ALLOCATION IN THE LOWER MAINLAND
OF BRITISH COLUMBIA.
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
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.
LIST OF FIGURES v i i
Chapter
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
The Effect of Land Use Controls on Land Values......................... 24
Discussion.. 25
i v
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 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
The Derived Shadow Prices of the Constraints 90
General Considerations 97
BIBLIOGRAPHY 1° 6
i .- APPENDIX....... 109
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
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
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
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
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.
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
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)
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
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
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)
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.
- 42 -
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)
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
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 =
- 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:-
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
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
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 D i s t r i c t
Langley C i t y
Langley Township
Chilliwack Township
Cultus Lake
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 . , great