UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An assignment model of urban housing demand Mason, Greg C. 1972

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
UBC_1972 A8 M38.pdf [ 2.44MB ]
Metadata
JSON: 1.0107076.json
JSON-LD: 1.0107076+ld.json
RDF/XML (Pretty): 1.0107076.xml
RDF/JSON: 1.0107076+rdf.json
Turtle: 1.0107076+rdf-turtle.txt
N-Triples: 1.0107076+rdf-ntriples.txt
Original Record: 1.0107076 +original-record.json
Full Text
1.0107076.txt
Citation
1.0107076.ris

Full Text

AN ASSIGNMENT MODEL OF URBAN HOUSING DEMAND by GREG C. MASON B.A., University of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Economics We accept this thesis as conforming to the required standard, THE UNIVERSITY OF BRITISH COLUMBIA December, 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allow e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of E^cz^rTVySr)W t C J The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada A B S T R A C T A c r i t i c a l view of the literature on land rent and r e s i - dential location is undertaken with special emphasis on the journey to work hypothesis. A housing demand model is constructed based upon the new demand theory advanced by K. J . Lancaster and an assignment model of housing developed by W. F. Smith. The model that is presented i s a simple integer program that attempts to analyze housing demand given the assumption that both household and houses have unique and separable characteristics. These attributes of both the product and consumer are thought to affect the demand for different parts of the housing stock. i i TABLE OF CONTENTS Page INTRODUCTION 1 CHAPTER I. SURVEY OF LITERATURE ON LAND RENT AND THE ECONOMICS OF HOUSING 3 THE RESIDENTIAL BID PRICE 14 A CRITIQUE OF THE JOURNEY TO WORK 17 II . THE ASSIGNMENT THEORY OF HOUSING 22 THE NEW THEORY OF DEMAND 22 APPLICATIONS TO HOUSING, , 27 W. F. SMITH AND A MATRIX ANALYSIS OF NEIGHBOURHOOD CHANGE . . . . 28 III. AN ASSIGNMENT MODEL OF URBAN HOUSING DEMAND 36 THE ASSIGNMENT PROBLEM 37 THE BID FUNCTION MATRIX 39 A MODEL OF URBAN HOUSING DEMAND 41 THE MODEL: AN EVALUATION 45 SOME STATISTICAL CONSIDERATIONS 45 PROBLEM OF DEMAND AND SUPPLY 46 BIBLIOGRAPHY 48 APPENDIX 1 A NORMAL COMPETITIVE MARKET AND RENT MAXIMIZATION. . . . 51 APPENDIX 2 AN ALGORITM TO SOLVE THE TRANSPORTATION PROBLEM 53 APPENDIX 3 PROGRAM LISTING OF THE MODEL 56 APPENDIX 4 PROGRAM LISTINGS 60 i i i INTRODUCTION Housing i s the most complex good any consumer has to purchase. This essay deals with the economics of housing demand. Nothing i s attemp- ted with respect to house finance, macro housing p o l i c y for a nation, or the economics of housing supply. Chapter I begins by reviewing the h i s t o r y of land rent from Ricardo through Von Thunen and the land economists of the 1930's. The work of recent authors such as Lowdon Wingo, J r . , William Alonso and J . F. Kain are surveyed i n d e t a i l and then c r i t i c i z e d . The main weakness that my model attempts to overcome i s the lack of analysis that deals w i t h housing as a complex of c h a r a c t e r i s t i c s . Most of the w r i t i n g has paid l i p service to the existence of t e c h n i c a l l y separate a t t r i b u t e s that a f f e c t the demand for housing; however, very l i t t l e has been attempted i n r e s t r u c t u r i n g the theory. Chapter I I lays the basic theory f o r my model. The work of K. J . Lancaster i s reviewed. He was the f i r s t to restate demand theory using the assumption that i t was the c h a r a c t e r i s t i c s of a good which are demanded and not the good i t s e l f . The pure theory presented by Lancaster i s unsuitable for the analysis of consumer durables and recourse must be made to an integer programming framework. At t h i s point the work of W. F. Smith i s introduced. Recently Smith has formulated a simple model of housing demand using the assignment approach. His model i s a s i g n i f i c a n t advance i n that the households i n a community are pictured as examining each element of the housing stock and placing a b i d on each house. Then 2 with an assignment algorithm, each household i s placed i n one and only one house so as to maximize the aggregate rent of the community. The r a t i o n a l e for rent maximization i s shown to be sound w i t h i n the context of t h i s model. A defect i n h i s theory i s that the basis of the b i d formu- l a t i o n s i s very vague. By using the conclusions from Lancaster's theory a more secure basis for the formation of rent o f f e r s of b i d can be made. The l a s t chapter presents the model and a test run using com- p l e t e l y imaginary data and b i d function matrices. At t h i s point the model i s exceptionally u n r e a l i s t i c and the concluding part of the chap- ter i s spent i n examining some s t a t i s t i c a l methods that are "quick and d i r t y " so that the model may be made operational i n a short time. Aside from the s t a t i s t i c a l problems, there are some questions of the manner i n which the model may best be made dynamic. As i t stands, the model i s simply a s t a t i c one-shot assignment. Nothing i s said about the supply or financing of housing and t h i s c e r t a i n l y i s a major defect. In a d d i t i o n , nothing i s mentioned about the i n t e r a c t i o n of demand and supply, since one of the assumptions of the assignment s o l u t i o n i s that the number of assigned households must equal the number of houses to which these house- holds are assigned. Considerably more e f f o r t must be made i n these l a s t areas be- fore the model can become a useful planning t o o l . CHAPTER I SURVEY OF THE LITERATURE ON LAND RENT AND THE ECONOMICS OF HOUSING In t h i s chapter, the h i s t o r y of land rent i s b r i e f l y traced from i t s o r i g i n s i n the 19th Century to the present day. The o r i g i n a l thinker on the matter of why economic a c t i v i t y locates where i t does and why land p r i c e s are what they are was, of course, David Ricardo.''" Ricardo showed that the most productive land was the f i r s t to be c u l t i v a t e d . As the demand f o r farm produce grew w i t h population, less productive land was used. Since the land already i n use yielded a higher return, the competitive process resulted i n d i f f e r e n t i a l prices of the land. The difference between the p r i c e of a p a r t i c u l a r p l o t and the pric e of the l e a s t f e r t i l e or marginal p l o t was c a l l e d the economic rent. Ricardo gives l i t t l e consideration to the other costs i n a g r i c u l t u r e such as transportation costs to and from the market place and most often he assumed that these costs were equal. As a r e s u l t , he came to no conclu- sions concerning the exact l o c a t i o n of various types of farming or r e - course extracting a c t i v i t y . Later, an economist i n Germany e x p l i c i t l y treated the problem of transport costs. J . Von Thunen assumed that f e r t i l i t y d i f f e r e n t i a l s 2 were non-existent. Like Ricardo, the competitive market process r e s u l t s David Ricardo, On The Principles of Political Economy and Taxation, 1817. 2 Johann H. Von Thunen, The Isolated State, 1826. 3 4 i n the highest land prices being paid f o r the most desirable p l o t s of land. In th i s case, however, the attractiveness of land Is based upon the savings due to l o c a t i n g as close to the market as pos s i b l e . The rent the land earns i s the transport cost saving from l o c a t i n g closer to the market place. This theory underlies v i r t u a l l y a l l the analysis of econo- mic l o c a t i o n theory. Once the costs of production are also included, the rent on a p l o t of land i s the value of the product less the costs of transport and production. The theory remained i n th i s r e l a t i v e l y crude s t a t e apart from 3 some embellishments by A l f r e d Marshall u n t i l the ea r l y 20th Century. Marshall made the d i s t i n c t i o n between the s i t u a t i o n or s i t e value and the a g r i c u l t u r a l rent. The s i t e value or the p r i c e of urban land i s the p r i c e i t would obtain as farm land plus the sum t o t a l of monetary advantages i t possesses by i t s l o c a t i o n i n the c i t y . Much l a t e r , w r i t e r s were to em- phasize the r o l e that external economies of scal e play i n conditioning the value of the land. The next contribution to land rent theory was made by R. M. Hurd who outlined a theory of urban land values p a r a l l e l to Von Thunen's. "Since value depends upon economic rent, and rent on l o c a t i o n and l o c a t i o n on convenience, and con- venience on nearness, we may eliminate the intermediate steps and say that value depends upon nearness." 5 A l f r e d Marshall, Principles of Economics (London: Macmillan and Co., 1917), e s p e c i a l l y Appendix G. ^R. M. Hurd, Principles of City Land Values (New York: The Record and Guide, 1903). ^Tbid., p. 11. 5 The next step i n the evolution of the theory was the f o r m a l i z a t i o n of Hurd's analysis i n t o a canon. R. M. Haig f i r s t advanced the proposition that there was a complementarity of rent and transportation costs. In other words, the rent on any s i t e was equal to the transport costs not paid. U n t i l very recently, t h i s "law" formed the basis of land rent theory and r e s i d e n t i a l l o c a t i o n theory. One of the most recent and w e l l known contributions i s Transpor- tation and Urban Land by Lowdon Wingo, J?J The assumption made by Wingo i s common to most of the work In the f i e l d since Hurd. A featureless p l a i n with no geographical or i n s t i t u t i o n a l b a r r i e r s to movement i s assumed. Wingo assumes that there i s perfect competition i n the labour markets and the workers i n the c i t y have completely homogeneous tastes and incomes. Also assumed i s the complementarity of transport costs and rent. Transport costs are composed of f i n a n c i a l costs which include the expenditure dependent upon the distance t r a v e l l e d and terminal costs which are a function of the congestion i n the c i t y . In addition there are the opportunity costs of the time t r a v e l l e d which cannot be computed d i r e c t l y . These opportunity costs are thought to be an extension of the working day. Since the labour market i s assumed to be perfect the worker can simply s h i f t them back to the em- ployer by demanding an increment to the pure wage or the wages of those who l i v e at the job s i t e . R. M. Haig, "Toward An Understanding of Metropolis," Quarterly Journal of Economies, Vol. 35, No. 2 CMay, 1926). 7Lowdon Wingo, J r . , Transportation and Urban Land (Washington, D.C.: Resources f o r the Future, Inc., 1961). 6 Three equations form the b a s i s of Wingo's t h e o r e t i c a l system. F i r s t , t h e r e i s the demand f o r c e n t r a l i t y o r l o c a t i o n : pq - k ( t o ) = k ( t m ) (1) where p i s the p r i c e per square f o o t of l a n d ; q the q u a n t i t y of l a n d con- sumed; k ( t ) the t r a n s p o r t c o s t s f u n c t i o n w i t h t the p o i n t of s e t t l e m e n t and t the d i s t a n c e from the c e n t r e of the c i t y to the f r i n g e . T h i s equa-m t i o n s t a t e s t h a t the t r a n s p o r t c o s t s p l u s the r e n t f o r a s i t e i s e q u a l f o r a l l workers i n the c i t y and i s equal to the t o t a l commutting c o s t s from the c e n t r e 0 f the c i t y to the f r i n g e . Second, t h e r e i s the demand f o r space which i s a s i m p l e p a r a m e t r i c e x p r e s s i o n : q = ( a / p ) b (2) where q and p are as b e f o r e and a and b are parameters. A t any g i v e n l o c a - t i o n e q u a t i o n (1) g i v e s the amount spent on l a n d w h i l e e q u a t i o n (2) i n d i - cates the amount of l a n d consumed by the worker. T h i r d , i f the a v a i l a b i l i t y of l a n d i s g i v e n by a s i m p l e e x p r e s - 2 s i o n such as S = irt where TT i s the c o n v e n t i o n a l e x p r e s s i o n and t the r a d - i u s of the c i t y , th_en Wingo c a l c u l a t e s the margin of s e t t l e m e n t a c c o r d i n g to the f o l l o w i n g e q u a t i o n : t n = 2TT / m t q ( t ) d t (3) where n i s the p o p u l a t i o n of the c i t y ; l / q ( t ) i s the d e n s i t y of s e t t l e m e n t and the ot h e r v a r i a b l e s are as d e f i n e d as b e f o r e . The o n l y unknown i n t h i s f o r m u l a t i o n i s tm wh i c h i s found s i m u l t a n e o u s l y w i t h equations (1) and ( 2 ) . 7 One of the important conclusions Wingo ar r i v e s at i s that the transport technology w i l l be r e f l e c t e d by the land prices i n the c i t y . I f the cost of movement i s high, then competition f o r the c e n t r a l s i t e s w i l l be keen and thus the land prices f o r the r e s i d e n t i a l s i t e s close to the job w i l l command a premium. The actual l e v e l of land p r i c e i s a function of the numbers of workers i n the c i t y . This i s elementary but important. Recently, there has been considerable empirical work done on the importance of the journey to work as a determinant of r e s i d e n t i a l l o c a t i o n . J . F. Kain i s perhaps the most known of researchers i n t h i s area. His theo r e t i c a l base i s s i m i l a r to Wingo and the land economists, i n that the compl mentarity of transport costs and rent i s assumed. The transport costs of the household are broken i n t o three cate- gories : 1. The costs of t r a v e l l i n g to and from service obtainable w i t h i n the r e s i d e n t i a l area. 2. The costs of t r a v e l l i n g to and from work. 3. The costs of t r a v e l l i n g to and from those services only a v a i l a b l e outside the r e s i d e n t i a l area. Kain i s vague on the d e f i n i t i o n of area and appears to use the word interchangeably with r i n g . He presents some preliminary s t a t i s t i c a l evidence to show the importance of the journey to work. For example, 43.9 J . F. Kain, The Journey to Work as a Determinant of Residential Location, Papers and Proceedings of the Regional Science Association, IX 1962. 8 per cent of a l l trips undertaken by households sampled from 39 major c i t i e s are journeys to work, while 21.4 per cent are social and recreational jour- neys. Since many of the recreational and social centers i n c i t i e s are close to employment centers, the social destination trips are l i k e l y to reinforce the journey to work. Kain takes the conclusions of the land economists i n developing his hypotheses that are to be tested: 1. Transport costs increase with distance from the workplace. 2. The price of land decreases as distance from the job s i t e increases. 3. The workplace of the individual i s fixed. 4. The household maximizes u t i l i t y . 5. Housing is a normal good. The assumption about the complementarity of the rent and trans- port costs is retained. Location rent i s the saving possible per unit of land consumed the household may achieve by moving farther from the place of work. If rents per unit area decrease as the household moves from the place of work, then the absolute savings depend upon the amount of space consumed. Kain describes this situation by isospace or bid price curves (BPC) which show the decline i n location rent for each amount of land con- sumed as the household moves away from the job s i t e . While the l o c a t i o n rent declines with movement away from the employment s i t e , the transport costs increase. T(x) Is tke transport cost function and the t o t a l transport costs paid i n l i v i n g at any one l o c a t i o n i s the area under the curve from the place of employment to the residen- t i a l s i t e . S i m i l a r l y , the transport cost savings or l o c a t i o n rent i s the area under the isospace curve that applies to the amount of land being consumed by the household. The minimum costs l o c a t i o n i s simply at the i n t e r s e c t i o n of the two curves. The locations which minimize the l o c a t i o n costs of the house- holds are now known. The t o t a l l o c a t i o n cost divided by the space consumed i s the p r i c e the household must pay for r e s i d e n t i a l space. Given the p r i c e of a l l other goods and services, the consumption of r e s i d e n t i a l space i s determined. Once the amount of space consumed i s known, then the residen- t i a l l o c a t i o n of each household i s determined. From t h i s a n a l y s i s , the author concluded that for households of d i f f e r e n t types, depending upon ethnic o r i g i n , income, age composition and family s i z e , there w i l l be d i f f e r e n t propensities to undertake a journey to 10 work of given length.. Also, r e s i d e n t i a l l o c a t i o n i s a function of the job s i t e . The empirical test consists of s t r a t i f y i n g D e t r o i t i n t o s i x r i n g s . The f i r s t f i n d i n g i s not s u r p r i s i n g . Residences as a proportion of land area increases as distance from the inner r i n g increase. The inner r i n g i s , of course, the prime employment area i n any urban area. I t was discovered that most of the journeys to work are from the outer to inner r i n g s . This supports the theory i n that the only way to reduce the l o c a t i o n rent paid i s by moving toward the perimeter of the c i t y . As the edge of the c i t y i s reached, the l o c a t i o n rent curve f l a t t e n s and the constraint on space consumption eases. Other findings of i n t e r e s t are that workers i n the CBD make con- siderably longer journeys to work. In ad d i t i o n , the length of the journey to work i s a function of income. The r i c h appear to have a h i g h preference for space and can aff o r d the transport costs to get i t . Very small house- holds and large households tend to make the shortest journeys to work w i t h middle sized households undertaking longer commuting journeys. At f i r s t blush, the empirics seem to substantiate the claims of the land economists. The household does appear to sub s t i t u t e savings i n land costs f o r transport expenditures. Before considering objections to the theory, I w i l l now consider the work of W. Alonso. The work of Alonso, Location and Land Use, i s a t h e o r e t i c a l im- provement over that of Wingo and the other land economists i n that where Q Wingo postulates separate demands for space and l o c a t i o n , Alonso integrates William Alonso, Location and Land Use (Cambridge, Mass.: Har- vard University Press, 1961). 11 them into a u t i l i t y maximizing framework. Both space and l o c a t i o n enter the u t i l i t y and budget equations of the household. Previous work was con- tent with asking only where the household w i l l l o c a t e ; or i f the amount of space consumed i s investigated, and a r b i t r a r y demand f o r space equa- t i o n with l i t t l e basis i n r e a l i t y i s employed. The basic assumptions used are common to most work i n land economics: a featureless p l a i n with no geographical b a r r i e r s or features which d i s t i n g u i s h one area from another. No i n s t i t u t i o n a l b a r r i e r s to the transfer of property between landlords are assumed to e x i s t . S i m i l a r l y , transportation i s uninhibited by natural b a r r i e r s and there are no unusual costs to overcoming the f r i c t i o n of space. Perfect knowledge abounds with the f i r m maximizing p r o f i t s and the consumer maximizing u t i l i t y . P r i c e as used by Alonso r e f e r s to the amount the household or fi r m pays for the r i g h t to use one u n i t of land. Under these terms are subsumed the costs of ownership, contract rent and sales p r i c e i n the long term, which given perfect competition tend to equality ( i n terms of d i s - counted present value). The study commences by assuming that a l l economic (commercial, r e t a i l , i n d u s t r i a l , etc.) a c t i v i t y takes place at the core of the c i t y . Therefore the household always faces the center of the c i t y when attempting to f u l f i l l various demands. Once commercial and a g r i c u l t u r a l users are ex- p l i c i t l y accounted for t h i s assumption i s lent some p l a u s i b i l i t y . Since the consumer i s asked to make the dual decision of how much to buy and where to s e t t l e t h i s must be included i n the income and u t i l i t y functions. The income equation consists of the d i r e c t costs of s i t e con- t r o l , the costs of commuting to and from the s i t e , plus the costs of a l l 12 other goods and s e r v i c e s . To prevent the study from becoming too complex, a l l goods a s i d e from l a n d are aggregated i n t o a composite commodity — z. The budget e q u a t i o n appears as f o l l o w s : y = P z z + p ( t ) q + k ( t ) C4) where p^ i s the p r i c e of the composite good; z i s the composite good; p C t ) i s the p r i c e of l a n d a t d i s t a n c e t from the c e n t e r of th_e c i t y ; q the amount of l a n d consumed; and k ( t ) the c o s t s of commutting a s s o c i a t e d w i t h t h a t p a r t i c u l a r s i t e . The u t i l i t y f u n c t i o n i s simple and s t r a i g h t f o r w a r d : U = u ( z , q, t ) (5) S u b j e c t i n g these e x p r e s s i o n s to the u s u a l t o o l s of d i f f e r e n t i a l c a l c u l u s , Alonso o b t a i n s the f o l l o w i n g s o l u t i o n s : u /u = p ( t ) / p (6) a z z u /u = (qdp/dt + dk/dt)/p (7) t z z The i n t e r p r e t a t i o n of these e x p r e s s i o n s i s s i m p l e . The f i r s t [ e q u a t i o n C6)] s t a t e s the m a r g i n a l r a t e of s u b s t i t u t i o n between l a n d and a l l o t her goods i s equal t o the r a t i o of t l i e i r r e s p e c t i v e p r i c e s . I n a s i m i l a r v e i n the second e q u a t i o n [ e q u a t i o n (7)1 s t a t e s t h a t th_e m a r g i n a l r a t e of s u b s t i t u t i o n between d i s t a n c e and a l l other goods i s a g a i n e q u a l to t h e i r p r i c e r a t i o s . The p r i c e of d i s t a n c e i s equal to the t o t a l amount spent on l a n d a t p o i n t t Cqdp/dt), and the c o s t s of commutting to p o i n t t ( d k / d t ) . The numerator i n the second e x p r e s s i o n [ e q u a t i o n (7)] i s impor- 13 tant since i t indicates that cost of a marginal movement Is equal to the change i n the amount paid for land plus the change i n commuting costs. Since consumption of the composite good i s pleasurable, therefore i s p o s i t i v e and since commuting fr^s d i s u t i l i t y attached to i t , the U i s negative. Commuting costs are r e l a t e d to the distance t r a v e l l e d and even i f there were no d i r e c t costs to movement there would always be the opportunity costs, t h i s makes dk/dt negative. Since q i s p o s i t i v e , t h i s implies that the p r i c e of land declines as one moves away from the center of the c i t y . The conclusion that Alonso derives from t h i s analysis i s that the consumer of urban housing s e t t l e s at the point where the costs of commut- ti n g are j u s t greater than the savings from consuming cheaper land. In other words, the point of equilibrium f o r the resident i s the point at which the costs of commutting incurred by moving incrementally from the centre are exactly balanced by the savings i n the p r i c e of land by such a move. Alonso proceeds to examine the actions of commercial and a g r i c u l - t u r a l users of land. Instead of maximizing u t i l i t y , they maximize p r o f i t s . A revenue and cost function are substituted for the u t i l i t y and budget con- s t r a i n t . Since the d e t a i l s are not germane to the a n a l y s i s , I w i l l skip them and proceed to consider the nature of.the b i d p r i c e function which forms a c r u c i a l l i n k i n Alonson's wrork. I t also i s an important step i n the construction of the simulation model and thus needs to be examined c a r e f u l l y . 14 THE RESIDENTIAL BID PRICE CURVE As was stated i n the int r o d u c t i o n , the bid p r i c e curve f o r an i n d i v i d u a l i s defined as "the set of land prices an i n d i v i d u a l would pay at various distances and s t i l l derive a constant l e v e l of s a t i s f a c t i o n . Several points need to be stressed. F i r s t , the curves f o r d i f f e r e n t house- holds could and most l i k e l y would be very d i f f e r e n t . Secondly, a p a r t i c u - l a r bid p r i c e curves refers to a given l e v e l of s a t i s f a c t i o n and as a r e s u l t each household has many bid p r i c e curves corresponding to d i f f e r e n t l e v e l s of s a t i s f a c t i o n . Third, the bid p r i c e curve i s i n no way re l a t e d to th_e pri c e that i s eventually paid. There i s no consideration of supply f a c - tors and therefore i n a sense t h i s i s a very u n r e a l i s t i c concept. This point w i l l a r i s e l a t e r i n the essay. Once the bid prices are established for 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 users, Alonso employs a game theoretic approach where the users compete f o r the land w i t h the sale going to the highest bidder. Since commercial and a g r i c u l t u r a l users are extran- eous to the essay I w i l l pass over t h i s point. The b i d p r i c e function i s derived very simply. By d e f i n i t i o n a bid p r i c e i s the amount the household pays for a given combination of l o c a - t i o n and u t i l i t y . Therefore both the u t i l i t y and distance are treated i n i t i a l l y as given w i t h u t i l i t y set at u and l o c a t i o n set at t . The o o u t i l i t y function now appears as follows: U q = uCz, q, t Q ) (8) Alonso, op. cit., p. 59. 15 and the budget c o n s t r a i n t w i t h y^, p z , k ( t ) a l l g i v e n : y o = p z z + P C O q + k C t o ) C9) The s i g n s of the p a r t i a l s remain the same as b e f o r e . I f the problem i s c a s t i n a Lagrangean framework the f o l l o w i n g s e t of equ a t i o n s r e s u l t , ^ U = u ( z , q, t Q ) - X[y - Cpz Z + p C t Q ) q + K O ] (10) 8u • | = u q - Ap(d o) = 0 C12) 8u and from (11) and (12): _E - P < d o } u p- z 2 (14) Equations ( 8 ) , (9) and (14) now form a system of t h r e e e q u a t i o n s w i t h u, t , p^, x, y and k ( t Q ) a H g i v e n and t h r e e unknowns [ z , q, p ( t Q ) ] . As w i t h any system of l i n e a r e q u a t i o n s , i t can be made p a r a m e t r i c s i m p l y by choos- i n g a " g i v e n " and denoting i t a parameter. I n t h i s case, i f t i s chosen as the parameter, then the p r i c e of l a n d p ( t ) can be s o l v e d f o r v a r i o u s d i f f e r e n t l o c a t i o n s and becomes the b i d p r i c e c u r v e , w h i c h i s a f u n c t i o n of d i s t a n c e t . 12 Some Important c o r o l l a r i e s a r e proved by Alo n s o : 11„ Alonso uses the t o t a l d i f f e r e n t i a l i n s t e a d of the Lagrangean- however, the r e s u l t s a r e the same. ' ' Al o n s o , op. ext., Appendix H. 16 1. The b i d p r i c e curve i s s i n g l e valued, implying that for any given u t i l i t y f u n c t i o n at any s p e c i f i e d l o c a t i o n there i s only one b i d p r i c e f or the household. 2. Lower b i d prices imply greater u t i l i t y since they s i g n i f y that the bids f or land i n the community are lower. 3. Bid p r i c e curves f o r the same household do not cross. The equilibrium of the household can be found by superimposing the p r i c e of land upon the mapping of the b i d p r i c e curves as shown i n Figure 1. I n c i d e n t a l l y , i t can be shown that the b i d p r i c e curves slope downward. In addition, Alonso shows that the p r i c e of land declines less r a p i d l y as the distance from the centre of the c i t y increases as i s also shown i n Figure 2. The point becomes clear when the next author's work i s considered and as w i l l be demonstrated has to do with the transport technology of the c i t y . $/sq. f t . BPC P(d) 1 t Figure 2 17 A CRITIQUE OE THE JOURNEY TO WORK The l i t e r a t u r e j u s t surveyed suffers from several serious de- fect s that stem from the assumptions employed by some or a l l of the research- ers . The f i r s t issue involves the assumption of p e r f e c t i o n i n e i t h e r the labour or land markets. The w r i t e r s named above a l l view the consumer or worker operating i n an environment of perfect competition. Wingo's assumption of perfect labour markets i s not d i r e c t l y r e l a t e d to land and housing economics but the assumption used by Wingo, Alonso and Kain that there are no impediments to the entry and e x i s t i n the housing or land mar- ket i s very r e s t r i c t i v e . The requirements for competition i n a market are w e l l known and are f u l f i l l e d by the following conditions. 1. Buyers and s e l l e r s must be numerous. 2. The transactions of any one economic u n i t must be small enough not to have any e f f e c t on the prices or quantities offered i n the market. 3. There i s no c o l l u s i o n . 4. Entry and e x i t i s free and unimpeded for both buyers and s e l l e r s . 5. A l l p a r t i c i p a n t s have complete and costless i n f o r - mation. 6. There are no i n s t i t u t i o n a l b a r r i e r s to transaction. 7. The product i s homogeneous. These points can be summarized by three conditions that there be homogeneous goods, many buyers and s e l l e r s and the costs of Information and a c q u i s i t i o n are n i l . Taking these points i n turn, i t becomes apparent that 18 the housing market may by d e f i n i t i o n be imperfect. V i r t u a l l y a l l goods are d i f f e r e n t i a t e d : even simple commodities such as cement and wheat are d i f f e r e n t i a t e d to the informed buyer normally i n these markets. Hous- ing i s perhaps the most complex of consumer goods. In a d d i t i o n most hous- ing possesses a f i x e d l o c a t i o n which automatically acts to give each- house a uniqueness. To judge a market as imperfect simply due to the very nature of the good appears to be misguided. I t can be argued that housing i s one economic good that i n f o r - mation i s e a s i l y obtained. C l a s s i f i e d ads and r e a l estate companies act to disseminate t h i s information i n an e f f i c i e n t manner. Information i n a market does not merely consist of easy knowledge of what goods are presently av a i l a b l e but what goods w i l l be demanded and i n supply i n the future. Due to the nature of housing, there tends to be a long production period. In addition, housing i s durable. D u r a b i l i t y , long production periods and f i x e d l o c a t i o n are the common reasons that are given as to why the housing market should by d e f i n i t i o n be imperfect. Surely the most important consideration when examining the per- f e c t i o n of any p a r t i c u l a r market must be the r e l a t i v e number of buyers and s e l l e r s . I f s i n g l e family units are considered alone then no v i o l a t i o n to r e a l i t y i s done i f the perfect competition assumption i s used. I f the mar- ket consists of m u l t i p l e dwelling u n i t s , then i t i s l i k e l y that the numbers of owners or s e l l e r s i s very much, less than the number of buyers. Since there i s l i t t l e empirical evidence on the structure of the housing market, a l l that can be said i s that a priori housing i s an imperfect market. 19 A second objection to the literature surveyed i s the treatment of the location decision of the household. At best the household i s viewed considering only the distance to the job s i t e and the amount of land con- sumed. Some evidence in the form of the frequency of the trips to various destinations about the city was given by Kain which showed that almost 50 per cent of a l l trips were work-oriented. This i s not enough to enable i t to be stated categorically that the journey to work is the sole determinant of residential location. None of the locational theorists surveyed main- tained that this was the case; however, l i t t l e work has been done i n estab- lishing other locational motivations to the residential decision. Some recent empirical work was attempted by J. Wolforth who examined Kain's .hypothesis concerning the substitution of journey to work 13 expenditures for s i t e expenditures. The alternate hypothesis advanced by Wolforth i s that the costs of commutting are not sufficient to affect the location of the residences. The consumer lives where i t can be afforded and meets the costs of commutting as best as possible. While this does not directly contradict Kain's hypothesis, however, i t s v e r i f i c a t i o n would i n - dicate that the journey to work was not such an important factor i n residen- t i a l location that other motivations can be ignored. The proposition was tested similarly to Kain. Vancouver was divided into six rings and the labour force was classified into six occupa- tional classes. The percentage of each occupational group i n each ring was J. Wolforth, Residential Location and Place of Work in Vancouver, (Vancouver; Tantalus Press, 1965). 20 compared to the mean income of the r i n g . Wolforth discovered that lower income workers tend to locate closer to the CBD than do more a f f l u e n t workers. A second t e s t was t r i e d i n which, the c i t y was divided i n t o areas assumed to have homogeneous housing costs. The median value of hous- ing i n each census t r a c t was assumed to be the cost of housing i n a p a r t i c - u l a r t r a c t . Spearman c o r r e l a t i o n c o e f f i c i e n t s were computed f o r the occu- pational groups ranked according to percentage i n each t r a c t and occupation groups ranked by income. The c o e f f i c i e n t s were s i g n i f i c a n t and p o s i t i v e . The conclusion of Wolforth's study i s that there i s considerably more v a r i a t i o n i n r e s i d e n t i a l l o c a t i o n patterns than would be expected i f proximity to work was the only motivation to choosing a house. Unfortunate- l y , l i t t l e more can be said from hi s study. The difference between Wolforth's and Kain's study can l a r g e l y be a t t r i b u t e d to differences i n the c i t i e s studied. Those c i t i e s that have been established f o r a long period of time (more than 100 years), grew using a transport technology that was c o s t l y to i n d i v i d u a l s . As a r e s u l t , a prem- ium was placed upon c e n t r a l i t y and the placement of i n d u s t r i a l and commer- c i a l a c t i v i t y was located at the core. Thus these c i t i e s , such as New York, D e t r o i t , Montreal, etc., conditioned the Inhabitants i n t o accepting c e r t a i n l o c a t i o n patterns. Admittedly, these constraints on r e s i d e n t i a l l o c a t i o n are weakening as the c i t i e s expand; however, compared to Vancouver which- grew using more i n d i v i d u a l and less c o s t l y transportation (the car] they s t i l l l i k e l y place more constraint upon the consumption a l t e r n a t i v e s of the worker. 21 The point that must be made here i s the household In a l l l i k e - lihood i s responsive to other t r i p s . The propensity to t r a v e l to various destinations such as shopping centres, schools, r e c r e a t i o n a l centres, nightclubs, etc. varies w i t h the structure of the household. The number of c h i l d r e n and t h e i r ages are important factors i n where to lo c a t e f o r those f a m i l i e s . S i m i l a r l y f o r s i n g l e person households, proximity to night l i f e i s important and r e f l e c t e d i n the amount the household would be w i l l i n g to pay to l i v e i n an area close to such f a c i l i t i e s . The households responds to many l o c a t i o n a l p u l l s . The theory surveyed while paying l i p service to t h i s has assumed that these t r i p s were i n s i g n i f i c a n t and no damage to the realism of the r e s u l t s was made i f the journey to work was assumed to be the only l o c a t i o n a l motivation. I propose a model that views the house as a c o l l e c t i o n of a t t r i b u t e s , not a l l l o c a t i o n a l , that households of d i f f e r e n t structure and income value d i f f e r e n t l y . In the next chapter the theory underlying the model i s out- l i n e d . Two separate strands — one from operations research, the other from modern demand theory — are united to form the base of the model. CHAPTER I I PROGRAMMING THEORY AND THE ECONOMICS OF HOUSING In the previous chapter the l i t e r a t u r e on r e s i d e n t i a l l o c a t i o n was surveyed and found d e f i c i e n t i n I t s assumption that the household de- cides on the basis of only two c h a r a c t e r i s t i c s — space and l o c a t i o n to work. The reason for t h i s i s that t r a d i t i o n a l economic theory i s quite constrained i n the analysis of consumer goods. In t h i s chapter I out- l i n e a theory of demand f i r s t developed by K. J . Lancaster. He employs a l i n e a r a c t i v i t y analysis to study the behaviour of consumers when goods are considered to have c h a r a c t e r i s t i c s which d i f f e r e n t i a t e them from one another. Once t h i s theory i s ou t l i n e d , i t becomes apparent that f o r con- sumer durables such as housing, the analysis given by Lancaster needs to be amended to an integer programming framework or assignment model. Once th i s i s established, the work of W. F. Smith i s reviewed. Smith has used an assignment approach to housing studies. From here i t i s a short step to my model of housing demand. THE NEW THEORY OF DEMAND Tr a d i t i o n a l economic theory has the consumer s l i d i n g up and down a smooth u t i l i t y curve choosing between two goods, often totally- unrelated and pinned to r e a l i t y only by a budget constraint. The consumer i s p i c - tured as making r a t i o n a l choices between shoes and cars or guns and butter with the trade-off varying ( i n two dimensions) from a s t r a i g h t l i n e f o r perfect substitutes to a r i g h t angle for perfect complements. An under- 22 23 current of economic theory has always argued that the choice i s more ordered. Cars are traded off with commuter service of various types, butter w i t h mar- garine and shoes with other clothes, Implying that consumers choose among c h a r a c t e r i s t i c s rather than goods. K. J . Lancaster has formalized t h i s view i n t o a f a i r l y rigorous theory and postulates three assumptions as the departure from conventional thinking on the matter."^ 1. I t i s the c h a r a c t e r i s t i c s inherent i n the good and not the good i t s e l f which y i e l d s u t i l i t y . 2. In general, goods possess more than one charac- t e r i s t i c . 3. Goods i n combination, or complements may give r i s e to more than one c h a r a c t e r i s t i c and d i f f e r e n t charac- t e r i s t i c s than goods s i n g l y . A c t u a l l y , the a p p l i c a t i o n of l i n e a r a c t i v i t y analysis to con- sumption theory i s merely the reverse of production theory. In production theory an a c t i v i t y involves the combination of several inputs i n the crea- t i o n of one product, while i n consumption theory the act of consuming involves one input or good and several j o i n t outputs or c h a r a c t e r i s t i c s . Therefore, associated with each good there i s a vector of charac- t e r i s t i c s . I f i s t h i s vector and b.. i s the amount of i t s c h a r a c t e r i s - t i c s possessed by good j , then a l l the vectors of c h a r a c t e r i s t i c s may be represented by a matrix B which Lancaster refers to as the consumption 14 K. J . Lancaster, Mathematical Economics (New York: Macmillan, 1968); "The New Theory of Consumer Demand," Journal of Political Economy, Vol. 74, No. 4 (July, 1966). 24 technology matrix. In general, the numbers of c h a r a c t e r i s t i c s and goods w i l l not be equal and Lancaster postulates that for advanced economies the number of goods i s greater than the number of c h a r a c t e r i s t i c s . I f the e n t i r e array of c h a r a c t e r i s t i c s i s represented by z and the array of goods by x then the fundamental r e l a t i o n s h i p of the theory follows: z = Bx . (15) Assumed i n the simply theory i s that the c h a r a c t e r i s t i c s are normal or no s a t i a t i o n of c h a r a c t e r i s t i c s i s possible. Also, i f the matrix B i s square, and can be decomposed in t o a diagonal matrix, t h i s new theory i s nothing more than a restatement of the old theory. Since the consumer operates i n c h a r a c t e r i s t i c space and not goods space, the u t i l i t y function i s of the following form: U = U(z ) (16) This i s maximized subject to the f o l l o w i n g constraints: z = Bx (17) px <_ k (18) x >o (19) where p i s a vector of goods prices and k i s the income of the household. The program as i t stands i s non-linear but can be transformed simply by s u b s t i t u t i n g Bx f o r z i n the u t i l i t y function. As long as a l l the elements of B are p o s i t i v e and that x i s non- negative, then the "attainable c h a r a c t e r i s t i c s s e t " i s i n the p o s i t i v e quadrant. Given are two c h a r a c t e r i s t i c s and four goods. Goods are repre- 25 sented by rays which, i n d i c a t e the mixture of c h a r a c t e r i s t i c s each possesses. The attainable c h a r a c t e r i s t i c s set i s shown by the shaded area. The quanti- t i e s of any one good that can be purchased by spending one's e n t i r e income on i t i s r e f l e c t e d by the length of the ray i n question. Some goods Cas good 3 i n Figure 3) may not be considered at a l l by any consumer. The per- sonal choice i s shown by the in d i f f e r e n c e curves 1^ - 1^• An optimal s o l u t i o n i s one which minimizes the expenditure of the household w i t h i n the constraints set by the atta i n a b l e c h a r a c t e r i s t i c s - s e t . There are two s u b s t i t u t i o n e f f e c t s i n operation here c a l l e d the e f f i c i e n c y s u b s t i t u t i o n and the convention s u b s t i t u t i o n , shown i n Figures 4 and 5. Suppose the pr i c e of good two r i s e s . This has the e f f e c t of mov- ing point X2 along the ray 2 (Figure 4). As soon as #X1, X2, X3 form a 26 F i g u r e 6 C e n t r a l i t y 27 s t r a i g h t l i n e the consumer w i l l maximize h i s welfare by switching to a combination of goods 1 and 3. This i s the e f f i c i e n c y s u b s t i t u t i o n . Con- ventional s u b s t i t u t i o n occurs when the e n t i r e constraint function s h i f t s inwards and the i n d i f f e r e n c e map governs the switch i n p o r t f o l i o of goods held as i n Figure 5. APPLICATIONS OF THE THEORY TO HOUSING Consider a straightforward approach to housing w i t h no m o d i f i - cation of the theory. Assume a r e s i d e n t i a l bundle with a w e l l defined set of c h a r a c t e r i s t i c s . These include such factors as proximity to work, play, the schools, the noise l e v e l of the area, the ethnic mix of the neighbour- hood, etc. Graphic p o r t r a y a l of the s i t u a t i o n i s shown i n Figure 6 where two c h a r a c t e r i s t i c s and two housetypes are shown. The theory implies that given the type of u t i l i t y function i n the f i g u r e , then the r a t i o n a l household would hold two types of houses. However, the formulation ignores the prob- lem of the costs of m u l t i p l e dwelling ownership. I t i s very l i k e l y that these are very high. Representation of t h i s s i t u a t i o n becomes very d i f f i - c u l t since i f the f e a s i b l e f r o n t i e r made non-convex by having the p r i c e l i n e bow i n as i n the dotted l i n e t h i s would indeed force the consumer to hold only one house, however the economic meaning of such geometry i s dub- ious. In e f f e c t , t h i s formulation implies that there i s some smooth func- t i o n a l r e l a t i o n s h i p between the type of house and the transaction costs. A l l that can be r e a l l y stated unequivocably i s that the f e a s i b l e region remains non-convex for most consumers of housing with incomes below a cer- t a i n l e v e l simply because the holding of m u l t i p l e dwelling units f o r pure consumption purposes i s very rare. 28 A second i m p o r t a n t i s s u e r e v o l v e s around the a c t u a l r e p r e s e n t a - t i o n of houses i n t h i s framework. By u s i n g a continuous v e c t o r , what i s i m p l i e d i s t h a t the consumer can f r e e l y a d j u s t the amount of ho u s i n g t h a t i s consumed. I n ot h e r words, the p r i c e l i n e may f a l l anywhere al o n g a g i v e n r a y and t h e r e would be a house t h a t would e x i s t w i t h the exact mix of c h a r a c t e r i s t i c s s p e c i f i e d . I n r e a l i t y t h i s i s u n l i k e l y . Large con- sumer goods such as houses need to be r e p r e s e n t e d by d i s c r e t e p o i n t s i n c h a r a c t e r i s t i c s space. Once t h i s i s r e c o g n i z e d then the problem becomes an i n t e g e r programming problem w i t h a l l the a c t i v i t i e s and c o n s t r a i n t s i n the form of i n t e g e r s . No f r a c t i o n a l s o l u t i o n s a r e p e r m i t t e d . The n e x t step i n the e v o l u t i o n of the housing model Is t o o u t - l i n e an assignment model of h o u s i n g demand developed by W. F. Smith. The assignment problem i s one form of i n t e g e r programming problem t h a t has r e - c e i v e d c o n s i d e r a b l e refinement i n r e c e n t y e a r s . W. F. SMITH AND A MATRIX ANALYSIS OF NEIGHBOURHOOD CHANGE The l a s t c h i n k i n the t h e o r e t i c a l atmosphere i s now to be f i l l e d . To r e c a p i t u l a t e , h o u sing has been conceived of as a bundle of c h a r a c t e r i s - t i c s r o u g h l y d i v i d e d i n t o those r e s u l t i n g from the s p a t i a l s i t u a t i o n of the house and those i n t r i n s i c to the good i t s e l f . Of course, q u a l i t y i s not independent of d i s t a n c e from v a r i o u s urban a c t i v i t i e s ; however, the concep- t i o n of such interdependence l e t a l o n e the measurement i s beyond me. I a l s o v i e w the urban landscape as the r e s u l t of a c o m p e t i t i v e b i d d i n g p r o - cess where p o t e n t i a l u s e r s f o r s p e c i f i c p l o t s of land compete w i t h one another and the p r o p e r t y goest to the h i g h e s t b i d d e r . This w i l l be f o r m a l - i z e d s h o r t l y i n the co n t e x t of the " o p t i m a l assignment model." 29 Smith's theory or urban r e s i d e n t i a l structure i s a d i r e c t 16 descendent of the sector theory formulated by Homer Hoyt i n the 1930's. At that time the controversy was over the concept of " f i l t e r i n g . " B a s i c - a l l y , the proposition was that f i l t e r i n g i s the process whereby the demand for durable goods — i n p a r t i c u l a r housing — was met f o r low income groups through a process of "hand-me-downs" or f i l t e r i n g . New r e s i d e n t i a l construction i s inhabited by the r i c h who "bequest" t h e i r old homes for the next lower status group. As an armchair empirical f a c t there appears to be l i t t l e to dispute, however, controversy existed as to whether enlightened s o c i a l p o l i c y consisted of b u i l d i n g high q u a l i t y housing to Induce the r i c h to move and thereby increase the supply of housing f o r poor people, or whether i t was preferable to construct low income housing projects. Even today housing p o l i c y i s very much divided on t h i s matter. The sector theory formulated by Hoyt seemed to in d i c a t e that the succession of houses from the r i c h to the poor was a natural f a c t of urban ecology. As the c i t y matures the r i c h areas were hypothesized to move outward i n rays resembling pie s l i c e s . The exodus of the r i c h leaves behind housing that i s quickly converted to m u l t i p l e occupancy. The middle income groups c l u s t e r about the r i c h forming an i n s u l a t i o n between the i n - come extremes. Hoyt gave several rules of migration. Generally the r i c h move to the high ground, along transportation routes and tend to avoid 15W. F. Smith, Filtering and Neighborhood Change (Berkeley: Center for Real Estate and Urban Economics, 1965). 16 Hoyt, c i t e d i n Smith, op. cit., p. 9. 30 s i t u a t i o n s where subsequent outward movement i s impeded. At the base of Hoyt's theory i s a recognition that the rate of change i n population i s an important determinant of the success of the r i c h migrating outward without being encroached upon by the poor. However, w i t h a great i n f l u x of low-income households and a sluggish supply response to new housing demand, the d i s t i n c t i o n s between the sectors may very w e l l become blurred. The sector theory i s not s u f f i c i e n t to predict or even adequate- l y explain the e x i s t i n g s p a t i a l structure of c i t i e s . In p a r t i c u l a r , the actions of the middle income portion of the population were never accounted fo r i n d e t a i l . C e r t a i n l y considerable portions of the new housing stock was aimed at the middle income groups simply because the r i c h did not form a s u f f i c i e n t part of the population to allow the lower income groups to i n - h e r i t enough houses. Smith's model of housing i s very simple and i s based upon the optimal assignment model from the theory of integer programming. I t i s a process whereby the e x i s t i n g population i s assigned according to some predetermined r u l e to the e x i s t i n g housing stock. Households are d i f f e r e n t i a t e d with respect to income while houses are characterized accord- ing to "value" or p r i c e . Several questions are asked of the model: 1. What i s the pattern of urban r e s i d e n t i a l structure produced by the purely competitive market? 2. What i s the impact w i t h i n the constraints of the model given a change i n the Income d i s t r i b u t i o n of the community? 3. I f the population of the model c i t y i s i n - v a r i a n t , what new housing w i l l be constructed i n the event of replacement construction? 31 4. Given the addition of such new stock, how w i l l the pattern of occupancy change? 5. I f there i s an increase i n the housing stock and population, what new housing w i l l be constructed? Several important assumptions are made which l i m i t the a p p l i c a - b i l i t y of the model considerably. Only f i v e f a m i l i e s are assumed to i n - habit t h i s c i t y and only f i v e houses of d i f f e r e n t p r i c e or q u a l i t y are a v a i l a b l e . P r i c e i s defined s i m i l a r l y to Alonso's d e f i n i t i o n . A very c r u c i a l point i n the construction of the model i s the c r e a t i o n of rent offe r s for various house types. Each household makes an o f f e r on each-house: an o f f e r which varies according to the d e s i r a b i l i t y of the house and income of the family. The e f f e c t s of various income l e v e l s and house q u a l i t y on the rent o f f e r s of f a m i l i e s can be shown simply as a matrix: HOUSEHOLDS HOUSES A B C D E 1 L +5 +10 +15 +20 2 +10 +15 +20 +25 +30 3 +20 +25 +30 +35 +40 4 +30 +35 +40 +45 +50 5 +40 +45 +50 +55 +60 Assumed i s that the ranking of the houses as to d e s i r a b i l i t y i s the same for a l l f a m i l i e s . L i s the lowest rent offered by any house- hold f o r any house; i t represents the basic demand f o r s h e l t e r . What i s important i s not the amount L but the differences from L that w i l l be offered for various houses by various households. 32 Smith, then examined the implications of t h i s matrix and concluded that i t i s not r e a l i s t i c to assume that the income e l a s t i c i t y of demand f o r q u a l i t y i s zero. Smith argues that an increase i n q u a l i t y would be worth something to a higher income household and that i t would be w i l l i n g to pay a premium f o r q u a l i t y . In other words, each increment up i n both income and q u a l i t y r e s u l t s i n an increase i n the rent o f f e r of one d o l l a r . There i s no reason f o r choosing t h i s f i g u r e since the analysis i s unchanged as long as the income and quality, e f f e c t s are p o s i t i v e . HOUSEHOLDS HOUSES A B C D E 1 2 +1 +2 +3 +4 3 +2 +4 +6 +8 4 +3 +6 +9 +12 5 +4 +8 +12 +16 The two matrices are now simply added together and r e s u l t i n a rent o f f e r which depicts the bid made by each household f o r each house. The problem i s now to assign households to house according to some r u l e . I f the houses and households are ranked according to some objective c r i - t e r i o n such as sales p r i c e and income, then theory from l i n e a r programming indicates that an optimal s o l u t i o n e x i s t s when houses are matched to house- holds along the main diagonal as i n the following: 33 HOUSEHOLDS HOUSES A B O D E 1 L 2 +16 3 +34 4 +54 5 +76 I t i s t h i s assignment that r e s u l t s i n a maximization of the rent o f f e r s of the community. Smith uses the r e s u l t s from the theory of pure competition to j u s t i f y the r e l a t i o n of a Pareto optimal s o l u t i o n and rent maximizing program.^ Note that l i k e the bid p r i c e of Alonso, these rent o f f e r s have no r e l a t i o n to the p r i c e a c t u a l l y paid. Before examin- ing some of the experiments that Smith subjects h i s model to, I might j u s t point out that any other assignment other than the one shown r e s u l t s i n a lower aggregate rent o f f e r on the part of the community. One of the assumptions that was made at the outset was that the supply of housing was already f i x e d , or i n other words, housing i s used without regard to the costs of production. I f household f i v e leaves the community and then a family of income l e v e l one moves i n there i s a r e - s h u f f l i n g w i t h households 1 and 2 occupying houses A and B and households 3, 4 and 5 moving to houses C, D and E. The aggregate rent o f f e r now dips to L + 130 which r e f l e c t s the loss of high income family. The fa c t that the aggregate rent o f f e r has declined i n no way has any Implication upon the standard of housing i n this simple model. What i s notable i s that f a m i l i e s 3, 4 and 5 now inhabit "better" housing. "^See Appendix 1 f o r a simple explanation. 34 Smith also points out that predictions as to what type of hous- ing i s required to meet anticipated demand can be made using t h i s frame- work. The basic technique i s to c a l c u l a t e the "economic value" of a p a r t i c u l a r type of house ( i . e . , i t s rent o f f e r i n the optimal assign- ment) and then compare t h i s with the construction cost. This i s done by considering the o r i g i n a l matrix and examining the change i n aggregate rent off e r s when d i f f e r e n t types of houses are added. For example, the change i n rent off e r s when houses of type A, B, C, D, or E area added are as follows: HOUSE ADDED CHANGE IN AGGREGATE VALUE A + 0 B + 5 C +11 D +18 E + 26 I f a house of type D were added, then households 1, 2, 3 and 4 would f i l t e r up with households 1 occupying house type B, household 2 occupying house type C, etc. I f the value curve Is p l o t t e d and compared with costs curves, i . e . , the costs of b u i l d i n g each type of house as i n the f i g u r e no new construction would be j u s t i f i e d unless the economic value equals or exceeds the construction costs. Here house type C or better i s warranted. The housing model that I present i s based d i r e c t l y on the work of Smith and extends i t i n several d i r e c t i o n s . In the f i r s t place, the 35 concept of b i d f u n c t i o n i s a m p l i f i e d to i n c l u d e b i d s by households f o r not j u s t " v a l u a b l e " houses. One of the weaknesses i n Smith's t h e o r y i s t h a t the concept of v a l u e i s v e r y s l i p p e r y . I t assumes t h a t v a l u e i s an ob- j e c t i v e c a t e g o r y , i . e . , what i s v a l u e d by one person w i l l a l s o be v a l u e d by another. Secondly, and r e l a t e d t o the f i r s t p o i n t , i s the assumption t h a t the b a s i s f o r such judgements i s upon income a l o n e . Once t h i s assumption i s q u e s t i o n e d and r e j e c t e d on the grounds t h a t seem i n t u i t i v e l y apparent, another more complex premise i s r e q u i r e d . I p o s t u l a t e t h a t f o r d i f f e r e n t types of households d i f f e r e n t v a l u a t i o n s of what i s d e s i r a b l e w i l l be made. Thus the d e s i r a b i l i t y of a p a r t i c u l a r house w i l l v a r y w i t h not o n l y income but the household s i z e , e t h n i c o r i - g i n , c l a s s o r i g i n and other i d i o s y n c r a t i c d e t a i l s . I n the next c h a p t e r I p r e s e n t a model which d i v i d e s the b i d f o r v a r i o u s houses a c c o r d i n g to what d i f f e r e n t households can be expected to o f f e r f o r v a r i o u s housing a t t r i b u t e s . At t h i s s t a g e w i t h v e r y l i t t l e e m p i r i c a l work done i n the a r e a of l a n d v a l u e s and t h e d e t e r m i n a t i o n , the f u n c t i o n s imputed to d i f f e r e n t households are pure c o n j e c t u r e . I a l s o assume, v e r y h e r o i c a l l y , t h a t the b i d s f o r d i f f e r e n t c h a r a c t e r i s t i c s are a d d i t i v e and form a b i d - f u n c t i o n m a t r i x which r e f l e c t s the b i d by each household f o r each house t y p e . Once t h i s m a t r i x i s o b t a i n e d , then an a l g o r i t h m based upon the o p t i m a l assignment problem i s used to a s s i g n each household t o a house a c c o r d i n g to a r u l e of maximizing the aggregate r e n t of the community. CHAPTER THREE AN ASSIGNMENT MODEL OF URBAN HOUSING DEMAND In t h i s s e c t i o n of the essay I o u t l i n e an e x t e n s i o n of the work of Smith, w h i c h i n c o r p o r a t e s some of the aspects of modern demand t h e o r y . As mentioned i n the p r e v i o u s c h a p t e r , one of the weaknesses of the model presented by Smith was the vague use of the n o t i o n of q u a l i t y of h o u s i n g . The model I p r e s e n t attempts to overcome t h i s d e f i c i e n c y by d i s a g g r e g a t i n g h o u s i n g i n t o i t s c h a r a c t e r i s t i c s and making these the b a s i s f o r the d e s i r - a b i l i t y of p a r t i c u l a r houses by the households i n the community. As a r e s u l t , i t i s no l o n g e r p o s s i b l e t o a priori rank the housing s t o c k by c l a s s e s w h i c h have d i f f e r e n t q u a l i t y . With t h i s m o d i f i c a t i o n made, i t i s no l o n g e r p o s s i b l e to have a ranked b i d f u n c t i o n m a t r i x s i m p l y have the o p t i m a l s o l u t i o n pop out as does Smith. A complex a l g o r i t h m i s needed to f i n d the o p t i m a l s o l u t i o n . The second m o d i f i c a t i o n then i s merely to use such an a l g o r i t h m to do the assignment. B e f o r e examining the s t r u c t u r e of the b i d f u n c t i o n m a t r i x as I employ i t , some examination of the p r o p e r t i e s of the assignment problem sho u l d be made. I t i s a s u b - c l a s s of l i n e a r programming problems i n t h a t the same assumption about o p t i m i z i n g w i t h l i n e a r o b j e c t i v e f u n c t i o n s and c o n s t r a i n t s i s needed; however, i n t h i s c ase, no f r a c t i o n a l answers are p e r m i t t e d . I t was evo l v e d by o p e r a t i o n s r e s e a r c h e r s to s o l v e the problem t h a t a r i s e s when a job has to be ass i g n e d to a s p e c i f i c f a c i l i t y and o n l y to t h a t f a c i l i t y . 36 37 THE ASSIGNMENT PROBLEM The assignment problem involves several factors whose productiv- i t y can be measured to several jobs i n such a manner as to maximize the 18 aggregate return. An example Is the matching of employees to tasks. The c r u c i a l requirement i s that one and only one fa c t o r be matched to one job. Mathematically, the problem can be stated as follows. Given an n x n matrix, A^j (the r a t i n g matrix) with a „ >_ 0 for a l l i , j , f i n d an n x n matrix X.. such that I X . . = 1 i 1 J EX.. = 1 J  1 3 I I a.. X.. = max 1 3 The f i r s t two conditions ensure that the value of X w i l l be 1 i f f a c i l i t y 1 i s assigned to job j and w i l l be 0 otherwise. Each column and row contain only one entry of u n i t value with a l l the others zero. The t h i r d condition s p e c i f i e s that the elements chosen from the r a t i n g matrix w i l l maximize the product. With a few amendments, the assignment problem can be transformed into one which has a very simple and straightforward s o l u t i o n . The housing See Appendix 2 for more d e t a i l s "on the algorithm. 38 market tkat Smith has solved by the assignment process can be described as * 11 - 1 9 follows: 1. Households can be ranked by income and houses can be ranked by q u a l i t y . 2. Each household offers a rent for each house. 3. Rent offers increase w i t h income and q u a l i t y . 4. A premium i s offered by each household fo r increases i n q u a l i t y . The s i t u a t i o n i s stated mathematically by Smitk as follows: "Let a., be the rent o f f e r of the i t h family f o r tke j t h house, tnen there i s a matrix of rent offe r s i n which i + 1 i s a higher income l e v e l than i and j + 1 i s higher q u a l i t y dwelling than j suck that, a ( i j ) < a ( i + l , j ) a ( i j ) < a ( i , j+1) [ a ( i , j+1) - a ( i j ) ] < [aCi+1, - a ( i + l , j ) J I f the rent o f f e r (bid function matrix) i s set up w i t h households and houses ranked, then assigning the Lth to the j t h house with j = I r e s u l t s i n a rent maximizing assignment." 20 With the simple structure of Smith's model, i t i s simple to prove that any assignment other than that which assigns the i t h household to the j t h house w i t h j = i r e s u l t s i n a maximization of the aggregate rent of the - 21 matrix. Smith, op. ait.. Appendix 1. )Ib£d.3 p. 68. 'Ibid., pp. 67-70 for proofs. 39 I n f a c t , t h e r e i s no need to rank the b i d f u n c t i o n m a t r i x . The purpose of my r e f o r m u l a t i o n i s to s t r e s s the b a s i s of the q u a l i t y of the v a r i o u s c l a s s e s of h o u s i n g . Once the housing s t o c k i s d i s a g g r e g a t e d i n t o economic goods w i t h c h a r a c t e r i s t i c s which each possess i n d i f f e r e n t degrees and the households are not s i m p l y c h a r a c t e r i z e d by Income but e t h n i c i t y , household s i z e , c h o i c e of l i f e s t y l e , e t c . , the a b i l i t y to rank the columns and rows i s i m p o s s i b l e . I now t u r n and examine the b a s i s f o r a more extended and com- p l e t e b i d f u n c t i o n m a t r i x . For t h i s I must r e p e a t , at the r i s k of o v e r - s t a t i n g , the b i d p r i c e curve as f o r m u l a t e d by A l o n s o . THE BID FUNCTION MATRIX Thi s s e c t i o n develops the n o t i o n of b i d f u n c t i o n m a t r i x which i s v e r y c l o s e to the b i d p r i c e curve employed by A l o n s o . To repeat the usage of A l o n s o : "A b i d p r i c e curve of a r e s i d e n t i s the s e t of l a n d p r i c e s the i n d i v i d u a l c o u l d pay a t v a r i o u s l o c a t i o n s w h i l e d e r i v i n g a c o n s t a n t l e v e l of s a t i s - f a c t i o n ; t h a t i s to say, i f l a n d p r i c e s were to v a r y i n the manner d e s c r i b e d by the b i d p r i c e curve, then ^ the i n d i v i d u a l would be i n d i f f e r e n t among l o c a t i o n s . " Three important p o i n t s t h a t were s t r e s s e d were: 1. The b i d p r i c e curve r e f e r s to the i n d i v i d u a l household. 2. The curves v a r y from i n d i v i d u a l to i n d i v i d u a l . 3. There i s no r e l a t i o n between the b i d p r i c e curve and the p r i c e t h a t i s a c t u a l l y p a i d f o r l a n d . 'Alonso, op. c i t . , p. 5,8. 40 The extension of Alonso's work I wish to make involves develop- ing b i d p r i c e curves not only for land but some of the objective character- i s t i c s of housing such as the space a v a i l a b l e i n the house, the distance to work, shopping, schools and other destinations. I am assuming that the household when i n the market for a house has a clear idea what exactly i t wants and can state w i t h a f a i r degree of p r e c i s i o n those c h a r a c t e r i s t i c s which i t values and those a t t r i b u t e s which are unimportant. Thus not only i s there a bid p r i c e curve for land but bid p r i c e curves.for each of the distance parameters mentioned, various amenities associated w i t h the neigh- bourhood and the c o m p a t i b i l i t y of the house design w i t h the chosen l i f e s t y l e of the household. Attempting to cast the problem i n a framework s i m i l a r to that employed by Alonso becomes very d i f f i c u l t and cumbersome. For the s i t u a t i o n that Alonso was considering i t was f a i r l y reasonable to pi c t u r e the housing consumer as moving to and from the centre of the c i t y u n t i l the optimum combination of land and distance was discovered; however, I demand that the consumer not only f i n d an equilibrium between land, and distance from the c i t y centre, but an equilibrium among a l l the c h a r a c t e r i s t i c s of houses. A s t a r t can be made, however, i f the same vari a b l e s as used by Alonso are retained except for a second distance or variable-distance from shopping areas. The problem now appears as follo w s . The urban housing consumer i s pictured as separating each pros- pective dwelling i n t o i t s constituent c h a r a c t e r i s t i c s upon which i t places a value. The bid that i s offered upon the house i s the sum of the bids offered upon each c h a r a c t e r i s t i c . This i s the c r u c i a l assumption of the 41 model and i s c e r t a i n l y the most suspect and u n r e a l i s t i c . I m p l i e d i s t h a t the u t i l i t y f u n c t i o n f o r these c h a r a c t e r i s t i c s i s s e p a r a b l e , and t h i s i s most c e r t a i n l y wrong. For example, the ease of d r i v i n g to work may be e n t i r e l y negated by the l a c k of p a r k i n g or the p r o x i m i t y to shopping areas c o u l d not be a f a c t o r s i m p l y because t h a t p a r t i c u l a r household does a l l i t s shopping to and from work. The interdependence of c h a r a c t e r i s t i c s i s a v e r y s e r i o u s q u a l i f i c a t i o n of L a n c a s t e r ' s theory of demand. The assump- t i o n of a d d i v i t y seems to me to he the o n l y workable e m p i r i c a l h y p o t h e s i s . I n f a c t , v i r t u a l l y a l l l a n d v a l u e s i n v e s t i g a t i o n s i m p l i c i t l y make t h i s assumption. U n t i l some method of s i f t i n g the impacts of c h a r a c t e r i s t i c s from one another i s d e r i v e d , i t appears t h a t t h i s assumption needs to be r e t a i n e d . A MODEL OF URBAN HOUSING DEMAND The model i s i n two p a r t s . The f i r s t takes v a r i o u s c h a r a c t e r i s - t i c s of households and houses and fo r m u l a t e s a b i d f u n c t i o n m a t r i x w h i l e the second p a r t of the model i s an a l g o r i t h m which a l l o c a t e s the housing s t o c k to the households i n such a way as to maximize the aggregate r e n t o f f e r e d by the community and such t h a t no household can be r e a l l o c a t e d w i t h o u t mak- i n g any one other household worse o f f . The f u n c t i o n s t h a t a r e employed have no b a s i s i n e m p i r i c a l work s i n c e l i t t l e work has been done w h i c h c o u l d shed i n s i g h t i n t o the way i n which d i f f e r e n t households v a l u e d i f f e r e n t c h a r a c t e r i s t i c s of hou s i n g . A r b i t r a r i l y , I chose 15 house types and 15 household t y p e s . The households are c h a r a c t e r i z e d by the number of people I n the household to a maximum of 42 t h r e e , and income of which t h e r e are f i v e c l a s s e s , r a n g i n g from $500 per year t o $15,000. A l l the numbers used are a d m i t t e d l y n a i v e and have l i t t l e b a s i s i n r e a l i t y . The a t t r i b u t e s of space and l o c a t i o n to the downtown a r e weighted or measured by an index number and how t h i s number i s o b t a i n e d w i l l be d i s c u s s e d l a t e r . I make the assumption t h a t g i v e n almost com- p l e t e i g n o r a n c e of the r e l a t i o n between the v a l u a t i o n , the demand f o r urban space and f a m i l y s i z e and income, the r e l a t i o n i s l i n e a r and takes the form: Bidspace ( i j ) = f[Income i , Space j , Pers i ] In t h i s manner a m a t r i x w i t h house type along the rows and household type along the column i s c o n s t r u c t e d which shows the b i d by each household f o r each house. T h i s I term the b i d space m a t r i x . The b i d l o c a t i o n m a t r i x i s b u i l t i n a s i m i l a r manner. I assume t h a t the importance of a " c e n t r a l " l o c a t i o n i n c r e a s e s as does the income; The f u n c t i o n t h a t i s b e i n g used a t the moment takes the form: B i d l o c ( i j ) = f[Income i , Loc j ] where Space j i s the space i n house j ; Income i i s the income of household i ; P ers i i s the number of people i n household i ; Loc j i s the index of c e n t r a l i t y f o r house j . 43 R e c a l l i n g the tenuous assumption about a d d i v i t y of the charac- t e r i s t i c s i n the u t i l i t y f u nction, the b i d l o c a t i o n matrix i s simply added to the b i d space matrix to form the b i d function matrix which i s the same matrix that was employed by Smith-in h i s study except that i t was somewhat more l a b o r i o u s l y derived. At t h i s point the households are assigned to houses by an algor- ithm which i s explained along w i t h a program l i s t i n g i n Appendices 2 and 3. Table 1 shows the b i d l o c a t i o n matrix while Table 2 i s the b i d space matrix. The f i n a l s o l u t i o n with aggregate b i d offered by the community i s shown i n Tables 3 and 4. The tables are presented i n Appendix 4. The model as i t now stands i s very u n r e a l i s t i c and contains many s i m p l i f i c a t i o n s which r e s u l t i n a d i r e c t contravention of what i s already known about r e s i d e n t i a l l o c a t i o n . I f the f i n a l s o l u t i o n i s examined c l o s e l y , i t i s indicated that low income fa m i l i e s locate i n the most remote housing while high income f a m i l i e s are clustered about the core of the c i t y . Every study on r e s i d e n t i a l structure and even casual empirical observations con- t r a d i c t t h i s r e s u l t . The problem l i e s i n an incomplete s p e c i f i c a t i o n of the b i d function matrix. A more complete analysis would no doubt add sever- a l more va r i a b l e s and have more sophisticated behavioural functions. More w i l l be said about t h i s l a t e r . However, perhaps the most glar i n g point at t h i s moment i s the omission of any consideration of the budget constraints faced by the urban housing consumer. In a way this, i s included i n the behavioural functions; nevertheless, some e x p l i c i t supply factors: need to be incorporated i f the model i s to approach some realism. 44 The strength of t h i s s t y l e of thinking i s that the consumer of r e s i d e n t i a l housing i s viewed not as simply choosing a good c a l l e d housing, but i n f a c t d i s c r i m i n a t i n g between types of house. From here the study of various l o c a t i o n c h a r a c t e r i s t i c s and not j u s t the journey to work. L i t t l e i s known about the importance of journeys other than that of work-oriented journeys i n the demand for housing and therefore u n t i l the empirical work has been completed, nothing other than conjecture can be advanced. The same must be said about other c h a r a c t e r i s t i c s than space. One c e r t a i n l y could say that elements such as design (ranch, apartment, townhouse or s p l i t l e v e l ) have for some households an importance that i s r e f l e c t e d i n the p r i c e these households are w i l l i n g to o f f e r to l i v e there. In the next section I consider a s t a t i s t i c a l method whereby the r e l a t i o n of house and household c h a r a c t e r i s t i c s and the demand f o r urban residences could be discovered. In a d d i t i o n , the way i n which the model could become a dynamic model i s discussed as i s the problem of in t e g r a t i n g a supply side to housing a l l o c a t i o n . I conclude the essay with a b r i e f examination of the p o l i t i c a l bias of the model — namely, i s the model r e l e - vant f o r income levels? 45 THE MODEL: AN EVALUATION As i t stands now, the model that I have presented i s very naive and res t r i c t e d . In the f i r s t place, the hid functions and behavioural e q u a t i o n s that I have used are very si m p l i s t i c and have no basis i n r e a l i t y . Secondly, the model i s not dynamic. SOME STATISTICAL CONSIDERATIONS The problem quite simply i s to discover how the bid for houses of different types varies with the structure of the household. A p r i o r i , the premium placed on space and privacy w i l l be a function of the number of persons i n the household and the age structure of the family. Similarly, the effect of in d u s t r i a l centres w i l l be less for those households whose members possess s k i l l s normally used i n the CBD, such as professionals. Two methods suggest themselves immediately. E i r s t , the house- holds could be asked d i r e c t l y what aspects and attributes of their present house they l i k e or d i s l i k e . This type of information would then be corre- lated with data on the family structure. Aside from the cost of such a survey, the chance that accurate responses could be obtained i s s l i g h t . An alternate method would be to use land values as a trap for the advantages and disadvantages of various locations and types of house. The relation between the sales value of a house and the characteristics of a house such as space, indices of privacy, indices of environmental quality, and indices of proximity to work, shopping, schools, etc. could be measured using regression techniques. 46 The problem w i t h the l a s t method i s that no i n d i c a t i o n of how bids vary w i t h the str u c t u r e of the household i s possible. In a l l l i k e - l i h o od, the market i s too sluggish to be very s e n s i t i v e to changes of family structure f o r a p a r t i c u l a r house. For two houses of the same q u a l i t y (exactly the same a t t r i b u t e s ) , the sales p r i c e l i k e l y would not r e f l e c t any difference i n the structure of the two f a m i l i e s l i v i n g i n these houses. Other considerations such as bargaining s k i l l s of the var- ious buyers and s e l l e r s would be important i f home ownership were the case. The only way to resolve t h i s d i f f i c u l t y appears to be to use both methods to gain some idea of how consumer demand v a r i e s , then to con- s t r u c t b i d function curves that r e f l e c t the r e s u l t s of the s t a t i s t i c a l tests but are not d i r e c t l y r e l a t e d to the parameters discovered. Thus a combination of questionnaire and regression should i n d i c a t e which variables or a t t r i b u t e s of a house and household are the most important. PROBLEMS OF DEMAND AND SUPPLY At the moment, no attempt i s made to consider a s i t u a t i o n i n which the numbers of households of houses i n a p a r t i c u l a r class are greater than one. C e r t a i n l y i f the model i s to be r e a l i s t i c , t h i s must be corrected. At the moment, only s i m p l i s t i c solutions come to mind. These classes of household or houses which are i n excess supply could simply be ignored. A better s o l u t i o n would be to have the model assign the various classes optimally. A c e l l w i t h an oversupply of households would have some of i t s members assigned to the next class down. I f one were to s t a r t at the 47 top and move through the e n t i r e s o l u t i o n , i f t h e r e was an o v e r s u p p l y of households, the households a t the bottom of the s c a l e would be f o r c e d out of the market. A t t h i s p o i n t , t h i s i s pure c o n j e c t u r e and n o t h i n g c o n c l u s i v e can be s a i d u n t i l a f i r m e r ground i s c o n s t r u c t e d . B I E L I 0 G E A P E I Alonso, William.. Location and Land Use. Cambridge, Mass.: Harvard U n i v e r s i t y Press, 1961. Ford, L.. R.. and D. R. Fulkerson. "Solving the Assignment Problem," Management Science ( J u l y , 1956]. Gale, D.. Theory of Linear Economic Models. New York: McGraw-Hill, 1961., Haig, R. M. "Toward An Understanding of Metropolis," Quarterly Journal of Economics, V o l . 35, No. 2 (May, 19261. Hurd, R. M. " P r i n c i p l e s of C i t y Land Values," The Record and Guide. New York, 1963. Kain, J . F . The Journey to Work as A- Determinant of Residential Location, Papers and Proceedings of the Regional Science Association, IX (1962). Kuhn, H. W. "The Hungarian Method f o r the Assignment Problem," Naval Logistics Quarterly, V o l . IV, No. 1 (1955). Lancaster, K. J . Mathematical Economics. New York: Macmillan, 1968., . "The New Theory of Consumer Demand," Journal of Political , Economy, Vol., 74, No. 4 (July, 19661. Marshall, A l f r e d . Principles of Economics., London: Macmillan a n.d Co., 1917. Ricardo, David. On the Principles of Political Economy and Taxation. 1817.. Smith, W.. F. Filtering and Neighborhood Change. Berkeley, Calif.,: Center f o r Real Estate and Urban. Economics, 1265. 48 49 Von Thunen, Johann H_. The Isolated State.. 1826. Wingo, Lowdon.. Transportation and Urban Land. Washington, D..C.,: Resources fo r the Future, 1961.. A P P E N D I C E S APPENDIX 1 A NORMAL COMPETITIVE MARKET AND RENT MAXIMIZATION In David Gale's book, The Theory of Linear Models, some e f f o r t i s spent on the theory of competitive markets and resource a l l o c a t i o n . Gale gives an example of pr i c e equilibrium using the housing market. The basic difference between the housing market described by Gale and the market and the one used by Smith and myself i s that suppliers are used by Gale. The use of assignment techniques i n the analysis i s v a l i d . The housing stock i s varied and households d i f f e r i n both t h e i r a b i l i t y to bid f o r various houses and t h e i r tastes. Because of long production lags the s o l u t i o n that flows from such analysis can be used for f a i r l y long periods. Changes i n the stock of houses i s also influenced by the condi- t i o n and numbers of the present housing stock. The market used by Gale has n i n d i v i d u a l s interested i n buying n houses. A value matrix a., i s constructed which shows the worth of each house to each household. Also, the suppliers have set sales prices p^ on each house. Naturally a household would not be interested i n purchasing unless i t s val u a t i o n of that house were higher than the sales p r i c e . Gale uses programming theory to show that such a market r e s u l t s In that impossible dream of the "greatest good for the greatest numbers." He shows that the p r o f i t s of the producers i s matched by the surpluses of the consumers and the assignment problem y i e l d s a value maximizing arrangement of house s e t t l e - ment . 51 52 From the theory of competitive markets i t can be said that these markets r e s u l t i n a value maximizing arrangement of the stock. The market that both Smith and I use are purely competitive i n nature since the housing stock and since the household bids can be ranked an optimal assignment where no a r b i t t r a g e or arrangement other than the s o l u t i o n could improve the p o s i t i o n of any member i s possible and i s consistent with the theory of pure competition. See D. Gale, Theory of Linear 'Economic Models (Toronto: McGraw H i l l Co. Inc., 1961). APPENDIX 2 AN ALGORITH TO SOLVE THE TRANSPORTATION PROBLEM The algorithm that i s used i n my housing model i s a v a r i a t i o n of a routine designed by L. R. Ford and D. R. Fulkerson 1 to solve the Hithcock transportation problem. The problem can be stated mathematically as follows: Find an m x <n array of numbers x = (x„), i = 1, 2, . . ., m and j = 1, 2,. . ., n that minimizes T. c..x.. subject i j 1 3 1 3 to the constraints X x. . - a. £ x. . = b . T-3 3 3 > 0 x. . — where a., b., c.. are non-negative integers and the sum of the i 3 1 J vector a = the sum of the vector b. I f m = n and a and b are equal to 1 t h i s becomes the optimal assignment problem. Usually the c.^ matrix i s a tableau of un i t shipping costs from point i to point j ; a. i s a vector that indicates the supply of goods at point i and b. r e f l e c t s the demand at point j . The purpose of the algor- ithm designed by Ford and Fulkerson i s to a l l o c a t e the movement of goods between supply and demand points so that transportation costs are at a minimum. L. R. Ford and D. R. Fulkerson, "Solving the Assignment Prob- lem," Management Science (July, 1956). 53 54 The algorithm i s modified to search for a maximum value merely by scanning the cost matrix, f i n d i n g the maximum value i n the e n t i r e matrix and subtracting t h i s value from each element of the matrix. Using t h i s aug- mented matrix w i t h i n the cost minimizing framework produces the s o l u t i o n for a cost maximum. A discussion of the algorithm requires d e t a i l and development that would be outside the scope of t h i s paper. A l l that r e a l l y need be said i s that the method i s a v a r i a t i o n of the simplex method that i s so widely used i n l i n e a r programming. Because of the unique feature of t h i s problem, such as integer values, square value matrix and no surpluses or shortages f o r any of the row or column entires of the value matrix, several shortcuts can be used to a r r i v e at the s o l u t i o n f a s t e r . Proofs and a de s c r i p t i o n of the method can be found i n Ford 2 and Fulkerson, and i n H. W. Kuhn.. Kuhn, H. W. "The Hungarian Method for the Assignment Problem," Naval Logistics Quarterly, Vol. IV, No. 1 (1955). A P P E N D I X 3 FORTRAN IV G COMPILER MAIN 03-02-72 Of 1 5 : 1 9 : 1 7 C ASSIGNMENT SOLUTION FOR HOUSING MODEL MARK 1 C BASED UPON THE HITCHCOCK TRANSPORTATION PR OB EL M, MOD I F T «= n C BY CONSTRAINING THE SURPLUSES AND S U P P L I E S TO 1 , U U l h U D PAGE 0001 * * 0001 0002 000 3 000 4 0005 000 6 000 7 0008 202 0009 203 ; 0010 204 0011 2041 • 0012 30 6 C 0013 ' 0014 701 0015 0016 123 J017 0018 77 0019 " 0020 1112 0021 0022 ' 00-2 3 ;0024 0025 0026 534 002 7 533 •0028 0029 900 -00 30 400 0031 40 2 'OO 32 0033 401 .00 34 0035 0036 40 5 ,0037 ,0038 40 6 00 39 0040 40 7 , 041 1-1042 3043 500 )044 40 8 * * * * * * * * * * * * * * * * * * * * THIS PROGRAM IS INPUT FOR THE ASSIGNMENT MODEL IT READS IN DATA ON HOUSEHOLD AND HOUSE CHARACTERISTICS CREATES COMPLETELY IMAGINARY BEHAVIOURAL FUNCTIONS ;ED IN THE CREATION OF THE BIDFUNC TION MATRIX REAL*8 A , B , C , D , E , I N C O M ( 1 5 ) , P E R S ( 1 5 ) , S P A ( 1 5 ) , L O C ! 15) * * AND TO BE REALMS B I D L O C ! 1 5 , 1 5 ) , B I D S P A ( 1 5 , 1 5 ) , A A , B B , C C , D D DATA A , B , C , D , E / 6 H ,6H * * , 6 H DIMENSION K ( 1 0 0 , 1 0 0 ) , L ( 1 0 0 , 1 0 0 ) DIMENSION IA(IOO) ,IW(100),IS( 1 0 0 ) ,IC ( 1 0 0 ) , J R ( 100) ,KR( ] nm DIMENSION J B ( 1 0 0 ) , J W ( 1 0 0 ) , J S ( 1 0 0 ) , I R ( 1 0 0 ) JC 00 * DIMENSION HEAD(20 ) 14,2X, '*' ) ) ) ) ' 'HOUSEHOLDS * * < , 1 7 ( 1 4 , 2 X , 1 H * ) / ( 1 1 X , ' ( 1 X , 2 A 6 , 1 7 ( A 1 , A 6 ) ) ^ 1 x ; 2 I A 6 * J 1 x ; ^ x : ; 6 n , I 4 ' 2 x ' 1 H , , , / , l l x ' 2 H " ' 1 7 , i ^ " ' ' » ' ' i » ' ( 1 3 H S U B - C O S T S * * , 1 7 ( I 5 , I X , I H * ) / ( 1 1 X , 2 H » * , 1 7 { 1 5 , i , , H » | : 1, M ) = 1,M) FORMA T( FORMA T FORMA T FORMA T FORMA T R E A D ( 5 , 7 0 1 ) M , N , A A , B B , C C , D D F 0 R M A T ( 2 I 5 , 4 F 5 . 0 ) R E A D ( 5 , 1 2 3 ) ( S P A ( J ) , L 0 C ( J ) , J ^ FORMA T ( 2 F 3 . 2 ) R E A D ( 5 , 7 7 ) ( I N C O M ( I ) , P E R S ( I ) , I F O R M A T ( F 6 . 0 , F 3 . 0 ) W R I T E ( 6 , 1 1 1 2 ) M , N , A A , B B , C C , D D FORMAT!• « , « I N P U T S ' , 2 1 5 , 4 F 5 . 0 ) DO 533 1=1,M DO 534 J = 1 , N B IDS P A ( I , J ) = ( I N C O M ( I ) *SPA(J )+ INCOM( I ) *PERS( I ) / 1 0 . B I D L O C ( I , J ) = A A / I N C O M ( I ) * L O C ( J ) * I N C O M ( I ) / 3 . 0 K ( I , J ) = B I D S P A ( I , J ) + B I D L O C ( I , J ) CONTINUE CONTINUE LA=1 W R I T E ( 6 , 4 0 0 ) ( (I ) , I=1 ,M) FORMAT! '1 ' , ' H O U S E T Y P E ' , 1 5 ! 5 X , I 2 ) ) FORMAT! • ' , 'LOCATION ' , 10X , F 3 . 2 , 14 ( 4 X , F 3 . 2 ) ) W R I T E ( 6 , 4 0 1 ) ( S P A ( J ) , J = 1 , N ) FORMAT! ' ' , ' S P A C E • , 1 1 X , F 3 . 2 , 1 4 ! 4 X , F 3 . 2 ) ) W R I T E ! 6 , 4 0 2 ) ( L O C ( J ) , J = 1 ,N ) W R I T E ( 6 , 4 0 5 ) * 1 . 0 / B B <MA T( ' i * * * * * * * * * * * * * * * * W R I T E ! 6 , 4 0 6 ) FORMAT! 1 • , 'HOUSEHOLD' ) WRITE(6 ,407 ) FORMAT! ' ' , ' INCOME PERSONS.') 1=1 I F ( L A - 2 ) 5 0 0 , 5 0 1 , 5 0 2 W R I T E ( 6 , 4 0 8 ) I * * * * * * •V- 'r- * * * * * * Jji * * * • ) * * FORMA T( , 7 X , 1 2 ) ;gRTRAN IV G COMPILER MAIN 0 3 - 0 2 - 7 2 0045 0046 0047 0048 049 •0O5O 0051 00 52 0053 00 54 00 55 0056 00 57 0O58 0059 0060 00 61 0062 0063 0064 0065 00 66 0067 0068 0069 .0070 .0071 )0 72 0073 00 74 0075 0076 00 77 00 78 0079 0080 10081 0082 00 83 00 84 00 85 0086 00 87 00 88 0089 "0090 *00 91 0092 0093 00 94 0095 00 96 40 9 501 502 740 410 700 102 103 105 200 201 W R I T E ( 6 , 4 0 9 ) INCOM( I ),PERS( I ) , ( BID S P A ( I , J FORMA T( ' ' , F 6 . 0 , 1 X , F 3 . 0 , 5 X , 1 5 ( F 6 . 0 , 1 X ) ) 1=1+1 I F ( I .GT.M) GO TO 410 GO TO 5 00 W R I T E ( 6 , 4 0 8 ) I W R I T E ( 6 , 4 0 9 ) I N C O M ( I ) , P E R S ( I ) , ( B I D L O C ( I , J 1=1+1 I F ( I . G T . M ) GO TO 410 GO TO 501 W R I T E ( 6 , 4 0 8 ) I WRITE(6,7 40) I NC OM ( I ) , PER S ( I ) , ( K ( I , J ) , J FORMA T( ' ' , F 6 . 0 , 1 X , F 3 . 0 , 5 X , 1 5 ( I 6 , 1 X ) ) 1=1+1 I F ( I . G T . M ) GO TO 4 1 0 GO TO 502 LA =LA +1 I F ( L A . G T . 3 ) GO TO 700 GO TO 900 DO 102 1=1,M IA ( I ) = 1 DO 103 J=1,N J B ( J ) = J MN=M*N 151 170 160 1 5 : 1 9 : 1 7 ,J=1,N) PAGE 0 0 0 2 ,J=1,N) 1,N) FUNCTION MATRIX') DATA P R I N T OUT W R I T E ( 6 , 2 0 0 ) FORMA T( W R I T E ( 6 , 2 0 1 ) FORMA T{ 1 ",9X, 'HOUSES' , 6 X , A 2 , 1 4 ( 5 X , A 2 ) ) W R I T E ( 6 , 2 0 2 ) ( J B ( J ) , J = 1 , N ) DO 170 J J = 1,2 IF (N .GT. 17) GO TO 151 WRITE ( 6 , 2 0 3 ) D , D , ( E , D , I I IF ( J J .EQ. 2 ) GO TO 160 GO TO 170 WRITE ( 6 , 2 0 3 ) D , D , ( E , D , I I CONTINUE DO 30 60I=1,M W R I T E ( 6 , 2 0 4 ) I A ( I ) , { K ( I , J ) , J = IF (N .GT. 17) GO TO 150 ( C, I I 1,N) 1, 17) = 1 ,N) A-WRITE ( 6 , 2 0 4 1 ) GO TO 3060 150 WRITE ( 6 , 2 0 4 1 ) A , B , ( C , I I = 30 60 CONTINUE DO 171 J J = 1,2 IF (N .GT. 17) GO TO 153 WRITE ( 6 , 2 0 3 ) D , D , ( E , D , I I = I F ( J J .EQ.2 ) GO TO 162 GO TO 171 153 WRITE ( 6 , 2 0 3 ) D , D,(E,D,I I.= 171 CONTINUE C CONVERSION TO MAX PROBLEM 162 M I N=K ( 1 , 1 ) DO 600 1=1,M 1,N) 1,17) 1,N) 1,17) FORTRAN IV G COMPILER MAIN 0 3 - 0 2 - 7 2 1 5 : 1 9 : 1 7 p A G E 0 0 0 3 0097 % 0 9 8 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0111 0112 0113 0114 0115 0116 0117 0118 0119 ' ) l 2 0 0121 :0122 012 3 :0124 :012 5 0126 0127 0128 0129 0130 0131 0132 0 1 3 3 0134 3 1 3 5 3136 0137 0138 3139 0140 3141 0142 0143 0144 |145 1146 )147 )148 M I N = K ( I , J ) DO 640 J=1,N I F ( K ( I , J ) .GT.MIN) 640 CONTINUE 600 CONTINUE W R I T E ( 6 , 7 4 1 ) MIN 741 FORMA T( • «, 17) DO 602 1=1,M DO 6 0 3 J=1,N K ( I , J ) = ( K ( I , J ) - M I N ) * ( 6 0 3 CONTINUE 602 CONTINUE W R I T E ( 6 , 7 0 2 ) ( ( K ( I 702 FORMA T( • « ,15 15) DO 608 J=1,N 60 8 J B ( J ) = 1 DO 60 7 I=1,M 607 IA ( I ) =1 W R I T E ( 6 , 1 1 1 1 ) ( J B ( J 1 , J = 1 , N ) W R I T E ( 6 , 1 1 1 1 ) { I A ( I ) , 1 = 1,M) 1111 FORMAT! ' ',1515 ) C THIS COMPLETES THE PROBELM CONVERSION '1) • J ) , J = l , N ) , I = 1 , M ) C * * * * * * * * * * * C GETTING STARTED 705 DO 1 1=1 ,M IS { I ) = I A (I ) J IG=K{I ,1 ) DO 2 J=1,N J S ( J ) = J B ( J ) L ( I , J ) = - l I F ( J I G - K ( I , J ) ) 2 , 2 , 3 3 J I G = K U , J ) 2 CONTINUE I W ( I ) = - J I G DO 4 J = l ,N IF ( J IG-K ( I , J ) )4 , 5 ,4 5 L ( I , J ) = 0 4 CONTINUE I CONTINUE DO 6 J = l , N DO 7 1=1 ,M I F ( L ( I , J ) ) 7 ,8 ,7 8 J W ( J ) = 0 GO TO 6 7 CONTINUE J I G = K ( 1 , J ) + I W ( 1 ) DO 11 1=1 ,M K R ( I ) = K ( I , J ) + I W ( I ) I F ( J I G - K R ( I ) ) 1 1 , 11,10 10 J IG =K R( I ) I I CONTINUE J W ( J ) = - J I G DO 46 I =1 ,M I F ( J I G - K R ( I ) ) 4 6 , 9 5 , 4 6 95 L (I , J )=0 46 CONTINUE * * * * * TO * >; MAXIMUM * * * > .OUTRAN IV G COMPILER MAIN "^.0150 151 (0152 ! 0153 0156 ! 0157 \ 0158 [0159 i 0160 0161 10162 0163 0164 0165 0166 0167 - 0168 10169 ' 3170 0171 0172 0173 - 0174 0175 I 0176 0177 90178 -0179 0180 0181 I 0182 '0183 ; 0184 - 01 85 v 018 6 i' 01 8 7 01 88 i 0189 -ij 0190 ' 0191 i 0192 ? 0193 6 C C 14 C 15 C 16 13 12 C C c c 57 19 21 20 36 C 23 25 26 24 22 C 29 31 32 33 CONTINUE * * * * 0 3 - 0 2 ~ 7 2 * * 1 5 : 1 9 : 1 7 PAGE 0 0 0 4 * * * DETERMINATION OF I N I T I A L ALLOCATIONS DO 12 1=1,M DO 13 J = l , N I F { L ( I , J ) ) 1 3 , 1 4 , 1 3 IF ( I S ( I ) - J S ( J ) ) 1 6 , 1 5 , 1 5 J S LESS THAN I S L ( I , J )=JS ( J ) I S ( I ) = I S ( I ) - J S ( J ) J S ( J ) =0 GO TO 13 IS LESS THAN J S L ( I , J ) = I S ( I ) J S ( J ) = J S ( J ) - I S U ) I S ( I ) = 0 CONTINUE CONTINUE GO TO 51 * * * * * I T E R A T I V E PROCEDURE * * * * sjc ?p ?js i £ ĵs J^C * * * * * * * * * * L A B E L I N G PROCEDURE DO 19 J = l , N J C ( J ) = - l IR ( J ) = -1 DO 20 1=1,M IC ( I ) = - l J R(I)=»1 IF ( IS ( I ) ) 2 0 , 2 0 , 2 1 IC ( I ) = I S (I ) J R ( I ) =0 CONTINUE IND = 0 LABEL ROWS DO 22 1=1,M IF ( I C ( I ) )2 2 ,22 ,23 DO 24 J = l ,N IF (L ( I , J ) 124,25,25 IF ( I R ( J ) )26 ,24,24 I R ( J ) = I C ( I ) JC ( J ) = 1 IND=1 IF ( J S ( J ) ) 2 4 , 2 4 , 2 7 CONTINUE CONTINUE LABEL COLUMNS DO 28 J = l ,N IF ( I R ( J ) DO 30 1=1 IF ( L ( I , J IF ( I C ( I ) J R ( I ) = J IF ( L ( I , J IC ( I )=L ( I ) 2 8 , 2 8 , 2 9 ,M ) ) 3 0 , 3 0 , 3 1 ) 3 2 , 3 0 , 3 0 ) - I R ( J ) ) 3 3 , 3 4 , 3 4 , J ) FORTRAN IV G COMPILER MAIN 0 3 - 0 2 - 7 2 15:19:17 PAGE 0 0 0 5 01 P194 u l 9 5 0196 0197 0198 0199 0200 0201 02O2 02O3 0204 0205 0206 0207 0208 0209 0210 0211 0212 0213 0214 0215 0216 $217 0218 34 30 28 C c 27 38 37 39 42 18 50 C 51 80 C C 1 0219 35 1 0220 11 02 22 60 0223 S 0224 61 0225 022 6 62 0227 59 .0228 590 0229 0230 0231 63 0232 0233 72 0234 64 0235 58 0236 5 80 f0237 0238 0239 66 2 40 0 2 41 0242 70 3243 91 IMD=-1 GO TO 30 I C ( I 1 = I R ( J 1 IND=-1 CONTINUE CONTINUE IF ( I N D ) 3 6 ,35 ,36 ifc =jc <̂ >',z >',< i|< it< if. & if if if if if if if if BREAKTHROUGH PROCEDURE IF ( J S ( J l - I R ( J ) 1 3 7 , 3 8 , 3 8 L H = I R ( J ) GO TO 39 LH=JS ( J ) J S ( J )=JS( J l - L H L ( I , J ) = L ( I , J ) + L H 11=1 IF ( J R ( I I ) ) 18 ,50,18 J 1 = J R ( I I ) L ( 11 , J 1 ) =L ( I I , J D - L H I1=JC ( J l ) L ( 1 1 , J 1 ) = L ( I 1 , J l l + L H GO TO 42 I S ( I 1 ) = I S ( I 1 ) - L H ARE A L L SHORTAGES S A T I S F I E D DO 80 J=1,N IF ( J S ( J 1 1 8 0 , 8 0 , 5 7 CONTINUE GO TO 44 * * * * * * * JJ5 * 3jC * * * * * jjc * NONBREAKTHROUGH PROCEDURE LK=99999 DO 5901=1,M IF ( I C ( I ) 1 5 9 0 , 5 9 0 , 6 0 DO 59 J = l ,N IF ( I R ( J ) 161,59 ,59 LY=K ( I , J 1 +IW ( I 1 + J W ( J 1 I F ( L Y - L K ) 6 2 , 5 9 , 5 9 LK=LY CONTINUE CONTINUE DO 5801=1,M I F ( I C ( I ) 1 5 8 0 , 5 8 0 , 6 3 DO 58 J=1,N IF ( I R ( J 1 172 ,58 ,58 IF ( K ( I , J ) + I W ( I ) + J W ( J ) - L K ) 5 8 , 6 4 , 5 8 L ( I , J ) = 0 CONTINUE CONTINUE DO 65 J=1,N IF ( I R ( J ) 165 ,66,66 J W ( J 1 = J W ( J 1 +L K DO 90 1=1 ,M IF (L ( I , J 1 190,70,90 IF ( I C ( I ) 1 9 1 , 9 0 , 9 0 L ( I , J )=-l * * * F O R T R A N I V G C O M P I L E R MAIN 0 3 - 0 2 - 7 2 1 5 : 1 9 : 1 7 PAGE 0 0 0 6 | 0244 0245 i: 0246 ~H 0247 | ^48 I 0249 ; 02 50 0 2 5 1 0 2 5 2 0 2 5 3 0 2 5 4 0 2 5 5 0 2 5 6 0 2 5 7 0 2 5 8 0 2 5 9 0 2 6 O 0 2 6 1 0 2 6 2 0 2 6 3 0 2 6 4 0 2 6 5 0 2 6 6 0 2 6 7 0 2 6 8 0 2 6 9 0 2 7 0 J 2 7 1 6 2 7 2 0 2 7 3 0 2 7 4 0 2 7 5 0 2 7 6 0 2 7 7 0 2 7 8 0 2 7 9 0 2 8 0 0 2 8 1 0 2 8 2 0 2 8 3 0 2 8 4 0 2 8 5 0 2 8 6 0 2 8 7 0 2 8 8 0 2 8 9 0 2 9 0 0 2 9 1 0 2 9 2 0 2 9 3 0 2 9 4 0 2 9 5 F 2 9 6 9 0 6 5 6 9 6 7 C C 4 4 C O N T I N U E C O N T I N U E D O 6 7 1 = 1 , M I F ( I C ( I ) ) 6 7 , 6 9 , 6 9 IW ( I ) = I W ( I ) - L K C O N T I N U E G O T O 5 7 2 0 6 2 1 0 3 0 5 2 0 5 6 1 1 6 1 2 2 0 7 2 0 8 1 5 2 1 7 2 1 6 1 1 5 5 3 0 7 1 5 4 1 7 3 1 6 3 9 5 9 >i< j(e # >;<: * P R O C E D U R E -v*f -sV N J ^ >fV >yV T E R M I N A T I O N L C = 0 D O 2 0 5 J = l , N KR(J ) = 0 D O 3 0 5 I = 1 , M I F ( L ( I , J ) ) 2 1 0 , 3 0 5 , 2 0 6 L Y = L ( I , J ) * K { I , J ) L C = L C + L Y K R ( J ) = K R ( J ) + L Y G O T O 3 0 5 L ( I , J ) = 0 C O N T I N U E C O N T I N U E D O 6 1 1 1 = 1 , M I A ( I ) = 1 D O 6 1 2 J = 1 , N J B ( J ) = J W R I T E ( 6 , 2 0 7 ) F O R M A T ( 1 H 1 , 3 0 X , 8 H S O L U T 1 0 N / 2 9 X , 1 2 H # * * * * * # # * # * # / ) W R I T E ( 6 , 2 0 8 ) L C F O R M A T ( 1 H , 1 2 H T O TA L C O S T = , 1 1 0 ) W R I T E ( 6 , 2 0 1 ) W R I T E ( 6 , 2 0 2 ) ( J B ( J ) , J = 1 , N ) D O 1 7 2 J J = 1 , 2 I F ( N . G T . 1 7 ) G O W R I T E ( 6 , 2 0 3 ) D , D , I F ( J J . E Q . 2 ) G O G O T O 1 7 2 W R I T E ( 6 , 2 0 3 ) D , D , C O N T I N U E D O 3 0 7 I = 1 , M W R I T E ( 6 , 2 0 4 ) I A ( I ) . G T . 1 7 ) G O ( 6 , 2 0 4 1 ) A , B, (C , I I = 1 , N ) 3 0 7 ( 6 , 2 0 4 1 ) T O 1 5 2 ( E , D , I I T O 1 6 1 ( E , D , I I = 1 , N ) 1 , 1 7 ) I F ( N W R I T E G O TO W R I T E C O N T I N U E D O 1 7 3 J J = 1 , 2 I F ( N . G T . 1 7 ) G O W R I T E ( 6 , 2 0 3 ) D , D , I F ( J J . E Q . 2 ) G O G O T O 1 7 3 W R I T E ( 6 , 2 0 3 ) D , D , C O N T I N U E W R I T E ( 6 , 3 0 6 ) ( K R ( J ) W R I T E ( 6 , 9 5 9 ) F O R M A T * 1 H 1 ) , ( L ( I , J T O 1 5 5 , ( , I I = ) , J = I , N : A , B , ( C , I I = 1 , 1 7 ) T O ( E TO 1 5 4 »Dr I I 1 6 3 1 , N ) ( E , D , I 1 = 1 , 1 7 ) , J = 1 , N ) 1 FORTRAN IV G COMPILER 0297 0298 0299 0300 0 301 0 30 2 0 30 3 0304 0305 0 30 6 0307 M A I N 743 744 745 746 747 0 3 - 0 2 - 7 2 1 5 : 1 9 : 1 7 PAGE 0 0 0 7 GO TO 7 4 4 DO 747 J=1,N DO 743 1=1,M I F ( L ( I , J ) . E Q . l ) CONTINUE W R I T E ( 6 , 7 4 5 ) I , J FORMA T( 1 ', 'HOUSEHOLD' , 1 2 , ' LOCATED W R I T E ( 6 , 7 4 6 ) I N C OM(I ) ,PERS( I ),S P A ( J ) , L O C ( J ) F ORMAT( 1 1,'INCOME •-2 , ' LOCATION COEF CONTINUE STOP END IN HOUSE',12) , F 6 . 0 , 2 X , ' F A M S I Z E = * , F 3 . 0 , 5 X , ' S P A C E C 0 E F = ' , F 3 , F 3 . 2 ) TOTAL MEMORY REQUIREMENTS 017EC2 BYTES COMPILE TIME = 17.4 SECONDS A P P E N D I X 4 TABLE 1 BIDSPACE MATRIX HOUSETYPE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 SPACE .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 LOCATION .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 HOUSEHOLD INCOME PERSONS $ 500 1 97 98 100 102 103 105 107 108 110 112 113 115 117 118 120 500 2 113 115 117 118 120 122 123 125 127 128 130 132 133 135 137 500 3 130 132 133 135 137 138 140 142 143 145 147 148 150 152 153 1,000 1 193 197 200 203 207 210 213 . 217 220 223 227 230 233 237 240 1,000 2 227 230 233 237 240 243 247 250 253 257 260 263 267 270 273 1,000 ,3 260 263 267 270 273 277 280 283 287 290 293 297 300 303 307 5,000 1 967 983 1000 1017 1033 1050 1067 1083 1100 1117 1133 1150 1167 1183 1200 5,000 2 1133 1150 1167 1183 1200 1217 1233 1250 1267 1283 1300 1317 1333 1350 1367 5,000 3 1300 1317 1333 1350 1367 1383 1400 1417 1433 1450 1467 1483 1500 1517 1533 10,000 1 1933 1967 2000 2033 2067 2100 2133 2167 2200 2233 2267 2300 2333 2367 2400 10,000 2 2267 2300 2333 2367 2400 2433 2467 2500 2533 2567 2600 2633 2667 2700 2733 10,000 3 2600 2633 2667 2700 2733 2767 2800 2833 2867 2900 2933 2967 3000 3033 2067 15,000 1 2900 2950 3000 3050 3100 3150 3200 3250 3300 3350 3400 3450 3500 3550 3600 15,000 2 3400 3450 3500 3550 3600 3650 3700 3750 3800 3850 3900 3950 4000 4050 4100 15,000 3 3900 3950 4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 Ul TABLE 2 BIDLOC MATRIX HOUSETYPE 1 2 3 4 5 6 7 8 9 10 11 .12 13 14 15 SPACE .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 LOCATION .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 HOUSEHOLD INCOME PERSONS $ 500 1 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 500 2 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 500 3 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 1,000 1 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 1,000 2 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 1,000 • 3 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 5,000 1 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 5,000 2 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 5,000 3 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 10,000 1 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 10,000 2 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 10,000 3 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 15,000 1 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 15,000 2 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 15,000 3 160 163 167 170 173 177 180 183 187 190 193 197 200 203 207 TABLE 3 * BIDFUNCT MATRIX HOUSEHOLD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 SPACE .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 LOCATION .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 HOUSEHOLD INCOME PERSONS $ 500 1 256 261 266 271 276 281 286 291 296 301 306 311 316 321 327 500 2 273 278 283 288 293 298 303 308 313 318 323 328 333 338 343 500 3 289 294 299 304 309 314 319 324 329 334 339 344 349 354 359 1,000 1 353 359 366 373 379 386 393 399 406 413 419 426 433 439 446 1,000 , 2 386 393 399 406 413 419 426 433 439 446 453 459 466 473 479 1,000 3 419 426 433 439 446 453 459 466 473 479 486 493 499 506 513 5,000 1 1126 1146 1166 1186 1206 1226 1246 1266 1286 1306 1326 1346 1366 1386 1406 5,000 2 1293 1313 1333 1353 1373 1393 1413 1433 1453 1473 1493 1513 1533 1553 1573 5,000 3 1459 1479 1499 1519 1539 1559 1579 1599 1619 1639 1659 1679 1699 1719 1739 10,000 1 2093 2129 2166 2203 2239 2276 2313 2349 2386 2423 2459 V: 96 2533 2569 2606 10,000 2 2426 2463 2499 2536 2573 2609 2645 2683 2719 2756 2793 2829 2866 2903 2939 10,000 3 2759 2796 2833 2869 2906 2943 2979 3016 3053 3089 3126 3163 3199 3236 3273 15,000 1 3059 3113 3166 3219 3273 3326 3379 3433 3486 3539 3593 3646 3699 3753 3806 15,000 2 3559 3613 3666 3719 3773 3826 3879 3933 3986 4039 4093 4146 4199 4253 4306 15,000 3 4059 4113 4166 4219 4273 4326 4379 4433 4486 4539 4593 4646 4699 4753 4806 Due to a truncation error this table may not be the exact sum of Tables 1 and 2. L n OO TABLE 4 ASSIGNMENT SOLUTION TOTAL COST = HOUSES HOUSEHOLDS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 15 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 SUB-COSTS 4550 4528 4507 4433 4393 4353 3227 3373 3520 2383 2013 1643 107 553 1000 VO

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 8 0
Canada 6 0
Philippines 1 0
China 1 0
City Views Downloads
Unknown 8 0
Ashburn 5 0
Redmond 1 0
Seattle 1 0
Beijing 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0107076/manifest

Comment

Related Items