O f f i c i a l Architecture  and Planning, V o l . 3, No. 2, (August, 1969), pp. 947 - 957. 27 and s i m i l a r l y densely populated areas i n other parts of the world, - p a r t i c u l a r l y Europe. In th i s author's opinion t h i s comparison i s inappropriate since the conditions under which the present patterns of land use developed i n Europe are d i s t i n c t l y d i f f e r e n t from the North American s e t t i n g . The t h i r d weakness ( c ) , i s the f a i l u r e to develop s p e c i f i c c r i t e r i a for formulating and guiding land a l l o c a t i o n decisions. Reference has been made e a r l i e r i n t h i s chapter to the f a c t that i n a planning s i t u a t i o n the economist can show how the market forces and the plans are r e l a t e d . I f t h i s was put in t o e f f e c t , a common denominator could be introduced into any plan. Furthermore, the f a c t that i n the f i n a l analysis the e f f e c t s of a proposed planning d e c i s i o n would be expressed i n d o l l a r terms, could have an important bearing on the vigour with which the decision makers examined the s i t u a t i o n . In e f f e c t the p r o f e s s i o n a l planners would have a consistent welfare function to maximize. This i n i t s e l f would be an improvement on the p r e v a i l i n g s i t u a t i o n where studies point to the absence of a consistent welfare functxon. This thesis attempts to r e c t i f y these defects with respect to land use planning i n the area of land i n the Lower Mainland region of B r i t i s h Columbia. ... 29 Reynolds, op. c i t . , p. 32. 30 N.J. Arrow, S o c i a l Choice and I n d i v i d u a l Values (New York: John Wiley Ltd., 1951). 28 Chapter 4 LINEAR PROGRAMMING AS AN ALLOCATIVE DEVICE, AND ITS APPLICATION IN LAND RESOURCE ALLOCATION Introduction It was stated at the end of the previous chapter, that i t i s the r e s p o n s i b i l i t y of economists to show how i n a given s i t u a t i o n , market forces and planning forces are r e l a t e d . Linear programming i s a mathematical technique which i f u t i l i s e d i n a c e r t a i n format may be applied to the economist's problem. This section therefore analyses the mathematical technique, i t s terminology, the basic s o l u t i o n of the method, i t s assumptions and l i m i t a t i o n s , and by a review of i t s a p p l i c a t i o n to problems to date, evaluates the a p p l i c a b i l i t y of l i n e a r programming to land use planning. 1. The Linear Programming Model Linear Programming i s a mathematical technique designed to enable analysis of the p o t e n t i a l i t i e s of alternate a c t i v i t i e s and to choose those that permit the best use of resources i n pursuit of a given o b j e c t i v e . I t has many uses. For example, i t can analyse a l l the factors of production and t r a n s l a t e i t s findings into minimum costs or maximum p r o f i t s for i t s user. Using t h i s method i t i s possible to a l l o c a t e , assign, schedule, s e l e c t or evaluate the a l t e r n a t i v e s which the l i m i t e d resources possess under a given set of circumstances.^ 1 E.O. Heady and W. Candler, Linear Programming Methods (Ames, Iowa: Iowa Uni v e r s i t y Press, 1958). 29 An economic problem to which l i n e a r programming may be applied, must contain the following components:-(a) An objective function which i s to be optimised - e i t h e r maximized or minimized. (b) A set of constraints which represent the l i m i t a t i o n s placed upon the objective function. (c) A set of a l t e r n a t i v e a c t i v i t i e s or production processes, Consider a production problem i n which i t i s desired to f i n d a production schedule Vj_ Vn, (where Vi Vn represent production processes or a c t i v i t i e s , ) which w i l l maximize the t o t a l income from the production system. Production of an a c t i v i t y r e s u l t s i n an associated return or income, designated Gi Gn. Furthermore assume that there i s a given f i x e d supply - B i - of the resource which i s used i n the production processes. Then mathematically, the problem can be represented as follows: Find a set of non negative numbers Vi......Vn which maximizes the l i n e a r function, | V i G J < « • Subject to the Conditions:-n ^ V j A ^ ^ B i for a l l i (2) and V i Vn ^ 0 30 where A-y = the amount of the i 1 - * 1 resource used to produce one unit of a c t i v i t y V j . for i = 1 — j = 1 " m (Footnote 2) For example, consider a problem where: m = 3 and n = 4, then i n matrix format the problem appears as follox^s: -Gl G 2 G3 G 4 V l v 2 v 3 v 4 A l l A 1 2 A13 A14 B A21 A22 A23 A24 B A31 A32 A33 A34 B The object i s therefore to maximize:-Z = + V 2G 2 + V3G3 + V 4G A (3) Subject to the following conditions:-V X A n + V 2 A 1 2 + V 3 . A 1 3 + V 4 A 1 4 ^ B X V X A 2 1 + V 2 A 2 2 + V 3 A 2 3 + V 4 A 2 4 ^ B 2 v l A31 + v2 A32 + v 3 A33 + v4 A34 ^ B 3 and V l f V 2, V 3, V 4 ^ 0 2. Assumptions of the Linear Programming Formulation (4) (5) (6) Although l i n e a r programming models allow an extremely thorough analysis of a l t e r n a t i v e s open to the planner, the rigorous mathematical framework imposes severe demands because i t i s associated with a r e s t r i c t i v e set of assumptions. 2 D. Gale, The Theory of Linear Economic Models (New York: McGraw-Hill, 1960), pp. 7 - 8. 3 H.G. Campbell, Matrices with Applications (New York: .Appleton Century Cr o f t s , 1968), p. 3. 31 Since many aspects and data encountered i n planning problems, preclude rigorous adherance to such assumptions, i t i s imperative that i f simple l i n e a r models are to be su c c e s s f u l l y employed, these assumptions upon which the technique i s erected, and the l i m i t a t i o n s imposed thereby, are adequately understood.^ (a) L i n e a r i t y and A d d i t i v i t y This i s the primary and most important assumption. The objective function and each constraint equation must be l i n e a r i n form. In many si t u a t i o n s t h i s may i •be u n r e a l i s t i c , e s p e c i a l l y with regard to costs. A d d i t i v i t y i s assumed with regard i ' - . . . . to the a c t i v i t i e s . An equivalent statement i s : the t o t a l amount of resources used by several enterprises must be equal to the sum of the resources used by each i n d i v i d u a l enterprise. Thus no i n t e r a c t i o n i s possible i n the amount of resources required per unit of output regardless of whether a c t i v i t i e s are produced alone or i n combination.~* (b) Returns to Scale The t e c h n i c a l c o e f f i c i e n t s , - the A-j^'s - are taken to be constant. Thus, constant returns to scale are assumed by the computational method. (c) D i v i s i b i l i t y s o l u t i o n i s not necessar i l y yielded by the computational to assume that f r a c t i o n a l parts of both resources and 4 R.J. Colenutt, "Building Linear P r e d i c t i v e Models for Urban Planning," Regional Studies, Vol. 2, (1968), pp. 139 - 143. 5 Heady and Candler, op. cit.., p. 17. Since an integer procedure, i t i s necessary 32 a c t i v i t i e s are p o s s i b l e . In i t s e l f however, the l i m i t a t i o n a l e f f e c t of t h i s assumption i s not too important, since "rounding o f f " the values shown i n the optimal plan can y i e l d a s o l u t i o n very close to the true optimum. (d) Knowledge of Parameters This assumption i s the most d i f f i c u l t to j u s t i f y . That i s ; perfect knowledge i s assumed with respect to input data, p r i c e s , resource supplies, and te c h n i c a l input - output c o e f f i c i e n t s . C e r t a i n l y t h i s may be a l i m i t i n g assumption, but i s one over which the researcher has very l i t t l e c o n t r o l . However, the usefulness of. the technique should be not reduced, since any other method of analysis i s subject to the same r e s t r i c t i o n s . 3• Solution of the Linear Programming Formulation In order to solve a l i n e a r programming problem, i t i s f i r s t necessary to transform the set of l i n e a r i n e q u a l i t i e s (of equations 4-6 above), into a system of l i n e a r e q u a l i t i e s . This i s done by introducing a further set of vectors, known as slack, (also known as di s p o s a l , s t r u c t u r a l , or l o g i c a l ) vectors. In the general case, where there are n r e a l a c t i v i t i e s and m constraints or rows, i t w i l l be necessary to introduce m slack vectors, - (one slack vector for each constraint row or i n e q u a l i t y , ) - i n order to transform the set of m i n e q u a l i t i e s into a set of m l i n e a r equations i n (n + m) unknowns. The s o l u t i o n of such a set of l i n e a r equations may be made by use of a standard set of mathematical procedures known as the Simplex Routine. This 33 6 7 routine has been adequately documented elsewhere, 5 so a b r i e f review w i l l s u f f i c e here. This Simplex Method i s based on the f a c t that i f there or rows i n the constant vector, then there i s a set of m columns vectors) which can be used to express any Right Hand Side . This c a l l e d a b a s i s . The Simplex Method works from one basis to another (by exchanging one column i n the basis with one column not i n the basis on each step or i t e r a t i o n , ) u n t i l a s o l u t i o n ( c a l l e d the basic f e a s i b l e solution) i s obtained that meets a l l the c r i t e r i a (apart from o p t i m a l i t y ) , i n c l u d i n g the requirements that a l l the column values be non-negative. A f t e r a basic f e a s i b l e solution, i s found the method proceeds by examining a s e r i e s of f e a s i b l e solutions using a c r i t e r i o n equation, to f i n d one that s a t i s f i e s the requirement that the value of the objective function be a maximum or a minimum. This i s c a l l e d the optimal s o l u t i o n . I t i s true however, that there i s not always an optimal s o l u t i o n to a l i n e a r programming problem. If there i s no s o l u t i o n i n non-negative v a r i a b l e s , or none that e x i s t s that keeps the v a r i a b l e s within the s p e c i f i e d constraints, the l i n e a r programming problem i s said to be i n f e a s i b l e . I f , on the other hand, a f e a s i b l e s o l u t i o n i s 6 Heady and Candler, op. c i t . ? pp. 53 - 108, are m constraints (variables or expression i s 7 R. Dorfmann and others, Linear Programming and Economic Analysis (New York: McGraw-Hill, 1958). : 34 found- but the constraint rows do not confine the value of the f u n c t i o n a l row to f i n i t e values, the problem i s said to be unbounded. If a s o l u t i o n i s found, the following output i s obtained: The optimal value of Z, (the objective function) expressed i n terms of the values or costs of the a c t i v i t i e s , i . e . Z = (V x . G x) + (V 2 . G 2) + (V 3 . G 3) + ( V 4 . G 4) (8) The i n d i v i d u a l acti.vity l e v e l s , i . e . V\ V 4 and values of the associated l o g i c a l or slack vectors. More information relevant to any p a r t i c u l a r problem can be derived, when desired, by post-optimal a n a l y s i s . 4. Post-Optimal Analysis Transformations may be executed on the objective function, or any s p e c i f i e d set of input-output c o e f f i c i e n t s or r i g h t hand side parameters, by post-optimal parametric programming. These analyses a s c e r t a i n the s t a b i l i t y of the optimum s o l u t i o n and the e f f e c t s of parametric transformations on the s o l u t i o n . The s t a b i l i t y of the parameters ( c o e f f i c i e n t s ) i s c r i t i c a l , since the c o e f f i c i e n t s are the predictors i n the model and w i l l r e f l e c t any errors that have accumulated through measurement or s p e c i f i c a t i o n . 8 a) b) Colenutt, op. c i t . , p. .142. 35 The s t a b i l i t y of the parameters i s measured by the range over which they may change without a f f e c t i n g the value of the objective function. Thus, i t i s possible to analyse r e l a t i o n s h i p s between v a r i a b l e s . 9 S i m i l a r l y , the s e n s i t i v i t y of the parameters i s of i n t e r e s t to the model b u i l d e r . The degree of s e n s i t i v i t y w i l l i n d i c a t e the v u l n e r a b i l i t y of the model to error, and w i l l provide clues about the nature of the re l a t i o n s h i p s i n the model, so that causal inferences can be made more confidently. I f the model i s very s e n s i t i v e to change i n the parameter values, i . e . i f the r e s u l t s produced by the model are s i g n i f i c a n t l y d i f f e r e n t i f marginal changes are made i n the value of the parameters, t h i s may in d i c a t e that a d d i t i o n a l v a r i a b l e s or re s t r u c t u r i n g of the problem are necessary. However, i f small changes make l i t t l e d i f f e r e n c e , then the model might be considered adequate. 5. Applications of Linear Programming i n Land Use Planning The a p p l i c a t i o n of l i n e a r programming models to problems i n planning f i e l d s has gained momentum during the l a s t ten years. In a g r i c u l t u r a l economics, problems of land a l l o c a t i o n , production and product d i s t r i b u t i o n have been increasingly subjected to examination by l i n e a r and l i n e a r s p a t i a l models.^' ^ 9 Ben Shahar et. a l . , op. c i t . , p. 113. E.O. Heady and H.H. H a l l , Linear and Non Linear S p a t i a l Models i n A g r i c u l t u r a l Competition, Land Use and Production P o t e n t i a l , Journal Paper J 5995 of Iowa A g r i c . and Home Econ. Exper. Sta., Project 1405, (1969). A.C. Egbert and E.O. Heady, Regional Analysis of Production Adjustments i n the Major F i e l d Crops, U.S.D.A. TecFirical B u l l . , No. 1294, (1964). 36 S i m i l a r l y i n the urban planning f i e l d , a l l o c a t i o n of land uses has been made using mathematical programming models. However, most of these attempts were oriented to fore c a s t i n g rather than to optimal planning of land u s e s . I 2 p o r example, a t h e o r e t i c a l model - the Herbert-Stevens model -prepared i n 1961, simply projects a pattern of r e s i d e n t i a l development.-^ Schlager, and Steger have provided s i m i l a r studies f o r t h e i r respective areas (South East Wisconsin and Pittsburgh).1^> x$ However, more soph i s t i c a t e d models have been evolved. The work of S i l v e r and Sloan pioneered the representation of zoning r e s t r i c t i o n s i n the l i n e a r programming formulation.16 McLaughlin advanced the l i n e a r programming technique i n planning recreation f a c i l i t i e s around Rochester, New Y o r k . l 7 By employing post-optimal techniques of analysis the trade o f f s between the variables and the constraints i n the problem were elucidated. Ben Shahar et. a l . have also investigated t h i s important feature of l i n e a r models.1^ Two 12 Ben Shahar, et. a l . , op. c i t . , pp. 105-113, 13 J.D. Herbert and B.H. Steven, "A Model for the D i s t r i b u t i o n of R e s i d e n t i a l A c t i v i t y i n Urban Areas", Journal of the Regional Science Assoc., Vol. 2, (1960) 21-36. 1^ K.J. Schlager, "Simulation Models i n Urban and Regional Planning", The S.E. Wisconsin Regional Planning Commission Technical Record, Vol. 2, No. 1, (1964). 15 W.A. Steger, "The Pittsburgh. Urban Renewal Simulation Model", Journal of  the American I n s t i t u t e of Planners, V o l . 31, No. 2,. (1965). 1^ A.L. S i l v e r and A.K. Sloan, "A Model Framework for Comprehensive Planning i n New York C i t y " , Journal of the American I n s t i t u t e of Planners, Vol. 31, No. 3, (1965). . - . . 17 J.F. McLaughlin, "Application of Linear Programming to Urban Planning" (unpublished M.V.P. d i s s e r t a t i o n , U n i v e r s i t y of I l l i n o i s , 1968). 18 Ben Shahar, et. a l . , op, c i t . , p. I l l , 37 further important features regarding the a p p l i c a t i o n of l i n e a r programming models to land use planning problems are that the s o l u t i o n indicates both the shadow prices of the relevant c o n s t r a i n t s , and the costs of deviation of the a c t i v i t i e s . The Shadow P r i c e of a Constraint The shadow p r i c e of a constraint i s a measure of how much of the t o t a l value of the objective function increases as a r e s u l t of a marginal r e l a x a t i o n of t h i s constraint.-'- 9 If the objective function i s expressed i n . . . . monetary terms the shadow prices i n d i c a t e by how much the d o l l a r value of the objective function increases or decreases as a r e s u l t of a r e l a x a t i o n or tightening the relevant constraints by one u n i t . It i s , therefore, a measure of the marginal opportunity cost of each constraint expressed i n d o l l a r terms. Many of these constraints are imposed by the planning authority, and therefore when the model i s used i n land use planning studies the knowledge of the shadow costs enables re-examination of these constraints and allows for t h e i r modification i f necessary. The more important^<!^th^ shadow prices involved i n a l i n e a r programming formulation of a land use plan are, f i r s t l y the shadow p r i c e of land, and secondly the shadow prices of the r e s t r i c t i v e design standards. The shadow p r i c e of land i n any area indicates the value of the marginal output of land i n each area. Naturally, i t i s to be expected that Ibid., p. 112. 38 t h i s value i s higher for c e n t r a l i z e d zones and lower for those areas on the periphery. The design standards under which planners operate i n an area are determined by the public a u t h o r i t i e s . For example these include maximum permissible density of b u i l d i n g , sizes of l o t s and the r e c r e a t i o n a l land requirement. The cost of these constraints i s derived from the shadow prices of land. These costs c l e a r l y c o n s t i t u t e a te s t on the effectiveness of the regulations and t h e i r marginal opportunity cost to the p u b l i c . If the shadow prices of these design.standards are low, i t i s apparent that the regulations are r e l a t i v e l y i n e f f e c t i v e and v i c e versa. Thus, for example, the s o l u t i o n provides the authority with information concerning the cost of incre a s i n g the area of parks by one acre or changing the maximum permissible b u i l d i n g density. Cost of Deviation A s e n s i t i v i t y analysis of the components of an optimal s o l u t i o n indicates the marginal increase or decrease i n the value of the objective function which r e s u l t s i f the plan deviates by one unit of a component from the optimum. This v a r i a b l e i s ref e r r e d to as "cost of dev i a t i o n " . Like the shadow prices of the con s t r a i n t s , "the cost of d e v i a t i o n " of the components provides s i g n i f i c a n t information for making r a t i o n a l decisions. For example, some measures of the r i s k inherent i n the plans can be deduced from the s o l u t i o n . By d e f i n i t i o n the plan showing a greater 39 s e n s i t i v i t y of t o t a l value to such deviations contains a greater element of r i s k . A d d i t i o n a l l y , the costs of deviation of each component can be used to assess the r i s k f a c t o r , since the higher the cost of deviation the greater the r i s k i n v o l v e d . ^ ' ^1 From the above discussion i t i s apparent that the i n t e r e s t i n the ap p l i c a t i o n of the l i n e a r programming technique i s being reorientated from simple p r e d i c t i v e aspects of planning towards c r i t i c a l a p p r a i sal and.reappraisal of planning p o l i c y . 6. Evaluation of the Linear Programming Technique Applied to Land Use Planning If the l i n e a r programming technique i s to be s u c c e s s f u l l y used i n land use planning i t i s imperative that one must be prepared to provide the input and to accept the assumptions of such a formulation. These assumptions have been s p e l l e d out above. Mention has been made of the d i f f i c u l t y i n rigorously adhering to such assumptions.' It i s true that i n many respects i t i s an o v e r s i m p l i f i c a t i o n to assume l i n e a r i t y i n the equation system. C e r t a i n l y with regard to costs, c o e f f i c i e n t s , design standards and p r i c e s , t h i s statement holds true. However, by post-optimally performing a s e n s i t i v i t y analysis on the d i f f e r e n t parameters, i t i s possible to i d e n t i f y and r e f i n e the c r i t i c a l elements. Furthermore, McLaughlin, op. c i t . Ben Shahar, et. a l . , op. c i t , , p. 112. 40 methods have been developed i n which economies of scale have been included 22 in t o l i n e a r programming models. Again, i t has been claimed that a l i n e a r programming model determines the optimum f or one point i n time with l i t t l e consideration of the period before or a f t e r , and that by doing so the models can only sub-optimize, whereas 23 planning aims to optimize the system as a whole. However, t h i s appears to be a conceptual problem associated with planning i t s e l f rather than the mathematical modelling system. The use to which the model i s to be put i s the key determinant of the degree of accuracy required, A highly accurate model may be necessary f o r some purposes. For others a l e s s e r degree of accuracy may be s u f f i c i e n t . The major contribution of l i n e a r models has been that by post-optimal analysis of the output and r e s t r u c t u r i n g of the problem, the planner may gain valuable i n s i g h t i n t o the form of the problem. Another valuable con t r i b u t i o n i s that i n p r a c t i c e the goals of the au t h o r i t i e s and planners have been assumed v a l i d and have been i m p l i c i t l y s u b j e c t i v e l y weighted without reference to p u b l i c c r i t e r i a , whereas i n a l i n e a r programming analysis the goals and the weights attached to the c r i t e r i a are e x p l i c i t l y stated i n the model and stand to be c r i t i c i s e d . H. Giaever and J . Seagraves, Linear Programming and Economies of Size, Journal of the A g r i c u l t u r a l Experimental Station, North C a r o l i n a . Paper 1026, (1959). M. M. Camina, "Plan Design Models - A Review", Town Planning, V ol. 1, (1969), pp. 119-130. 41 In summary, i t appears that the over r i d i n g advantages of l i n e a r models are that used properly they can work w e l l and they do force the model bu i l d e r and planner to think c a r e f u l l y about t h e i r problem. One cannot yet say that a s i n g l e run of a l i n e a r programming model w i l l provide the r i g h t answer. However, i t may be the development of the model, rather than the model i t s e l f , which i s valuable. Nevertheless, i n the development of a l i n e a r programming model, every s i n g l e exception cannot be considered i n d i v i d u a l l y and i t i s necessary to s i m p l i f y the problem. On t h i s point Camina has commented that c o n f l i c t may w e l l a r i s e between the s o c i a l s c i e n t i s t who appreciates that there are advantages to be gained by g e n e r a l i s a t i o n , and the more i n t u i -24 t i v e worker who i s u n w i l l i n g to make such s i m p l i f y i n g assumptions. The r e c o n c i l i a t i o n of such attitudes may provide a stimulus for continuously examining and questioning the v a l i d i t y of such a model and i t s generalisations, and i n i t s e l f t h i s c o n f l i c t may prove to be an e f f e c t i v e check on the a p p l i c a b i l i t y of the model. Camina, o p . c i t . , p. 128. - 42 -Chapter 5 A LINEAR PROGRAMMING MODEL FOR LAND USE PLANNING Introduction The preceding two chapters have described the general concepts involved i n land use planning and l i n e a r programming. In th i s chapter a l i n e a r programming model, having s p e c i f i c a p p l i c a t i o n to land use planning i n the Lower Mainland region of B r i t i s h Columbia, w i l l be developed. I t w i l l be r e c a l l e d from the previous chapter that the general format for a l i n e a r programme i s : Maximize ^ V..G.. (9) j= l n Subject to > V^An-, < B i (10) for a l l i = 1,2, m and Vj ^  0 (11) The l i n e a r programming problem thus has three s t r u c t u r a l components: the objective function (9), a matrix of input-output c o e f f i c i e n t s , j- the A^j's. and a set of constraints or r i g h t hand sides, -j B i . Each of these components - 43 -i s developed i n t h i s chapter for a l i n e a r programming land use planning model applicable to land use planning i n the Lower Mainland of B r i t i s h Columbia. The Lower Mainland of B r i t i s h Columbia has previously been the subject of planning studies which culminated i n the formulation of the " O f f i c i a l Regional Plan" i n 1966.^ D e t a i l s of the area, such as' acreage i n d i f f e r e n t land uses i n d i f f e r e n t m u n i c i p a l i t i e s and unincorporated areas were a v a i l a b l e from the records of the Lower Mainland Regional Planning Board, New, Westminster, B r i t i s h Columbia, and t h i s information provided the data base for the l i n e a r programming planning model now to be described. 1. The Objective Function The purpose of land use planning i s , as has been stated previously, to formulate objectives towards which the s p a t i a l organization of land based functions may be directed and c o n t r o l l e d , i n such a manner as to promote the highest and 'best use of the land resource. Necessarily, therefore, the plan which i s devised i s normative, i n that i t describes what should be done i n order to a t t a i n the objectives. In order to c l a r i f y the r a t i o n a l e and the concepts of the model which follows, i t i s necessary to trace the r e l a t i o n s h i p s between land use, land use-capacity, land rents and values, and the highest and best use of land resources. , The concept of land use-capacity r e f e r s to the a b i l i t y of any given u n i t of land to produce a net return above the production costs associated with Lower Mainland Regional Planning Board, O f f i c i a l Regional Plan New Westminster, B.C. - 44 -i t s use. The amount of this net return provides an index of use-capacity. Areas of high use-capacity and high income producing p o t e n t i a l therefore produce high economic land rent. As man resorts to lands of lower use-capacity, economic rents on t h i s marginal land tend to d e c l i n e . This i s i l l u s t r a t e d i n Figure 2. The diagram assumes a continuum of lands of decreasing land use-capacity, ranging from areas of highest use-capacity at A, to lands of much lower use-capacity at C. Thus at A, the economic rent accruing to the land i s large, ( t h e o r e t i c a l l y equal to the net return above the production cost associated with i t s use - namely AB), whereas at C there i i s no net return above the production costs, and therefore no rent accrues to i the land at C. However, land owners normally have more than one use to which they may put t h e i r land. In evaluating the choice of enterprise, the owner i s interested i n comparing the income producing p o t e n t i a l of these various p o s s i b i l i t i e s . In economic theory, the operators seek to maximize returns at t h e i r p a r t i c u l a r l o c a t i o n with t h e i r p a r t i c u l a r combination of productive f a c t o r s . Thus, they w i l l evaluate the choices open to them by e i t h e r general observation or by c a l c u l a t i o n of the probable amount of economic rent accruing to each choice. From an economic standpoint, comparisons of t h i s type, p a r t i c u l a r l y those i n v o l v i n g both uses and l o c a t i o n s , may be thought of i n terms of over-lapping rent t r i a n g l e s . These economic rent t r i a n g l e s vary considerably i n s i z e and shape depending on the land use i n question. Barlowe, op. c i t . , p. 157. - 45 -Figure 2 I l l u s t r a t i o n of Relationship Between Economic Rent and Land Use Capacity - 46 -Figure 3 shows the rent t r i a n g l e s which depict the competition between four types of land use. For i l l u s t r a t i v e purposes, the four uses may be designated: i n d u s t r i a l , commercial, r e s i d e n t i a l and a g r i c u l t u r a l . Those uses which produce the highest economic rent have f i r s t claim upon the areas of highest use-capacity. The lower comparative rent producing capacity of the lower uses makes i t impossible for them to compete with the more productive uses and, as a r e s u l t , are situated at other locations which have a lower use-capacity. In the example above, one would expect the ordering . of land a l l o c a t i o n to s t a r t from the highest use-capacity areas - i n d u s t r i a l , and proceed through commercial, r e s i d e n t i a l and a g r i c u l t u r a l land uses. F i n a l l y , at any one l o c a t i o n some use w i l l always return a higher economic rent compared to any a l t e r n a t i v e use, and from the standpoint of the i n d i v i d u a l operator this i s always the highest and-best use for h i s land. Furthermore, the t o t a l welfare of the population w i l l be highest when each piece of land i s being u t i l i z e d at i t s highest and best use (assuming a d d i t i v i t y ) . Thus, i n planning f o r the optimal a l l o c a t i o n of land to various uses, the administration should attempt to maximize the sum of the economic rents accruing to the land base of the region. We can think of this maximization of the aggregate economic rent accruing to land, i n a s o c i a l i s t system, where land i s f r e e , as providing a maximum of consumer's surplus. In a s i m p l i f i e d case, consider a region, i n which a l l land uses may be a l l o c a t e d to a l l parcels of land. The resource i s divided i n t o sub-regions, designated k, containing d i f f e r e n t grades or zones of land, designated i . Land uses, designated j , compete for the land resource. - 47 -FIGURE 3 I l l u s t r a t i o n of Relationship of Economic Rent and the A l l o c a t i o n of Land between Four Competing Uses Economic Rent I n d u s t r i a l Commercial R e s i d e n t i a l A g r i c u l t u r a l A l l o c a t i o n of Land By Decreasing Land Use Capacity - 48 -The optimal land a l l o c a t i o n ( i . e . , the highest and best use of land,) between the various competing uses can therefore be expressed as follows: Maximize: - 1 m n $, £ £ M j k (12) i = l j = l k=l Where Rijk i s the Economic rent accruing to use j on zone i i n sub-area k For i = 1,2, 1 3 = 1,2, m k = 1,2, n From the notion of maximization of the sum of the i n d i v i d u a l economic rents, i t i s possible to extend the analysis by r e l a t i n g the Economic Rent of land to the land value. From a t h e o r e t i c a l point of view, the value of a land resource always equals the sum of i t s future economic rents discounted back to the present. The formula used i n the computation of land values can be expressed as: Present Value = a + a + "''a'' +.... (1 + r) (1 + r ) 2 (1 + r ) 3 (1 + r ) n where r = the i n t e r e s t rate n = the number of years where a = average annual economic rent and when n = . then Present Value = a. ' r - 49 -Thus, i f i t i s assumed that a l l land prices r e f l e c t the same c a p i t a l i z a t i o n rate of the economic rents, an optimal a l l o c a t i o n of land to the competing uses may be made by maximizing the sum of the i n d i v i d u a l land values, i . e . , p r i c e - bids as determined by the competing uses, the highest p r i c e - b i d n a t u r a l l y capturing'that s i t e for i t s p a r t i c u l a r bidder. Therefore, i n terms of the above example, optimization of land a l l o c a t i o n can be achieved as follows:-! Maximize: 1 m n I & & i Gijk (13) i = l j = l k=l I Where Gijk i s the value of Land i n use j on zone i i n sub-area k For i = 1,2, 1 ^ j = 1,2, m k = l , 2 , n where i , j , k , r e f er to the same variables as i n Statement (12) A point has therefore been reached i n the above analysis of the land a l l o c a t i o n process from which i t can be observed that by applying the l i n e a r programming technique, an optimal a l l o c a t i o n of land resources between any number of competing uses, can be achieved. By adopting Statement (13) to serve as the objective function of a land use planning model, the optimum economic pattern of land uses can be derived for any land resource base. - 50 -The land use planning model for the Lower Mainland of B r i t i s h Columbia was. .based on .this objective function. The s p e c i f i c a t i o n s of the model involved maximizing the aggregate value of land a l l o c a t e d to a l l uses i n the region, for s p e c i f i c points i n time subject to the constraints described below. Solution of the model therefore gave normative land use plans for the Lower Mainland region, i n which the land base was a l l o c a t e d to i t s highest and best use. 2. The Constraints A condition was imposed on the model, that over the time period studied, (1966-1986), the a l l o c a t i o n of land resources must be s u f f i c i e n t to meet the demand for r e s i d e n t i a l s i t e s generated by the increase i n the population of the Lower Mainland region over t h i s period. Therefore, the maximization of the objective function subject to the condition that r e s i d e n t i a l land resource needs must be met, resulted i n the p r o v i s i o n of r e a l i s t i c normative patterns or plans of land use, which i f followed would r e s u l t i n land being put to i t s highest and best use. The model was required to locate the projected population for each sub-area i n r e s i d e n t i a l accommodation. For the purposes of this model two sub-areas were defined: a) the Metropolitan sub-area, and b) the Valley sub-area. The constituent m u n i c i p a l i t i e s of these sub-areas are l i s t e d below. Within e i t h e r of these two sub-areas, the population was allowed to r e s i d e n t i a l l y locate at any s i t e . - 51 -L i s t of M u n i c i p a l i t i e s included i n the Metropolitan Sub-Area Vancouver Burnaby New Westminster Coquitlam Port Coquitlam Port Moody Buntzen West Vancouver North Vancouver Ci t y North Vancouver D i s t r i c t Richmond Delta Surrey White Rock L i s t of M u n i c i p a l i t i e s included i n the Va l l e y Sub-Area P i t t Meadows P i t t Polder .. .. Maple Ridge Mission C i t y Mission D i s t r i c t Langley C i t y Langley Township Matsqui Abbotsford Sumas Chilliwack C i t y Chilliwack Township Cultus Lake Harrison Hot Springs Kent Hope - 52 -The population constraints were e q u a l i t i e s . That i s to say that the number of the population who were r e s i d e n t i a l l y located by the model i n each sub-area had to exactly equal an exogenously determined s i z e . The population, both i n each of. the sub-areas, and i n t o t a l , could not be l e s s than or more than a predetermined number. The constraints i n these equations were e q u a l i t i e s , since i n e q u a l i t i e s of e i t h e r sense ( i . e . , greater than or equal Co, or less than or equal to) would not have f i t t e d the o v e r a l l requirements that the population be housed. For example, suppose that the population constraints were wr i t t e n i n such a way that the model was prevented from l o c a t i n g more than the projected number of households. This would have been i o g i c a l since we were in t e r e s t e d i n the s i t u a t i o n where a p a r t i c u l a r number of households were to be located, and not where the model could continue a l l o c a t i n g land to r e s i d e n t i a l uses i n unlimited supply u n t i l a l l the a v a i l a b l e land had been used up. On the other hand, however, i t was j u s t as l o g i c a l to write the constraints i n such a way that the model was required to a l l o c a t e s u f f i c i e n t land to meet the r e s i d e n t i a l land requirements of the projected population increase. For these reasons i t was d i f f i c u l t to e s t a b l i s h a general r u l e , and therefore i t was preferable and e n t i r e l y reasonable to make these constraints e q u a l i t i e s . The second set of constraints describe the r e s t r i c t i o n s p e r t a i n i n g to the land supply. These constraints restrained the various land uses to the amount of land a v a i l a b l e i n each land zone category and subr-area. The signs attached to these iand supply constraints were of the form which indicated that the t o t a l amount of the land resources which were a l l o c a t e d to the various uses had to be l e s s than, or equal to, the supply of the resource. Thus over-commitment of the land resources was avoided. - 53 -3. The Input-Output C o e f f i c i e n t s The input-output c o e f f i c i e n t s of a l i n e a r programme represent the numerical r e l a t i o n s h i p s between the a c t i v i t i e s and the resources which are a l l o c a t e d to the a c t i v i t i e s . S p e c i f i c a l l y i n this model, the input-output c o e f f i c i e n t s described two main sets of r e l a t i o n s h i p s . The f i r s t set represented the r e l a t i o n s h i p between the land resources a v a i l a b l e and the actual land using a c t i v i t i e s . These are known as the land absorption c o e f f i c i e n t s . The second set denoted the r e l a t i o n s h i p between the land resources and the population, - i n other words, the density of population which could be accommodated on the d i f f e r e n t types and sub-areas of land i n d i f f e r e n t housing categories. 4. The Mathematical Model Mathematically, the primal l i n e a r programming model used f o r a l l o c a t i n g land between a l t e r n a t i v e uses i s : Maximize: 1 m n I I ^ ( V i j k . Gijk) (14) i = l j = l k=l which i s the aggregate of the values of land a l l o c a t e d to a l l uses i n the Lower Mainland region. - 54 -Subject to the constraints (a) 1 m ^ £ (Vi j k . Aijk) = Population k (15) i = l j = l for a l l k (b) 1 m C £ ( V i j ) <f Land k for a l l k (16) 1=1 j = l V i j k ^ 0 (17) fo r a l l i , j , k Where: i = Zoning category of land j = Land use a c t i v i t y ' k = Land sub-area and, V i j k = Land area o f land ( acres)used f o r Land use a c t i v i t y j on land zoned i , i n sub-area k. Gi j k = The demand p r i c e of land per acre used f o r land use a c t i v i t y j on land zoned i , i n sub-area k. A i j k - The density of population (persons per ac r e ) , associated with land use a c t i v i t y j on land zoned i i n sub-area k. - 55 -Land k = The supply of land i n sub-area k. Population k = The (exogenously determined) population i n sub-area k. 5. D e t a i l s of the Model I n i t i a l l y i t was hoped to have vectors i n the matrix which would represent a l l p o s s i b l e uses of land i n the Lower Mainland Region ( i . e . , i n d u s t r i a l , commercial, r e t a i l , r e s i d e n t i a l , f o r e s t r y , a g r i c u l t u r e and r e c r e a t i o n a l land uses). However, there were severe l i m i t a t i o n s to doing t h i s , created by the amount of data a v a i l a b l e , and consequently the number of major a l t e r n a t i v e s was reduced to three: s i n g l e family r e s i d e n t i a l use, multi family r e s i d e n t i a l use, and a g r i c u l t u r a l land use. The model was thus reduced from i t s conceptual s i z e of j=8 to j=3, for t h i s study. a) Constraint Vectors The land supply constraint vector set, (Equation 16), was constructed i n accordance with the zoning regulations of the O f f i c i a l Regional Plan i n the hopes of making the output of the model give normative land use plans which would r e f l e c t the zoning ordinance. The c l a s s i f i c a t i o n of land resources indicates land use p o l i c i e s , ( i n e f f e c t land use r e s t r i c t i o n s ) , and modification p o l i c i e s for land use i n the Lower Mainland region, upon which the land supply constraints of the model were erected.^ D e t a i l s of the zoning c l a s s i f i c a t i o n i n t h i s plan are as follows: Lower Mainland Regional Planning Board op. c i t . , p. 3. - 56 -URBAN Urban I. - Established Urban Areas: a. Purpose: The Established Urban Areas designate lands that becuase of:-1. established small l o t urban development, 2. s t r a t e g i c proximity to regional transportation f a c i l i t i e s , : 3. assured freedom from flooding, and | 4. u n s u i t a b i l i t y f o r , or pre-emption of, productive a g r i c u l t u r e , are best s u i t e d f or the current stage of small l o t urban development. b. Use P o l i c y : The Established Urban Areas may be used only f o r urban uses. c. Area M o d i f i c a t i o n P o l i c y : An established urban area may be extended. Urban I I ; - Developing Urban Areas: a. Purpose: Tb_e Developing Urban Areas designate land that because of:-1. p a r t i a l s u b - d i v i s i o n i n t o near urban l o t s , 2. a clear destiny f or future small l o t urban development, 3. l o c a t i o n adjacent to e x i s t i n g established urban areas, 4. assured freedom from flooding, and - 57 -5. u n s u i t a b i l i t y for," or pre-emption of, productive a g r i c u l t u r e , are best s u i t e d for large l o t suburban development with gradual absorption into established urban areas and future small l o t urban development. b. Use P o l i c y : The Developing Urban Areas may only be used f o r urban uses other than multi-family r e s i d e n t i a l uses. c. Area M o d i f i c a t i o n P o l i c y : i I 1. A developing urban area s h a l l not be extended. i ; 2. A developing urban area may be redesignated as an established urban area. RURAL Rural I. - Acreage Rural Areas: a. Purpose: The "Acreage Rural Areas" designate lands that because of:-1. predominant small holding p a r c e l s i z e , 2. pre-emption from, or u n s u i t a b i l i t y for extensive s o i l bound a g r i c u l t u r e , 3. some p o t e n t i a l f o r or l o c a t i o n adjacent,to future urban development. - 58 -4. assured freedom from flooding, ' are best s u i t e d f o r current and future intensive a g r i c u l t u r a l small holding development, with the possible long-term absorption of some small holdings i n t o established urban areas. b. Use' P o l i c y : The Acreage Rural Areas may be used only f o r r u r a l and transportation us es. c. Area M o d i f i c a t i o n P o l i c y : 1. An acreage r u r a l area may be extended when a d d i t i o n a l land for i n t e n s i v e a g r i c u l t u r a l or small holding development i s c l e a r l y warranted. 2. An acreage r u r a l area may be redesignated as an established urban area. Rural I I . - Upland Rural Areas: a. Purpose: The "Upland Rural Areas" designate lands that because of:-1. predominant large p a r c e l s i z e , 2. general a r a b i l i t y , i s o l a t i o n from urban development, and - 59 -3. complete freedom from flooding, are best s u i t e d f o r extensive upland a g r i c u l t u r e , f o r e s t r y , or other large holding r u r a l development, with the possible designation of some upland r u r a l areas as acreage r u r a l areas. b. Use P o l i c y : The Upland Rural Areas may be used only for r u r a l and transportation uses. c. Area M o d i f i c a t i o n P o l i c y : An upland r u r a l area may be redesignated as an acreage r u r a l area. Rural I I I . - Lowland Rural Areas: a. Purpose: The "Lowland Rural Areas" designate lands that because of:-1. l o c a t i o n i n a f l o o d p l a i n , 2. predominant large p a r c e l s i z e , and 3 . general a r a b i l i t y , or i s o l a t i o n from urban development, are best s u i t e d f o r extensive lowland a g r i c u l t u r e or other large holding r u r a l development that w i l l s u f f e r l e a s t from flooding. b. Use P o l i c y : A Lowland Rural Area may be used only f o r r u r a l and transportation uses. - 60 -c. Area M o d i f i c a t i o n P o l i c y : A lowland r u r a l area may be redesignated as an established urban area only where an e x i s t i n g established urban area located i n a f l a t p l a i n , i s committed to further urban development through early settlement and requires a d d i t i o n a l land f o r development. The t o t a l number of land supply constraints i n the model i s the sum of the i n d i v i d u a l sets of the land supply constraints which p e r t a i n to each of the m u n i c i p a l i t i e s x^ithin the Metropolitan sub-area ( l i s t e d on page 5 1 ) } and the set of constraints p e r t a i n i n g to the t o t a l of the Valley sub-area. This l a t t e r set, the Valley sub-area, i s an aggregation of the i n d i v i d u a l sub sets of the land supply constraints of each of the Valley m u n i c i p a l i t i e s . This was  done so as to enable the Valley sub-area to be treated as one large m u n i c i p a l i t y . Thus the model represents a t o t a l of 16 areas i n the whole Lower Mainland region, (15 m u n i c i p a l i t i e s i n the Metropolitan sub-area, plus the Valley sub-area), each with a pos s i b l e maximum of 5 land constraints, (Established Urban, Developing Urban, Acreage Rural, Lowland Rural and Upland Rural). However, not every municipality or sub-area contains a l l the f i v e categories of land, and therefore from a possible maximum of 80 land constraints, only 41 are required to describe the land resource supplies. These 41 vectors consist as follows: Land Category "Urban Land Established Developing T o t a l No. of Land Constraints 14 Rural Land Acreage Upland• Lowland - 61 -b. A c t i v i t y Vectors: The matrix consisted of 113 a c t i v i t y vectors. Of t h i s number, 70 vere s t r u c u r a l vectors, and the remaining 43 were disposal vectors, (one disposal , 4 vector f o r each associated c o n s t r a i n t row). The 70 s t r u c t u r a l vectors consisted as follows: • • • •  • • • • •  . Land Category •' •'. .-' .•' • • Land Using Established Developing Acreage Upland Lowland • A c t i v i t y Urban Urb an Rural Rural Rural Single Family Housing 14 9 9 6 Mu l t i p l e Family Housing 14 A g r i c u l t u r a l Land Use 9 3 6 c. The Input-Output Coefficients: The parameters used to denote the land absorption c o e f f i c i e n t s and population d e n s i t i e s between the a c t i v i t i e s and the respective land resources were obtained from the Lower Mainland Regional Planning Board and r e f l e c t further current land use controls of the m u n i c i p a l i t i e s r e l a t i n g to maximum permissible b u i l d i n g d e n s i t i e s . The d e r i v a t i o n of the s p e c i f i c input-output c o e f f i c i e n t s i s shown i n the data s e c t i o n following (Chapter 6). d. The P r i c e C o e f f i c i e n t s : The b a s i c p r i c e c o e f f i c i e n t s used f o r weighting the objective function were the current market prices of the land resources. This data was obtained 4 The 43 constraint rows include the two population constraints discussed on pp. 50-51. - 62 -from representatives of various organizations concerned with r e a l estate business i n the Lower Mainland. However, i n attempting to construct the model i n accordance with the land c l a s s i f i c a t i o n and c o n t r o l p o l i c i e s as outlined above, c e r t a i n s i m p l i f y i n g assumptions and adjustments regarding the values or prices of the various r e s i d e n t i a l a c t i v i t i e s were made. Inspection of the c l a s s i f i c a t i o n of land resources shows that r e s i d e n t i a l development was to be r e s t r i c t e d i n the f i r s t instance to e i t h e r Established Urban or Developing Urban zoned land. Secondly, development of Acreage Rural Land would folloxj. L a s t l y , the urban development of Lowland Rural zoned land would follow, should the pressure of population growth warrant urbanization. I t was presumed that r e s i d e n t i a l development of land resources could be scheduled, i n keeping with the regulations., by d i f f e r e n t i a l p r i c i n g of ,the r e s i d e n t i a l a c t i v i t i e s . The r a t i o n a l e behind t h i s idea was that the highest p r i c e d a c t i v i t y would enter the s o l u t i o n f i r s t , followed by the other a c t i v i t i e s i n order of t h e i r descending p r i c e s . Thus, by assigning Established Urban land a high p r i c e , and assigning lower, descending prices to Developing Urban, Acreage Rural, and Lowland Rural land zones, i t was hoped to simulate the area m o d i f i c a t i o n p o l i c i e s s p e c i f i e d i n the Regional Plan. These sources, are shown i n the Data s e c t i o n of the t h e s i s . - 63 -The a v a i l a b l e data did not i n d i c a t e r e s i d e n t i a l land prices v/ith regard to land zone. However, for each municipality, a range of r e s i d e n t i a l land values was given, and the following scheme was used i n assigning land values to the r e s i d e n t i a l a c t i v i t i e s on d i f f e r e n t zones of land:-a) The highest quoted value for a " t y p i c a l l o t " i n sub-area k, was used as the weighting f a c t o r f o r Single Family Housing a c t i v i t i e s on Established Urban land. , b) The average value for " t y p i c a l l o t s " i n sub-area k, was used as the weighting f a c t o r f o r Single Family Housing a c t i v i t i e s on Developing Urban land. i c) The lowest value quoted for a " t y p i c a l l o t " i n sub-area k was used as the weighting factor for Single Family Housing a c t i v i t i e s on converted Acreage Rural land. d) A value marginally lower - that i s by say $0.10 per acre - than the weighting f a c t o r used f o r Single Family Housing a c t i v i t i e s on converted Acreage Rural land, was used as the weighting f a c t o r for Single Family Housing a c t i v i t i e s on converted Lowland Rural land. In e f f e c t , therefore, i t was assumed that within one sub-area, the most valuable r e s i d e n t i a l land was situated i n Established Urban zoned areas. Developing Urban zoned land was the next most valuable, followed by converted Rural Acreage land and l a s t l y converted Lowland Rural land was assumed to be the l e a s t valuable. - 64 -This i s not an unreal assumption when viewed from the point of view 6 of the amenity l e v e l which p a r t i c u l a r r e s i d e n t i a l s i t e s possess. In t h i s instance i t was assumed that amenity l e v e l s would be higher i n Established Urban zones owing to proximity to shopping centres, schools, work centres, etcetera, and the presence of b e t t e r roads, sewage disposal, and other p u b l i c s e r v i c e s . R e s i d e n t i a l locations other than Established Urban zones were assumed to have les s amenity, decreasing i n the order from Developing Urban s i t e s to converted Acreage Rural and Lowland Rural land. e. Time Periods for the Model: i The l i n e a r programming model was solved for four time periods: 1966 to 1971 , 1966-1976, 1966-1981, and 1966-1986. St a r t i n g from 1966 and using an inventory of vacant l o t s , the projected incremental increase i n population between 1966 and 1971 was a l l o c a t e d land for r e s i d e n t i a l purposes. A second s o l u t i o n (or run) of the model st a r t e d from the same (1966) o r i g i n a l inventory of vacant l o t s , and a l l o c a t e d land to r e s i d e n t i a l purposes for a projected incremental increase i n population for the time period 1966 to 19 76. The t h i r d and f o u r t h runs followed the same procedure, using the projected incremental increases i n population for 1966 to 1981 and 1966 to 1986 r e s p e c t i v e l y . Each s o l u t i o n shows for the end of the respective time period, ( i . e . , 1971 1976, 1981, or 1986) an optimal configuration of land use - a g r i c u l t u r a l and r e s i d e n t i a l - for the Lower Mainland Region as a whole. Amenity l e v e l i s defined as the l e v e l of psychic s a t i s f a c t i o n which a household has an opportunity to enjoy because of c e r t a i n c h a r a c t e r i s t i c s of that site.. - 65 -f. Output The optimum solution, obtained i n the output of the model showed the maximum t o t a l value accruing to land i n the region which was pos s i b l e under the imposed conditions. This was inte r p r e t e d as being the optimum pattern of land use i n the region at the end of the respective time period. - 66 -•Chapter 6 The Data Introduction This s e c t i o n s p e c i f i e s the sources of data used i n constructing the l i n e a r programming model matrix. The data can be broken down in t o three major sets of parameters. (1) The Components of the Obj e c t i v e Function  : These are: a) The r e s i d e n t i a l land prices ' b) The a g r i c u l t u r a l land prices (2) The Right Hand Side Elements Thes e are: a) Land supply S t a t i s t i c s b) Projected Population S t a t i s t i c s (3) The Input-Output C o e f f i c i e n t s These are: a) R e s i d e n t i a l land Absorption C o e f f i c i e n t s b) Rural Land Absorption C o e f f i c i e n t s c) Rural Land Conversion r a t i o s d) Population d e n s i t i e s (1) Single Family Housing ( i i ) Apartment Housing - 67 -1. The Objective Function The objective function must be consistent. That i s to say, the values attached to each of the variables must be expressed i n the same terms, i n this case, i n $ per acre of a c t i v i t y . Thus, i n the output the economic r e l a t i o n s h i p s between the a c t i v i t i e s can be studied. R e s i d e n t i a l Land P r i c e s : Single Family Housing: The values of these parameters, expressed i n $ per acre, were derived by the following formula:-Value per Acre of Value per Single Number of Single Family Housing = Family Housing L°t^ x Lots per Acre^ R e s i d e n t i a l Land where k = sub-area. The values per l o t were obtained from the 1968 e d i t i o n of Real Estate Trends i n Metropolitan Vancouver.* The p u b l i c a t i o n l i s t s "prices of t y p i c a l l o t s " , ( f o r 1967), which are defined as being - "average as to 2 d e s i r a b i l i t y , view and l o c a l amenities". I t was hoped to use 19.66 p r i c e s , but these were not a v a i l a b l e . These data are presented i n Appendix A 1. As was explained i n the previous chapter, four values were used f o r weighting the objective function, depending on the land zoning category i n which the r e s i d e n t i a l a c t i v i t y takes place. To repeat, these four categories * Greater Vancouver Real Estate Board, S t a t i s t i c a l and Survey Committee. Real Estate Trends i n Metropolitan Vancouver, 1968. 2 I b i d . , p. A-2. - 68 -were as follows: a) The highest value quoted f o r a " t y p i c a l l o t " i n sub-area k x No. l o t s / a c r e k, was used as the weighting f a c t o r f o r Single Family Housing a c t i v i t i e s on Established Urban land. b) The average value quoted for a " t y p i c a l l o t " i n sub-area k x No. l o t s / a c r e k, was used as the weighting f a c t o r f or Single Family . Housing a c t i v i t i e s on Developing Urban land. c) The lowest value quoted f o r a " t y p i c a l l o t " i n sub-area'k x No. l o t s / a c r e k, was used as the weighting f a c t o r f o r Single Family Housing a c t i v i t i e s on converted Acreage Rural land. d) A value marginally lower, that i s by say $0.10 per acre, than the weighting f a c t o r used f o r Single Family Housing a c t i v i t i e s on con-verted Acreage Rural land, - was used as the weighting f a c t o r f o r Single Family Housing a c t i v i t i e s on converted Lowland Rural land. The number of l o t s per acre were taken d i r e c t l y from Lower Mainland Regional Planning Board data, and represent the current (that i s up to 1966) d e n s i t i e s of housing i n the various m u n i c i p a l i t i e s , on d i f f e r e n t zones - i . e . , Established Urban and Development Urban. Since these are not generally a v a i l a b l e , they are reproduced i n Appendix A 2. Apartment Housing: Unlike the data r e l a t i n g to the market p r i c e s of Single Family Housing l o t s , which i s given i n $ per l o t , the p r i c e s of Apartment 3 Housing s i t e s are given i n d o l l a r terms per front foot of l o t . . However, since 3 The term " f r o n t f o o t " i s common planning.jargon, and r e f e r s to the frontage of a l o t contiguous to the s t r e e t . - 69 -4 the average frontage of l o t s f o r apartment housing i s 66 l i n e a r f e e t , the following formula was used to estimate the value of land f o r these a c t i v i t i e s : Value per Acre of No. of M u l t i p l e Housing = 66 x P r i c e per front f o o t ^ x Lots per Acre^ R e s i d e n t i a l Land The prices used were again obtained from the Greater Vancouver Real Estate Board,"* and the data r e l a t i n g to l o t density i s as used i n computing the Single Family Housing value.parameters. The data are presented i n Appendix A 1. A g r i c u l t u r a l Land P r i c e s : Attempts to contact a r e a l estate agent conversant with a g r i c u l t u r a l land prices i n the Lower Mainland, were unsuccessful. Thus, approaches were made to two other sources, - the planning department of a l o c a l m u nicipality and the Greater Vancouver Real Estate Board, i n an e f f o r t to secure data of as high c a l i b r e as p o s s i b l e f o r i n c l u s i o n i n the model.^'^ However, much i n t e r p o l a t i o n was necessary and i t must be emphasized that the data can only therefore be regarded as being an approximation. The data used i s shown i n Appendix A 3 . 2. The Right Hand Side (Constraint) Elements: Land Supply s t a t i s t i c s :" The Land inventory was supplied by the Lower Mainland Regional 4 Lower Mainland Regional Planning Board. Personal Communication, 1970. ^ Greater Vancouver Real Estate Board, op. c i t . , p. B-3. ^ Jorden, M., A s s i s t a n t Planner, Corporation of Delta, B.C., Personal Communication, 1970. Williamson, M., Chief S t a t i s t i c i a n , Greater Vancouver Real Estate Board, Personal Communication, 1970. ' - 70 -Planning Board, New Westminster. Again, since these figures are not generally a v a i l a b l e , they are summarized and presented i n Appendix A 4. The data i s p r i m a r i l y an inventory of land l i s t e d according to zoning designation and categorized by m u n i c i p a l i t y . The inventory thus r e f l e c t s the zoning conditions which held i n 1966, (the temporal s t a r t i n g point f o r the a n a l y s i s ) . The input to the model was i n terms of vacant r e s i d e n t i a l l o t s f o r both the Established Urban and Developing Urban land areas. The Rural Land Supply was l i s t e d and u t i l i z e d i n terms of acres. Population S t a t i s t i c s : These are derived from the s t a t i s t i c s provided by the Lower Mainland 8 Regional Planning Board. The population data used as r i g h t hand side elements was i n the form of the incremental increase between 1966 and the year f o r which the analysis was being run, i . e . , 1971, 1976, 1981, or 1986. This expressed the incremental increase i n population from 1966 which would have to be e f f e c t i v e l y housed by new r e s i d e n t i a l construction. To make th i s c l e a r , consider the f i r s t run of the analysis which i s to determine r e s i d e n t i a l land which i s to be used f o r housing between 1966 and 1971. The population data used i n this case i s the d i f f e r e n c e between the population i n the area i n 1966 and 1971. In the second run, the population data i s the diff e r e n c e between the years 1966 and 1976. The population data are presented i n Appendix A 5. 3. The Inputs-Output C o e f f i c i e n t s : R e s i d e n t i a l Land Absorption C o e f f i c i e n t s : Since the Right Hand Side elements were expressed i n terms of vacant g Lower Mainland Regional Planning Board, Populations Trends i n the Lower  Mainland, New Westminster, B.C. 1968. - 71 -Established Urban or Developing Urban l o t s , and the units i n which the r e s i d e n t i a l a c t i v i t i e s were expressed was i n acres, the input-output coef-f i c i e n t s which were used as a " t i e " between resources and a c t i v i t i e s were the r e s i d e n t i a l l o t d e n s i t i e s , expressed i n l o t s / a c r e , already mentioned and l i s t e d i n Appendix A 1. Rural Land Absorption C o e f f i c i e n t s : Since the land supply constraints were expressed i n acres, and the output of a c t i v i t i e s was expressed i n acres, the input-output c o e f f i c i e n t s are obviously unity. Rural To Urban Conversion Ratios: In order to represent this a c t i v i t y as r e a l i s t i c a l l y as p o s s i b l e , i t was decided to use the number of l o t s per acre i n the Developing Urban Areas as the conversion rate of Rural acres i n t o Urban l o t s . Thus, f o r example, i f there was a l o t density of 1.5 l o t s / a c r e on the Developing Urban zone of sub-area k, and Rural zoned land i n the same region was being converted to urban use, then the f a c t o r used i n c a l c u l a t i n g new l o t supply i n sub-area k would also be taken as 1.5 new l o t s per acre of Rural land converted. I f , however, there was no Developing Urban land i n an area - only Established Urban land, - the conversion r a t i o factor used was taken as being the l o t density of the Established Urban land of the area. This was considered r e a l i s t i c in'the sense that the density of future development i n an area i s to a large extent determined by the e x i s t i n g density conditions, i . e . , the more densely used an area i s , the higher w i l l be the density of new development on newly a v a i l a b l e land. These data are presented i n Appendix A 6. - 72 -Population Densities: ': The data for population densities was of two types. F i r s t l y , the population density of apartment blocks was derived using the following formula: Density of Pop'n/Apt. Block^= (Residents/bachelor s u i t e x % bachelor s u i t e s ) + CResidents/1 bedroom s u i t e x %1 bedroom s u i t e s ) + CResidents/2 bedroom s u i t e x %2 bedroom suites) + — CResidents/3 bedroom s u i t e x %3 bedroom su i t e s ) x Average No. of suites per Apartment Block^ The f i g u r e so derived from t h i s c a l c u l a t i o n i s a weighted average of the population housed i n apartment blocks i n sub-area k. r The o r i g i n a l data was taken from the "Real Estate Trends" manual which l i s t s number of s u i t e s , type of s u i t e and apartment blocks i n the m u n i c i p a l i t i e s 9 of the area i n question. There was no data a v a i l a b l e regarding Residents per d i f f e r e n t type of s u i t e . Therefore, i t was assumed that the density per: Bachelor s u i t e was 1 person 1 Bedroom s u i t e was 2 persons 2 Bedroom s u i t e was 3 persons 3 Bedroom s u i t e was 4 persons Secondly, the population density f o r Single Family Housing was required. The f i g u r e used here was taken d i r e c t l y from the Canada Year Book."^ The pop-u l a t i o n density f o r Single Family Households i n B r i t i s h Columbia i n 1966 was 3.6 persons per household. The data are presented i n Appendix A 7. 9 Greater Vancouver Real Estate Board, op. c i t . , p. B-7 ^ Dominion Bureau of S t a t i s t i c s , Canada Year Book, 1969, Ottawa, Queen's P r i n t e r , p. 183. Chapter 7 The Results The assembled l i n e a r programming matrix was solved using the U n i v e r s i t y of B r i t i s h Columbia I.B.M. 360 computer. This chapter presents the solutions for each of the four time periods investigated. In each case, the s o l u t i o n shows an optimum (as s p e c i f i e d i n the'model) land a l l o c a t i o n pattern, and the marginal value p r o d u c t i v i t i e s (Shadow prices) of the l i m i t i n g resources. 1. The Optimum Solution The aggregate r e s u l t s of projected increases i n urban land use, and the associated projected decrease i n a g r i c u l t u r a l use between 1966 and 1986 are presented i n Tables 7.1 and 7.2. I t can be seen from Table 7.1 that from 1966 to 1986, there i s a projected increase i n the r e s i d e n t i a l l y urbanized area of 128,176 acres or approximately 200 square miles. This represents an average increase i n urbanized area of 6409 acres per annum. However, since a large proportion of t h i s increase represents an ' i n f i l l i n g ' of land space i n Established and Developing Urban areas for r e s i d e n t i a l development, the annual increase i n r e s i d e n t i a l l y urbanized acres does not accurately r e f l e c t the land use change patterns from a g r i c u l t u r a l to urban uses. However, Table 7.2 does give a more accurate r e f l e c t i o n of decreasing a g r i c u l t u r a l acreage. From the s t a r t i n g point i n the a n a l y s i s , where i n 1966 there was an estimated 364,022 acres of A g r i c u l t u r a l Zoned land i n the Planning Board Area, there i s a projected urbanization rate of 3,772 acres annually over the twenty year period to 1986. At this date, i f the projections were accurate, - 74 -Table 7.1 Cumulative Increase i n Urbanized Area - Projected as Occurring Between 1956 and the Dates Shown, 1971 - 1986 Zone 1971 1976 1981 1986 Acres Single Family Housing on Established Urban Zoned Areas Single Family Housing on Developing Urban Urbanized . Acreage Rural Area j Urbanized Lowland Rural Area 7,818 11,994 20,380 35,359 15,001 17,380 17,380 17,380 1,690 6,470 33,038 8,380 41,677 24,470 49,282 26,155 Tot a l 30,979 70,792 103,907 128,176 - 75 -Table 7.2 Area of Land i n A g r i c u l t u r a l Uses Projected For 1971 - 1986 and Compared to 1966 Zone 1966 1971 1976 1981 1986 Acres Acreage Rural (Rural I) 52,771 51,081 19,733 11,094 3,489 Upland Rural (Rural II) 145,365 145,365 145,365 145,365 145,365 Lowland Rural (Rural III) 165,886 159,416 157,505 141,416 139,731 To t a l 364,022 355,862 322,603 297,875 288,585 - 76 -there would be remaining i n the area 288,585 acres of r u r a l zoned land, - a reduction of 75,432 acres from 1966. I t can be seen that the greatest proportion of t h i s loss i s composed of converted Acreage Rural land - 65.3% of the t o t a l reduction i n a g r i c u l t u r a l land. From 1966 to 1986 there i s a projected conversion of 2,464 acres per year of Acreage Rural land to other uses. At this rate, i t i s projected that only 3,489 acres of Acreage Rural would remain i n a g r i c u l t u r a l use i n 1986. Since Upland Rural land was r e s t r i c t e d to a g r i c u l t u r a l use by the incorporation of the zoning regulations i n the model, t h i s area remained t o t a l l y i n a g r i c u l t u r a l use. However, the Lowland Rural zoned land was permitted to become urbanized, and from 1966-1986, the^acreage of t h i s land was projected as d e c l i n i n g at an average annual rate of 1,308 acres, from 165,880 acres i n 1966 to a remaining 139,731 acres i n 1986. Figures 4 and 5 i l l u s t r a t e these trends. Tables 7.3 to 7.6 show i n d e t a i l the projected increase i n urbanized area by municipality for f i v e year i n t e r v a l s from 1966 to 1986. The general trend appears that i n maximizing the objective function, the f i r s t land zones occupied by r e s i d e n t i a l expression are the Developing Urban areas, followed by conversion of Lowland Rural and Acreage Rural areas in t o urban uses. F i n a l l y " i n - f i l l i n g " i s projected to take place on the Established Urban areas mainly i n the l a t e r stages of the time period studied, although i t can be observed that some " i n - f i l l i n g " does occur from 1966 to 19 76. It can thus be assumed that i n respect of the "ordering" of the r e s i d e n t i a l FIGURE 4 Projected Increase i n T o t a l Urbanized Area, (Acres) 1966 - 1986 150,000 f 100,000 75,000 ACRES 50,000 1966 1971 1976 1981 19 Year - 78 ~ Figure 5 50,000 40,000 10,000 Projected Increase i n Urbanized Area by Zoning C l a s s i f i c a t i o n , 1966-1986 / / / Acreage Rural Developing Urban - - - Established Urban Lowland Rural 1986 - 79 -a c t i v i t i e s , the model did not meet the intended s p e c i f i c a t i o n s . Discussion of t h i s point w i l l be held over u n t i l the next section. 2. Shadow Pri c e s Tables 7.7 and 7.8 show the shadow pr i c e s of r e s i d e n t i a l s i t e s i n the Established and Developing Urban areas r e s p e c t i v e l y . These p r i c e s which are the marginal value p r o d u c t i v i t i e s of the l i m i t i n g resources are expressed i n 1 d o l l a r s per l o t . N a t u r a l l y , only those resources which are f u l l y u t i l i z e d have a shadow p r i c e . Tables 7.9 and 7.10 give s i m i l a r information regarding the Acreage Rural and the Lowland Rural land resources. These shadow prices are expressed i n terms of d o l l a r s per acre. Table 7.11 shows the marginal value p r o d u c t i v i t y of the population i n the Metropolitan and Val l e y sub-areas. I t i s of i n t e r e s t to note that these are the only shadow p r i c e s which decrease over time. The land resource shadow p r i c e s , i t w i l l be observed, either are constant, or increasing with time. There are v a l i d economic reasons f o r t h i s , and they w i l l be discussed i n the following appraisal of the model and the r e s u l t s . The reason f o r the M.V.P.'s being expressed i n terms of d o l l a r s / l o t i s due to the manner i n which the problem was formulated. For an explanation of t h i s , see for example Heady and Candler, op. c i t , , pp. 85-90, - 80 -Table 7.3 Increase i n Urbanized Area by Mun i c i p a l i t y or Area, 1966 - 1971 Area Established Urban Developing Urban Acreage Rural Lowland Rural T o t a l West Vancouver N. Vancouver C i t y N. Vancouver D i s t r i c t Vancouver C i t y Burnaby New Westminster Richmond Delta Port Moody Coquitlam Port Coquitlam Surrey White Rock Buntzen Valley Area 4133 1847 1690 148 acres 39 654 752 1095 634 4320 5 1000 6502 1690 6470 4133 39 1847 1690 148 8814 752 1095 634 4320 5 1000 6502 Tota l 7818 15001 1690 6470 30979 - 81 -Area Table 7.4 Increase i n Urbanized Area by Mu n i c i p a l i t y or Area 1966 - 1976 Established Urban Developing Urban Acreage Rural acres Lowland Rural T o t a l West Vancouver N. Vancouver C i t y N. Vancouver D i s t r i c t Vancouver City Burnaby New Westminster Richmond Delta Port Moody Coquitlam Port Coquitlam Surrey White Rock Buntzen Va l l e y Area 4133 4133 1847 5866 148 39 654 752 1095 634 4320 5 1000 8881 1690 26435 6470 1910 4913 39 1847 5866 148 8814 752 1095 634 32665 5 1000 13794 To t a l 11994 17380 33038 8380 70792 - 82 -Table 7.5 Increase i n Urbanized Area by Mun i c i p a l i t y or Area, 1966 - 1981 Area Established Urban Developing Urban Acreage Rural Lowland Rural T o t a l West Vancouver N. Vancouver Cit y N. Vancouver D i s t r i c t j Vancouver City Burnaby New Westminster Richmond Delta Port Moody Coquitlam Port Coquitlam Surrey White Rock Buntzen Val l e y Area 4133 837 4897 1847 5866 148 1925 727 acres 39 8881 654 752 1095 634 4320 5 1000 11857 1690 1695 26435 6470 1320 16680 4133 837 4936 1847 5866 148 10739 752 727 4110 634 47435 5 1000 20738 To t a l 20380 17380 41677 24470 103907 - 83 -Table 7.6 Increase i n Urbanized Area by Mun i c i p a l i t y or Area, 1966 - 1986 Area Established Urban Developing Urban Acreage Rural Lowland Rural T o t a l West Vancouver N. Vancouver C i t y N. Vancouver D i s t r i c t Vancouver City Burnaby New Westminster Richmond Delta Port Moody Coquitlam Port Coquitlam Surrey White Rock Buntzen Valley Area 4133 837 4897 1847 5866 148 6803 4509 727 2840 1625 690 437 acres 39 654 752 1095 634 4320 5 1000 8881 1690 1095 26435 610 18852 6470 1320 16680 1685 4133 837 4936 1847 5866 148 15617 5261 727 6950 2259 48125 442 1610 22423 To t a l 35359 17380 49282 26155 128176 - 84 -Table 7.7 Derived Shadow Prices of Established Urban Lots i n designated years 1971 - 1986 Area 1971 1976 1981 1986 per l o t West Vancouver $3,768 $4,408 $5,554 $7,124 North Vancouver C i t y - - $ 251 $1,900 North Vancouver D i s t r i c t $ 177 $1,879 Vancouver City $4,465 $5,137 $6,342 $7,99 2 Burnaby - $ 672 $1,877 $3,527 New Westminister $ 345 $1,017 $2,224 $3,874 Richmond - - $ 908 $1,650 Delta - - $ 149 Port Moody - - $2,516 Coquitlam - - - $1,399 Port Coquitlam - -• - - $1,49 7 White Pvock - - - $1,401 Valley Sub-Area - 85 -• Table 7.8 Derived Shadovz Prices of Developing Urban Lots i n designated years 1971 - 1986 Area 1971 1976 1981 1986 North Vancouver Dis t r i c t $13,480 $ per $14,153 l o t $15,358 $17,008 Richmond $ 7,845 $ 8,517 $ 9,722 $11,373 Delta $ 2,409 $ 3,082 $ 4,288 $ 5,940 Coquitlam $ 9,861 $10,535 $11,741 $13,394 i Port Coquitlam j $ 4,284 $ 4,956 $.6,161 $ 7,811 j-Surrey $ 4.009 $ 4,681 $ 5,885 $ 7,533 White Rock $23,312 $13,986 $25,192 $26,844 Buntzen $ 8,872 $ 9,544 $10,749 $12,400 Valley Area - $ 2,500 $ 2,500 $ 2,500 - 86 -Table 7.9. Derived Shadow Prices of Acreage Rural Areas i n designated years 1971 - 1986 Area 1971 1976 1981 1986 $ per Acre Vancouver $10,000 $10,000 $10,000 $10,000 Burnaby $10,000 $10,000 $10,000 $10,000 Richmond $ 8,715 $ 9,623 $11,250 $13,478 Delta $ 8,000 $ 8,000 $ 8,000 $ 8,000 Buntzen $ 8,000 $ 8,000 $ 8,000 $ 9,400 Port Moody $ 8,000 $ 8,000 " $ 8,000 $ 8,000 Coquitlam $ 8,000 $ 8,000 $ 8,298 $ 9,719 Surrey $ 2,000 $ 2,000 $ 3,132 $ 4,681 Valley $ 1,500 $ 1,500 $ 1,500 $ 1,500 - 87 -Table 7.10 Derived Shadow Prices of Lowland Rural Areas i n designated years 1971 - 1986 Area 1971 1976 1981 1986 $ per acre Richmond $8,715 $9,823 $11,250 $13,478 Delta $8,000 $8,000 $ 8,000 $ 8,000 Coquitlam $8,000 $8,000 $ 8,298 $ 9,719 Port Coquitlam $8,000 $8,000 $ 8,000 $ 8,000 Surrey , $2,000 $2,000 $ 3,131 $ 4,681 Valley $1,500 $1,500 $ 1,500 $ 1,500 Table 7.11 Derived Shadow Prices of Population i n designated years 1971 - 1986 Area 1971 19 76 1981 1986 $ per person Metropolitan Area $2,258 $2,258 $1,736 $1,278 Valley Area $2,778 $2,083 $2,083 $2,083 - 88 -Chapter 8 Discussion 1. The Optimum Land Use Plans The model provides f e a s i b l e plans for r e s i d e n t i a l l o c a t i o n , which i f followed, would r e s u l t i n maximum economic e f f i c i e n c y of a g r i c u l t u r a l and r e s i d e n t i a l land use, given the values used i n the a n a l y s i s . In other words, for the assumed s i t u a t i o n , no other plan of r e s i d e n t i a l development would r e s u l t i n the aggregate of r e s i d e n t i a l and a g r i c u l t u r a l land values being greater, for the Lower Mainland Region as a whole. The assumed s i t u a t i o n s are f i r s t l y , the constraints imposed upon the model, and secondly the s p e c i f i c a t i o n s of the model. The task of assessing and evaluating the research and r e s u l t s i s complicated. What has been derived i s a normative plan for securing highest and best use of the r e s i d e n t i a l and a g r i c u l t u r a l land resources of the region. However, the general trend of land development as shown by the r e s u l t s of the a n a l y s i s , turns out to quite consistent with the actual development of the land resources that i s taking place today i n the region. That i s to say that the present trends of land use development appear to conform to a high degree to the conditions necessary for maximum economic e f f i c i e n c y of u t i l i z a t i o n of the a g r i c u l t u r a l and r e s i d e n t i a l land resources of the region. The present-day r e a l l i f e trend of increasing numbers of households l o c a t i n g i n suburbia, or newly developing r e s i d e n t i a l areas, rather than l o c a t i n g i n apartment housing i n the c i t y , c l o s e l y p a r a l l e l s the normative plan shown i n the r e s u l t s . Furthermore, some of the differences - 89 -between the r e s u l t s obtained and the developing patterns i n the r e a l world may be due to estimation errors for some v a r i a b l e s , or omission of other relevant v a r i a b l e s as a r e s u l t of the s i m p l i f y i n g assumptions that were made. This might be taken to suggest that, contrary to much public opinion, the mechanism and resultant end - state of the land market, (or at l e a s t part of i t ) , i s r a t i o n a l when viewed from an economic standpoint, and i s i n f l u e n t i a l i n securing a pattern which approximates the highest and best use of land resources. This suggestion i s not s u r p r i s i n g , since the nature of the r e a l estate market to some extent resembles a c e n t r a l l y planned economy, i n which the economic p r i n c i p l e s of highest and best use are applied to the development of the land resource base by the r e a l t o r s . I t can be observed from the r e s u l t s that the model did not exactly simulate the zoning c l a s s i f i c a t i o n and modification p o l i c i e s p e r t a i n i n g to use of the land resources. It w i l l be remembered that by adjustment of the prices attached to the various housing a c t i v i t i e s , an attempt was made to "order" r e s i d e n t i a l development of the region as follows:- f i r s t l y Established Urban, secondly Developing Urban, followed by Acreage Rural and Lowland Rural land development:. The i n a b i l i t y of the model to cope s u c c e s s f u l l y with t h i s desired ordering owes to the f a c t that i n the s o l u t i o n procedure, the c r i t e r i o n equation implements the intr o d u c t i o n of a p a r t i c u l a r a c t i v i t y or v a r i a b l e into the basi s . The c r i t e r i o n equation takes account of not only the increase ( i n the maximizing example) i n the objective function by introduction of an a c t i v i t y , but the associated a c t i v i t i e s which have to be foregone i n order - 90 -to make the i n t r o d u c t i o n of this new v a r i a b l e possible. This, as may be appreciated i s the opportunity cost of introducing this new a c t i v i t y into the s o l u t i o n . The f a c t that the i n t r o d u c t i o n of a c t i v i t i e s i s dependent not only upon the: land resource, but also on a l i m i t e d population resource also, complicates any attempt to derive p r i c e s with which to weight the objective function so as to ensure the i n t r o d u c t i o n of the a c t i v i t i e s into the basis i n any desired order. Thus the s i m p l i s t i c approach, to simulating the zoning S c l a s s i f i c a t i o n , i s seen to be inadequate. However, the u t i l i t y of the model ! i s not s e r i o u s l y impeded, because although the model f a i l s to simulate the zoning r e s t r i c t i o n s , i t s t i l l produces r e s u l t s which are apparently p o s i t i v e - i n the sense that they show what i s happening. This could be due to r e a p p r a i s a l of zoning decisions which constantly takes place. That i s to say, that i n keeping with the tenets of the Regional Plan, redesignation of zoning boundaries may take place i f warranted. In many cases, economic considerations receive high p r i o r i t y from the planners and warrant such redesignation. 2. The Derived Shadow Prices of the Constraints Another important part of the r e s u l t s are the shadow prices of the constraints. The shadow prices of the land constraints i n d i c a t e by how much the t o t a l present value of the plan could r i s e by increasing the t o t a l area of a zone by one u n i t , - i n the case of a g r i c u l t u r a l land by one acre, or by one r e s i d e n t i a l l o t i n urban areas. The shadow pr i c e represents the best combination of the benefits which would accrue to the use of the extra - 91 -u n i t of the land resource t with a l l other conditions being held constant. Therefore, the increase i n the value of the objective function which would r e s u l t from a marginal r e l a x a t i o n of any land constraint i s a measure of the value of the marginal output of land i n each zone. How, therefore, should, the shadow prices given i n the s o l u t i o n be interpreted? F i r s t l y , i t w i l l be observed that the .shadow p r i c e of a g r i c u l t u r a l land i s at l e a s t as great as the market p r i c e of the land (Tables 7.9 and 7.10 on pages 86-87). Indeed, i n the l i g h t of the f a c t s above which show that the shadow p r i c e of any land constraint i s a measure of the value of the marginal output of land, t h i s i s obvious. An increase i n the plan, by one acre, of a g r i c u l t u r a l land would r e s u l t i n the objective function increasing by the market p r i c e of that acre. The marginal "output" of land i n t h i s case i s equivalent to i t s market p r i c e . However, where the r u r a l land resource has been used f o r r e s i d e n t i a l a c t i v i t i e s , and i s exhausted, i . e., been used to i t s l i m i t , the value of i t s marginal p r o d u c t i v i t y i s greater than when used s o l e l y for a g r i c u l t u r a l use. This owes to the f a c t that r u r a l land i n r e s i d e n t i a l uses i s more valuable than the same land i n a g r i c u l t u r a l use. Thus a marginal increase i n r u r a l land supply, which would be used for r e s i d e n t i a l purposes, would increase the objective function by a figure i n excess of the marginal value product of the same land i f i t were used for a g r i c u l t u r a l purposes. On the other hand, the marginal value p r o d u c t i v i t y of Urban land, used for r e s i d e n t i a l development, i s not n e c e s s a r i l y equal to, or greater than - 92 -i t s p a r t i c u l a r market p r i c e . In some cases, the shadow p r i c e i s less than the market value of the r e s i d e n t i a l land resource. At f i r s t sight t h i s may appear paradoxical i n that an increase i n the supply of r e s i d e n t i a l land which i s l i m i t e d , would be expected to increase the value of the objective function by at l e a s t i t s market value. However, the paradox i s explained by the fac t that r e s i d e n t i a l a c t i v i t y i n the s o l u t i o n , i s dependent not only on the l i m i t e d r e s i d e n t i a l land resources, but also on another li m i t e d resource, - population. That i s to say, that an increase i n r e s i d e n t i a l a c t i v i t y requires not only an increase i n land supply, but an accompanying increase i n population to be housed. Thus, i n the s i t u a t i o n which i s described by the shadow p r i c e of a resource, ( i . e . , the increase i n the value of the objective function brought about by a marginal increase i n the supply of one resource, with a l l other resources  being kept at t h e i r o r i g i n a l l e v e l , ) i t becomes apparent that i n order f o r the increased supply of land to be p r o f i t a b l y used f o r housing, (and not l i e i d l e , ) population must be a v a i l a b l e to be housed, from some source wi t h i n the optimum plan. The computation of the shadow pr i c e s of the resources takes t h i s f actor i n t o account. The source of population necessary to increase r e s i d e n t i a l a c t i v i t y on the extra u n i t of land becomes a v a i l a b l e by s a c r i f i c i n g r e s i d e n t i a l a c t i v i t y on some other land. Since t h i s " s a c r i f i c e " detracts from the value of the objective f u n c t i o n , the net increase i n the value of the objective f u n c t i o n brought about by i n c r e a s i n g the r e s i d e n t i a l a c t i v i t y l e v e l on the increased resource, w i l l therefore be equal to the dif f e r e n c e between the value of the r e s i d e n t i a l a c t i v i t i e s foregone, ( i n order to make increased r e s i d e n t i a l a c t i v i t y on the increased r e s i d e n t i a l land resource p o s s i b l e ) , and the value by which the o b j e c t i v e function would be increased by introducing an extra u n i t of r e s i d e n t i a l a c t i v i t y on the expanded r e s i d e n t i a l - 93 -land resource. Thus, this net increase i n the value of the objective function does not n e c e s s a r i l y have to be equal to the market pr i c e of the resource. By observation of Tables 7.7 to 7.10 i t can be seen that the marginal value p r o d u c t i v i t i e s of the l i m i t i n g land resources, which have been used for r e s i d e n t i a l development, increase oyer time. The reason for t h i s i s that as the population of the region increases, less p r o f i t a b l e (when viewed from the standpoint of the accrued returns) land must be a l l o c a t e d to r e s i d e n t i a l a c t i v i t i e s i n order to ensure that the population constraint of the model, (which dic t a t e s that the population must be housed,) i s met. Thus as lands of lower value are u t i l i z e d i n order to s a t i s f y t h i s population constraint, the p o t e n t i a l value of a d d i t i o n a l , more valuable land, increases. On the other hand, i t w i l l be observed from Table 7.11 that the shadow prices of the population resources decrease over time. In the case of the Metropolitan sub-area, the shadow p r i c e of population diminished for the whole time period studied - 1966 to 1986. The shadow p r i c e of population of the Valley sub-area on the other hand decreases only between 1966 and 1976. From 1976 to 1986 the shadow p r i c e remains constant. The f a c t of diminishing marginal p r o d u c t i v i t y of the population resources, i s explained as follows. With increased population pressure, the model a l l o c a t e s land of lower value to r e s i d e n t i a l a c t i v i t i e s i n order to meet the population constraint. Thus, s t a r t i n g from the s i t u a t i o n where there i s a r e l a t i v e l y small population, the value of the increase i n the - 94 -objective function, brought about by one extra person being brought into the region, i s high, since t h i s person w i l l be r e s i d e n t i a l l y located on a v a i l a b l e land of high value. But as the population increases, so the most valuable land w i l l become urbanized, and the plan resorts to a l l o c a t i n g lands of lower value to r e s i d e n t i a l a c t i v i t i e s . Therefore, the increase i n the objective function brought about by an extra person becomes l e s s , as the t o t a l population increases. In the case of the shadow prices of the Valley sub region, the i . • shadow prices f a l l and then remain constant. This r e s u l t s from the fac t that i n i • the plan for 1966 to 1971, a marginal increase i n population would be al l o c a t e d to the Developing Urban land, r e s u l t i n g i n a high shadow p r i c e . However at the end of the 1971 to 1976 period, the plan shows a lower marginal value product per u n i t of population, than for the previous period. This owes to the fac t that i n t h i s case a marginal increase i n population would be al l o c a t e d to lower valued Acreage Rural land, which would r e s u l t i n a low net be n e f i t to the objective function. The s i t u a t i o n where increases i n population i n the Valley sub region are allocated to Acreage Rural land p e r s i s t s u n t i l the period 1981 to 1986, when the supply of Rural Acreage land becomes f u l l y u t i l i z e d for r e s i d e n t i a l purposes. Land of the next highest value, i . e . . Lowland Rural i s then u t i l i z e d for r e s i d e n t i a l development to meet the 1986 population constraint. However i n the computational procedure for determining the shadow p r i c e s , the marginal difference between the value of Lowland Rural land and Acreage Rural does not s i g n i f i c a n t l y a f f e c t the value of the shadow pr i c e of the population constraint, and i s "rounded o f f " .to the nearest integer. The value of the shadow pr i c e for the 1986 population constraint i n the Valley sub region i s therefore the same as for the equivalent 1981 population constraint. • - 95 -The shadow prices are of i n t e r e s t i n the following ways. F i r s t l y , the shadow prices of land constitute a guide to the Regional Planning a u t h o r i t i e s as to how much they should be prepared to pay to land owners i n order to assure the execution of the most e f f i c i e n t plan. These payments could be made by e i t h e r purchasing the land for development and r e s e l l i n g i n the future, or by d i r e c t subsidies to encourage the execution of the optimal plan. By buying land at p r i c e s not higher than the present value of i t s shadow p r i c e , the pub l i c a u t h o r i t i e s would be expected to incur no loss by these transactions. By the same token, the d e r i v a t i o n and evaluation of the shadow pr i c e s of land could constitute a powerful t o o l for planning land development strategy i n the r e a l estate business. The values attached to the shadow prices of population i n the d i f f e r e n t sub-areas are of equal i n t e r e s t and importance. The shadow pr i c e s of the resources are the respective marginal value p r o d u c t i v i t i e s of these resources. When viewed from the standpoint of the economic theory r e l a t i n g to resource a l l o c a t i o n , the evaluation of these shadow prices assume a new importance. The theory states that optimum a l l o c a t i o n of a scarce resource (population i n t h i s case) between two fi x e d resource f a c t o r s , (land i n Metro-p o l i t a n sub region, and land i n the Val l e y sub.region), occurs when the marginal value p r o d u c t i v i t y of the scarce resource - (population), i s equal i n every use.* A. J . B r a f f , Microeconomic Analysis (John Wiley, New York, 1969) p. 237. - 96 -It can be seen from Table 7.11 that at the end of each time period, (1971, 1976, 1981, 1986), the marginal value p r o d u c t i v i t y of population i s higher i n the "Valley" sub-area, than the same figure for the Metropolitan sub-area. The d i f f e r e n c e between these two shadow prices indicates the maximum subsidy which could be. paid by the planning authority to a new prospective resident of the Lower Mainland region to r e s i d e n t i a l l y locate i n the Val l e y sub region, i n the i n t e r e s t s of promoting the highest and best use of the land base of the region as a whole. More generally, i n any regional land use plan, i n which a population target for the region i s set by the planners, the comparison of the shadow prices of population within the region and that of i other, surrounding regions, y i e l d s information of value i n assessing the o r i g i n a l population target. If the shadow p r i c e of population of the planned region i s higher than that i n the surrounding regions, then this constitutes a d i r e c t i v e to increase the population target. I f , on the other hand, the shadow p r i c e of population of the planned region i s lower than that of the adjacent regions, t h i s constitutes a d i r e c t i v e to reduce the planned population target. However, should the public a u t h o r i t i e s i n s i s t on maintaining the o r i g i n a l population target, a programme of subsidies to a t t r a c t new population to the region might be adopted. Again, the minimum volume of subsidies to encourage in-migration from other regions, i s indicated by the differ e n c e i n the shadow prices of population. - 97 -3. General Considerations The r e s u l t s of the study must be approached with caution. This owes to the f a c t of c e r t a i n error i n the model and i t s output. General evaluation of such error i n simple l i n e a r models has been b r i e f l y mentioned i n a previous section, but i t remains to examine t h i s s p e c i f i c model i n the same l i g h t . It w i l l be r e c a l l e d from Chapter 4 that error encountered i n simple l i n e a r models i s of two types. F i r s t l y there i s s p e c i f i c a t i o n error, or errors incorporated i n the model by way of the assumptions made and the equation system, used. Secondly there i s measurement or data error which appears, due to poor data being used i n the model. Both of these error types as they r e l a t e to the "Lower Mainland Regional Land Use Planning Model" w i l l be studied and t h e i r e f f e c t s evaluated. (a) S p e c i f i c a t i o n Error  Evaluation of the model's s p e c i f i c a t i o n s n e c e s s a r i l y includes discussion of the form of the objective f u n c t i o n . Two objective functions could have been used to achieve normative plans of highest and best use of the regional land resources. The f i r s t , (which was used i n the study), maximized the aggregate land value of the regional resource base. Secondly, optimization of land a l l o c a t i o n could have been achieved by using the converse of th i s function: that i s to say, a function which minimized the aggregate costs associated with a l l o c a t i n g land to d i f f e r e n t uses i n the region. The minimization of th i s l a t t e r function would have been subject to the same constraints that were applicable to the maximizing model. ' T h e o r e t i c a l l y , s o l u t i o n of the minimizing model would y i e l d the same r e s u l t s ( i . e . , a l l o c a t i o n patterns), as the maximizing model. In l i n e a r programming terminology, the minimizing model i s known as the dual of the - 98 -maximizing model, and v i c e versa. The reason f o r the f a c t that both would y i e l d the same r e s u l t s stems from the r e l a t i o n s h i p set out i n Chapter 5, between costs of production, land rents, and land values. To r e i t e r a t e b r i e f l y , the lower the costs of production associated with any piece of land - "X", i n a p a r t i c u l a r use, compared to the costs of production associated with employing an a l t e r n a t i v e piece of land - "Y", f o r the same purpose, the higher w i l l be the economic rent accruing to land resource "X". S i m i l a r l y the yalue of the land resource "X" w i l l be proportionately greater than that of "Y". In other words, high land values i n d i c a t e low associated production costs. Thus the equivalency of the maximizing and the minimizing models becomes c l e a r . •. i - - . -The reason that the maximizing model was chosen over the minimizing, was b a s i c a l l y r e l a t e d to ease of formulation. This stems from the f a c t that the gathering and processing of "cost of production" data i n an urban development s e t t i n g i s p r o h i b i t i v e . Cost of production f a c t o r s i n the urban f i e l d include a wide range of variables such as transport costs, b u i l d i n g costs, and general service costs (hydro, water, sewerage, e t c . ) . Within the time a v a i l a b l e for the study i t would have been impossible to c o l l e c t a l l the necessary data. Furthermore i d e n t i f i c a t i o n problems would have ar i s e n i n attempting to reduce these "costs of production" to some sort of common time base. For example two d i f f e r e n t resource developments might provide general services which were of d i f f e r e n t q u a l i t i e s , r e s u l t i n g i n d i f f e r e n t depreciation and renewal rates. In such cases the c a l c u l a t i o n of annual costs would have been impossible. Thus i t can be seen that the complexity involved i n the preparation of a minimizing l i n e a r programming model would have been immense. - 99 -Therefore the maximizing model was adopted since the value of the land resources are a function of these, many i n t e r a c t i n g f a c t o r s and provides an acceptable i n d i c a t i o n of the use-capacity of the d i f f e r e n t resources. S p e c i f i c a t i o n error i s undoubtedly an important factor i n the formulation of a model which s i n g u l a r i l y attempts to consider only a small number of land use competitors. This consideration of only the a g r i c u l t u r a l and r e s i d e n t i a l land use processes may therefore be misleading unless the model and the output are treated with care. Land uses are interdependent,"as i s witnessed by the patterns of land use i n any region. I n d u s t r i a l , Commercial, R e s i d e n t i a l , and A g r i c u l t u r a l land use functions i n any region are intimately r e l a t e d . Thus i n s i n g l i n g out a subset of th i s group, there i s almost c e r t a i n l y a m i s i n t e r p r e t a t i o n of the s i t u a t i o n . To allow only r e s i d e n t i a l a c t i v i t i e s to bid f o r land which i n r e a l l i f e would be the subject of numerous p r i c e bids from d i f f e r i n g uses,is a d i s t o r t i o n of the f a c t s of the s i t u a t i o n , and therefore l i m i t s the a p p l i c a b i l i t y of the r e s u l t s of th i s p a r t i c u l a r model to being used for planning recommendations. Further s p e c i f i c a t i o n e r r o r , which also e f f e c t i v e l y l i m i t s the a p p l i c a t i o n of the model, i s re l a t e d to the fac t that the l i n e a r equation set does not cater to change i n the system. In other words, the model i s non dynamic. As has been stated previously, the assumption of l i n e a r i t y i s i n many ways d i f f i c u l t to j u s t i f y . Changes i n the parameters which express r e l a t i o n s h i p s within the model are ignored. I t i s inconceivable that these r e l a t i o n s h i p s w i l l remain constant over time, and therefore by ignoring such changes, the output of the model w i l l become u n r e a l i s t i c , e s p e c i a l l y for the plans devised for the l a t e r time periods ( i . e . , 1981 and 1986). Whereas i t was shown i n an e a r l i e r chapter, that i n many cases l i n e a r models have worked w e l l , i t must be pointed out that i t - 100 -i s mainly i n s i t u a t i o n s where a "once-and-for-all-time-plan" has been devised and c a r r i e d out, e.g., the design and planning of a new town, that such s t a t i c 2 models have been succuessful. However, i n dynamic s i t u a t i o n s i t would be possible to constantly r e v i s e the parameters to br i n g the model "up to date", but t h i s would be a very poor method f or the purposes of formulating and guiding land use p o l i c i e s , because e f f e c t i v e long range plans would be almost impossible to devise due to uncertainty of knowledge of the values of the parameters. L a s t l y , with regard to s p e c i f i c a t i o n e r r or, i s the error of the evaluation of the r e s u l t s . It w i l l be remembered that from the r e s u l t s i t was possible to hypothesize that the model was to some extent p o s i t i v e as well as normative. This was based on the f a c t that the normative patterns f o r securing the highest and best use of land approximated the present trends i n land development i n the region. However, i t i s imperative, when evaluating the model and i t s output, not to lose sight of the o r i g i n a l goals of the study and the construction of the model, namely to provide normative plans. Over-emphasis of the p r e d i c t i v e powers of the model must be avoided since a bias i n the i n t e r p r e t a t i o n of the output i n t h i s d i r e c t i o n , can a l l too e a s i l y lead to overlooking the o r i g i n a l assumptions. (b) Data Error ' The main source of data error i s concerned with the values used f o r describing the land based a c t i v i t i e s - ( i n e f f e c t the prices of the land f o r d i f f e r e n t functions, i . e . , A g r i c u l t u r a l or R e s i d e n t i a l ) . With regard to Ben Shahar et. a l . , op. c i t . , p. 108. - 101 -a g r i c u l t u r a l land p r i c e s , the values used do not represent a true p i c t u r e of the c a p i t a l i s e d values of the economic rent, - (when the rent i s viewed as a surplus over production costs) . Instead, these prices may be speculative values, which are f a r i n excess of t h e i r true worth i n the case of a g r i c u l t u r a l land and r e f l e c t the value that the same land would assume i f the zoning controls were l i f t e d and other uses (e.g., R e s i d e n t i a l , I n d u s t r i a l , or Commercial uses) were permitted to locate on the land. The values or prices of the r e s i d e n t i a l land are s i m i l a r i l y subject to e r r o r . This i s not so much as a r e s u l t of speculation, but stems from the oligopo l i s t i c nature of the r e s i d e n t i a l r e a l estate market. The market for r e s i d e n t i a l land, at l e a s t i n the Lower Mainland of B r i t i s h Columbia cannot be c a l l e d perfect. The monopolistic element p a r t l y i s due f i r s t l y to the f a c t that the supply of land i s i n e l a s t i c , and secondly that the buyers are r e s t r i c t e d i n t h e i r choice of r e s i d e n t i a l s i t e s by l o c a l authority regulations, (e.g., zoning, sewage and other health standards,) and by f i n a n c i a l i n s t i t u t i o n s - the banks and mortgage companies. • In the case of the r e s i d e n t i a l land p r i c e s , however, i t i s d i f f i c u l t to see where, and how, a more accurate value of r e s i d e n t i a l land p r i c e s could be a r r i v e d at. On the other hand, the true values of the a g r i c u l t u r a l lands could be assessed more accurately by c a p i t a l i z i n g rent figures derived from a g r i c u l t u r a l cost figures f o r the region. The above disc u s s i o n has shown the l i m i t a t i o n s of the model and i t s assumptions. As an overview however, a point i n favour of the approach i s i t s o v e r a l l s i m p l i c i t y . This conclusion i s based on comparison with other more - 102 -complex modelling techniques which can become extremely inaccurate due to the 3 compounding of error through using many sub models i n the system. The l i n e a r programming system gives a simultaneous approach to the problem of resource a l l o c a t i o n , whereas other methods (e.g., simulation) attempts resource a l l o c a t i o n i n stages. Alonso, W. The Quality of Data and the Choice and Design of Models. Working Paper 72. Berkeley U n i v e r s i t y , C a l i f o r n i a . Centre for Planning and Development Research, 1968. - 103 -• Chapter 9 Summary and Conclusions This study examined the trends of land resource development i n the Lower Mainland of B r i t i s h Columbia from the mid 1800's to the mid 1960's. The patterns of increasing a g r i c u l t u r a l land development up to the 1940's followed by decreasing a g r i c u l t u r a l acreages to the present day (1970), were found to be intimately r e l a t e d to the population expansion of the region over the same time period. Furthermore i t was found that increasing argument has been directed against the urbanization of r u r a l land i n the region, and that there has been a I commensurate increase i n . the demand for some sort of planning mechanism to ensure.. i i that the use of the land resources i s i n the best i n t e r e s t s of the population. The study therefore continued' by examining the objectives, r a t i o n a l e and concepts involved i n land use planning. The ba s i c weaknesses of the concept were i d e n t i f i e d as being c l o s e l y r e l a t e d to a f a i l u r e on the part of the planners to: ' f i r s t l y , appreciate a l l the factors involved i n the use of land resources, and secondly, to develop s p e c i f i c c r i t e r i a with which to formulate and guide land a l l o c a t i o n decisions. The remainder of the study was directed towards devising and applying a conceptual framework which would, to some extent, r e c t i f y these weaknesses. The conceptual framework of the study was,based on the premise that the objectives of land use planning should be to a l l o c a t e land to i t s "highest and best use". The framework was evolved i n an economic context, and i t was stated that the economic highest and best use of land, pertains when i t i s used - 104 -f o r the purpose which y i e l d s the highest monetary returns. The c o r o l l a r y to t h i s was stated to be that the highest land value indicates the best use. Therefore by maximising the aggregate of the land values i n a region, highest and best use of the region's land resources would be attained. The mathematical technique of l i n e a r programming i s a most s u i t a b l e and convenient method with which to approach the problem of maximizing such a function.. By application, of the above concepts, an economic, l i n e a r programming land use planning model was constructed for the Lower Mainland region of B r i t i s h Columbia. Using the l i n e a r programming technique applied to the maximization of the sum of the land values of a g r i c u l t u r a l and r e s i d e n t i a l land i n the Lower Mainland region, normative land use plans were found for these land resource sectors for four time periods - 1966 to 1971, 1976, 1981, and 1986, The optimum land use plans produced by the model show from an economic standpoint, the most e f f i c i e n t possible land uses of a g r i c u l t u r a l and r e s i d e n t i a l land i n the whole region, for the four time periods studied.. The model, apart from being normative, also appears p o s i t i v e i n that i t shows the a c t u a l present day trend of continuing urban expansion onto r u r a l land. The p o s i t i v e aspects of the model were a t t r i b u t e d to the mechanism by which the r e a l estate market operates, and i t was hypothesized that by being to some extent p o s i t i v e , the model could be used for p r e d i t i v e purposes. - 105 -The main l i m i t a t i o n s of the model were associated with problems of s p e c i f i c a t i o n and subsequently was considered l i m i t e d i n i t s a p p l i c a b i l i t y . In conclusion, however, i t does not appear unreasonable to claim that t h i s study has provided an economic approach to r a t i o n a l multiple-use regional land use planning. This author f e e l s that the concepts are sound, but finds the a p p l i c a t i o n of the technique i s d i f f i c u l t owing to lack of r e l i a b l e data, and suggests further research i n the f i e l d to lead to the formulation of land use plans which are applicable to planning the development of a regional land base. - 106 -BIBLIOGRAPHY Alonso, W., The Quality of Data and the Choice and Design of Models. Working Paper No. 72, Berkely University, C a l i f o r n i a , Centre for Planning and Development Research, 1968. Arrow, N.J., Social Choice and Individual Values. Englewood C l i f f s , N.J.: Prentice H a l l , 1958. Ben Shahar, H., Mazor, A., and Pines, D. "Town Planning and Welfare Maximization: A Methodological Approach," Regional Studies, Vol. 3, 1969, pp. 105-113. Benson, P.A., North, N.L., and Ring, A.A., Real Estate P r i n c i p l e s and Practise. Englewood-Cliffs, N.J.: Prentice-Hall, 1954. Camina, M.M., "Plan Design Models - A Review," Town Planning, Vol. 1, 1969, pp. 119-130. Campbell, H.G., Matrices with Applications. New York: Appleton Century-Crofts, 1968. Canada, Dominion Bureau of S t a t i s t i c s , Census of Canada. Ottawa: Queen's Pr i n t e r , various years. Canada, Dominion Bureau of S t a t i s t i c s , Canada Year Book. Ottawa: Queen's Pr i n t e r , various years. Colenutt, R.J., "Building Linear Preditive Models for Urban Planning." Regional  Studies, Vol. 2, 1968, pp. 139-143. Davis, O.A., "The Economics of Municipal Zoning." Unpublished doctoral d i s s e r t a t i o n , University of V i r g i n i a , 1964. Dorfmann, R., Samuelson, P.A., and Solow, R.M., Linear Programming and Economic  Analysis. New York: Rand Series, McGraw-Hill, 1958. Egbert, A.C., and Heady, E.G., Regional Analysis of Production Adjustments i n the Major F i e l d Crops. U.S.D.A. Tch. B u l l . , No. 1294, 1964. - 107 -Ely, R.T., Outlines of Land Economics. 2 Vols. Ann Arbor, Michigan: Edwards Bros., 1927. Gale, D., The Theory of Linear Economic Models.. New York: McGraw-Hill, 1960. Glaever, H., and Seagraves, J . , Linear Programming and Economics of Size. Journal of the A g r i c u l t u r a l Experimental Station, N. Carolina. Paper 1026, 1959. Greater Vancouver Real Estate Board, S t a t i s t i c a l and Survey Committee., Real Estate Trends i n Metropolitan Vancouver, 1968. Vancouver: 1969. Heady, E.O., and Candler, W., Linear Programming Methods. Ames, Iowa: Iowa University Press, 1958. i i • . . I Heady, E.O., and H a l l , H.H., Linear and Non-Linear S p a t i a l Models i n A g r i c u l t u r a l  Competition, Land Use and Production P o t e n t i a l . Journal Paper J . 5995 of Iowa A g r i c u l t u r a l Experimental Station, Project 1405, 1969. Herbert, J.D., and Steven, B.H., "A Model for the D i s t r i b u t i o n of R e s i d e n t i a l A c t i v i t y i n Urban Areas". Journal of the Regional Science Association, Vol. 2, 1960, pp. 21-36. ' ' Howell-Jones, G.I., The Urbanization of the Fraser V a l l e y , The Lower Fraser V a l l e y :  Evolution of a C u l t u r a l Landscape, ed. A.H. Siemens, Vancouver: Tantalus Research, 1966. No. 3, 1965. Klassens, E.'j "Theories of Settlement Evolution". Unpublished paper, Department of Georgraphy, University of B r i t i s h Columbia, 1968. Kluckhohn, C , "Values and Value Orientation i n the Theory of Action". Towards a  General Theory of Action, eds. T. Parsons and E. S h i l s , Cambridge: Harvard Univ e r s i t y Press, 1954. L i c k f i e l d , N., The Economics of Planned Development. London: Estates Gazette, 1958. Lower Mainland Regional Planning Board, O f f i c i a l Regional Plan. New Westminster, B.C., 1966. Lower Mainland Regional Planning Board, Population Trends i n the Lower Mainland. New Westminster, B.C., 1968. - 108 -McConnel, S., "Economic Aspects of Land Use Planning". O f f i c i a l Architecture and Planning. Vol. 3, No. 2, 1969, 947-957. McLaughlin, J.F.., "Application of Linear Programming to Urban Planning". Unpublished M.V.P. di s s e r t a t i o n , University of I l l i n o i s , 1968. Parker, V.J., Problems and Progress i n Rationalizing the Use of Resources of the Fraser Valley, The Lower Fraser Valley: Evolution of a Cultural Landscape, ed. A.H. Siemens, Vancouver: Tantalus Research, 1966. R a t c l i f f e , R.V., The Metropolis i n Modern L i f e . New York: Fisher Co., 1955. Reiner, T.A., Journal of the American Ins t i t u t e of Planners, Vol. 32, No. 2, 1966. Reynolds, D.J., Economics Town Planning and-Traffic. London: I n s t i t u t e of Economic A f f a i r s , 1966. Schlager, K.J., "Simulation Models i n Urban and Regional Planning," S.E. Wisconsin  Regional Planning Commission Technical Record, Vol. 2, No. 1., 1964. S i l v e r , A.L., and Sloan, A.K., "A Model Framework for Comprehensive Planning i n New York Ci t y " , Journal of the American I n s t i t u t e of Planners, Vol. 31, No. 3, 1965. Steger, W.A., "The Pittsburgh Urban Renewal Simulation Model," Journal of the  American I n s t i t u t e of Planners, Vol. 31, No. 3, 1965. Ukeles, J.B., The Consequenes of Municipal Zoning. New York: Urban Land Economics Research I n s t i t u t e , 1966. * Uthwatt Report, Report on the Expert Committee on Compensation and Betterment. Cmnd. 6386. London: H.M.S.O., 1942. Willhelm, S.M. , Urban Zoning and Land Use Theory. New York: Free Press of Glencoe, 196 2. APPENDIX Appendix A l Re s i d e n t i a l Land Prices Used i n the Study Apartment Housing Lots Established Urban Single Family Housing Lots Established Developing Acreage Lowland Urban Urban - Rural Rural $ per acre West Vancouver 52,998 16,790 - -North Vancouver C i t y 137,440 23,205 North Vancouver D i s t r i c t 133,210 22,923 20,528 T" Vancouver 175,830 39,533 24,348 Burnaby 81,445 25,513 16,485 New Westminster 94,860 25,513 V-Richmond 37,340 23,438 21,563 19,688 Delta 32,343 17,243 14,339 11,435 Buntzen 17,000 14,000 Port Moody 46,893 19,390 15,235 Coquitlam 54,173 17,280 15,480 13,680 Port Coquitlam 53,818 20,000 17,000 Surrey 27,720 13,800 11,400 9,000 White Rock 57,089 23,040 18,240 Vall e y Sub Area 26,100 9,000 8,000 7,500 14,000 8999.90 7499.90 Note: The d e r i v a t i o n of these parameters i s described i n Chapter 6 of the text. - no -Appendix A2 Housing Densities of Established Urban and Developing  (Urban Zones by Mu n i c i p a l i t y or Area) Mu n i c i p a l i t y or Area Established Urban Developing Urban ' ' Lots per Acre West Vancouver 1.46 — North Vancouver Ci t y 3.57 -North Vancouver D i s t r i c t 3.46 0.95 Vancouver 3.14 -Burnaby 3.14 r -New Westminster 3.01 -Richmond 3.75 1.35 Delta 3.63 1.36 Buntzen - 1.00 Port Moody 2.77 ' -Coquitlam 2.88 0.86 Port Coquitlam 3.20 1.37 Surrey 3.00 0.94 White Rock 3.84 0.58 Valley Sub Area 2.90 0.80 Source: The Lower Mainland'Regional Planning Board of B.C. - I l l Appendix A3 A g r i c u l t u r a l Land Prices used i n the Study Vancouver Ci t y Burnaby Richmond Delta Buntzen Port Moody Coquitlam i Surrey Val l e y Rural Acreage $10,000 $10,000 $ 7,000 $ 8,000 $ 8,000 $ 8,000 $ 8,000 $ 2,000 $ 1,500 Upland Rural ($/Acre) Lowland Rural $5,000 $1,500 $1,000 $7,000 $8,000 $8,000 $8,000 $2,000 $1,500 Source: The Lower Mainland Regional Planning Board of B.C. - 112 -Appendix A4 Land Supply S t a t i s t i c s used i n the Linear Programming Study Established Developing Acreage Upland Lowland Urban Urban Rural Rural Rural No. Lots Acres Vancouver 5,800 - 835 Burnaby 18,419 - 275 -New Westminster 445 - - -Coquitlam 8,179 942 1,695 1,100 1,320 Port Coquitlam 5,331 869 - 565 Port Moody 2,013 - 845 -Buntzen - 1,000 610 -West Vancouver 6,034 - - -N. Vancouver C i t y 2,988 - - -N. Vancouver D i s t . 16,942 37 - -Richmond 25,510 881 1,690 6,470 Delta 16,366 1,023 1,534 17,148 Surrey 21,457 4,061 26,435 9,210 16,680 White Rock 1,677 3 - -Valley 35,912 7,105 18,852 135,050 123,700 Source: The Lower Mainland Regional Planning Board of B.C. - 113 -Appendix A5 Population Statistics Used in the Study Projected Incremental Increase in Population, 1966 to years shown. 1966^71 1966-76 1966-61 1966-^ -86 Metropolitan Area 133,616 276,616 442,616 631,616 Valley Area 18,727 38,727 59,727 84,000 Source; Lower Mainland Regional Planning Board,- Population Trends in  the Lower Mainland 1921^-1986, New Westminster, B.C., 1968. - 114 -Appendix A6 Rural Urban Conversion Ratios used i n the -Study No, of l o t s 'created' Area by conversion of 1 acre of Rural land to Urban Uses .  Vancouver City 3.14 Burnaby 3.14 Richmond 1.35 Delta 1.36 Buntzen 1.00 Port Moody 2.77 Coquitlam 0.86 Surrey 0.94 Valley 0.80 Note: The d e r i v a t i o n of these parameters i s descibed i n Chapter of the text. - 115 Appendix A7 Population Densities Used i n the Study Apartment Housing Established Urban Single Family Housing Established Developing Acreage Lowland Urban Urban Rural Rural West Vancouver North Vancouver Ci t y North Vancouver D i s t r i c t Vancouver Burnaby New Westminster Richmond Delta Buntzen Port Moody Coquitlam Port Coquitlam Surrey White Rock Valley Sub Area 119 141 137 137 148 160 231 223 133 138 157 184 236 159 persons per acre 5 . 12.8 12.8 11.3 11.3 10.84 13.5 13.1 9.7 10.4 11.8 10.8 13.8 10.4 3.4 4.8 4.9 3.6 3.1 4.9 3.4 2.09 2.9 11.3 11.3 4.8 4.9 3.6 9.7 3,1 3.4 2.9 4.8 4.9 3.1 4.9 3.4 2.9 Note; The d e r i v a t i o n of these parameters i s described i n Chapter 6 of the text. 

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