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An assignment model of urban housing demand Mason, Greg C. 1972-12-31

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AN ASSIGNMENT MODEL OF URBAN HOUSING DEMAND  by  GREG C. MASON B.A., U n i v e r s i t y 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  i n the Department of Economics  We accept t h i s thesis as conforming required standard,  to the  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 a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e the L i b r a r y s h a l l make i t f r e e l y  that  a v a i l a b l e f o r r e f e r e n c e and s t u d y .  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  thesis  f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood that copying 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 written  g a i n s h a l l n o t be a l l o w e d w i t h o u t my  permission.  Department o f  E^cz^rTVySr)W t C J  The U n i v e r s i t y o f B r i t i s h C o l u m b i a Vancouver 8, Canada  ABSTRACT  A c r i t i c a l view of the l i t e r a t u r e on land rent and r e s i dential location i s undertaken with special emphasis on the journey to work hypothesis. A housing demand model i s 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 i s 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 c h a r a c t e r i s t i c s . These attributes of both the product and consumer are thought to affect the demand for different parts of the housing stock.  ii  TABLE OF CONTENTS Page INTRODUCTION  1  CHAPTER I.  II.  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  THE ASSIGNMENT THEORY OF HOUSING  22  THE NEW THEORY OF DEMAND  22  APPLICATIONS TO HOUSING,  ,  W. F. SMITH AND A MATRIX ANALYSIS OF NEIGHBOURHOOD CHANGE  III.  27 . . . . 28  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:  45  AN EVALUATION  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 APPENDIX 3 PROGRAM LISTING OF THE MODEL  56  APPENDIX 4 PROGRAM LISTINGS  60  iii  53  INTRODUCTION  Housing i s the most complex good any consumer has to purchase. This essay deals w i t h the economics of housing demand.  Nothing i s attemp-  ted w i t h respect to house f i n a n c e , macro housing p o l i c y f o r a n a t i o n , or the economics of housing supply. Chapter I begins by r e v i e w i n g the h i s t o r y of land r e n t from Ricardo through Von Thunen and the land economists of the 1930's.  The work  of recent authors such as Lowdon Wingo, J r . , W i l l i a m Alonso and J . F. K a i n 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 t h a t my model  attempts to overcome i s the l a c k of a n a l y s i s 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 p a i d l i p s e r v i c e to  the e x i s t e n c e of t e c h n i c a l l y separate a t t r i b u t e s t h a t a f f e c t the demand f o r 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 l a y s the b a s i c theory f o r my model. K. J . Lancaster i s reviewed.  He was  u s i n g the assumption t h a t i t was  The work of  the f i r s t to r e s t a t e demand theory  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 u n s u i t a b l e f o r the a n a l y s i s of consumer durables and recourse must be made to an i n t e g e r programming framework. F. Smith i s introduced.  At t h i s p o i n t the work of W.  Recently Smith has formulated a simple model of  housing demand using the assignment approach. advance i n that the households  His model i s a s i g n i f i c a n t  i n a community are p i c t u r e d as  examining  each element of the housing stock and p l a c i n g a b i d on each house.  Then  2  w i t h an assignment a l g o r i t h m , each household  i s placed i n one and only  one house so as to maximize the aggregate r e n t of the community.  The  r a t i o n a l e f o r rent maximization i s shown to be sound w i t h i n the context of t h i s model.  A d e f e c t i n h i s theory i s t h a t the b a s i s of the b i d formu-  l a t i o n s i s very vague.  By u s i n g the conclusions from L a n c a s t e r ' s theory a  more secure b a s i s f o r the formation of r e n t o f f e r s of b i d can be made. The l a s t chapter presents the model and a t e s t run u s i n g comp l e t e l y imaginary data and b i d f u n c t i o n m a t r i c e s .  At t h i s p o i n t the  model i s e x c e p t i o n a l l y u n r e a l i s t i c and the concluding p a r t of the chapter  i s spent i n examining  some s t a t i s t i c a l methods t h a t are "quick and  d i r t y " so that the model may be made o p e r a t i o n a l i n a s h o r t 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. simply a s t a t i c one-shot assignment.  As i t stands, the model i s  Nothing i s s a i d about the supply or  f i n a n c i n g of housing and t h i s c e r t a i n l y i s a major d e f e c t .  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, s i n c e 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 households are assigned. Considerably more e f f o r t must be made i n these l a s t areas f o r e the model can become a u s e f u l planning t o o l .  be-  CHAPTER I SURVEY OF THE LITERATURE ON LAND RENT AND THE ECONOMICS OF HOUSING  I n 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 t h i n k e r on the matter of why economic a c t i v i t y l o c a t e s where i t does and why land p r i c e s are what they a r e was, of course, David Ricardo.''"  Ricardo showed that the most p r o d u c t i v e 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 p o p u l a t i o n , l e s s productive land was used.  Since the land a l r e a d y i n use y i e l d e d a  higher r e t u r n , the c o m p e t i t i v e process r e s u l t e d i n d i f f e r e n t i a l p r i c e s of the l a n d .  The d i f f e r e n c e 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  p r i c 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 r e n t . Ricardo gives l i t t l e c o n s i d e r a t i o n to the other costs i n a g r i c u l t u r e such as t r a n s p o r t a t i o n costs to and from the market place and most o f t e n he assumed that these costs were equal. sions concerning  As a r e s u l t , he came to no c o n c l u -  the exact l o c a t i o n of v a r i o u s types of farming or r e -  course e x t r a c t i n g a c t i v i t y . L a t e r , an economist i n Germany e x p l i c i t l y t r e a t e d the problem of t r a n s p o r t c o s t s .  J . Von Thunen assumed that f e r t i l i t y  differentials  2 were non-existent.  Taxation,  L i k e R i c a r d o , the competitive market process r e s u l t s  David R i c a r d o , On The Principles 1817. 2 Johann H. Von Thunen, The Isolated 3  of Political  State,  Economy and  1826.  4  i n the h i g h e s t land p r i c e s being paid f o r the most d e s i r a b l e p l o t s of land.  I n t h i s case, however, the a t t r a c t i v e n e s s of land I s based upon  the savings due t o l o c a t i n g as c l o s e to the market as p o s s i b l e .  The rent  the land earns i s the t r a n s p o r t cost saving from l o c a t i n g c l o s e r t o the market p l a c e .  This theory u n d e r l i e s v i r t u a l l y a l l t h e a n a l y s i s of econo-  mic l o c a t i o n theory.  Once the costs of production a r e a l s o i n c l u d e d , the  rent on a p l o t of land i s the value of t h e product l e s s the costs of t r a n s p o r t and p r o d u c t i o n . The theory remained i n t h 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 M a r s h a l l u n t i l the e a r l y 20th Century. M a r s h a l l 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 agricultural  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 o b t a i n 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 t o em-  phasize the r o l e t h a t e x t e r n a l economies of s c a l e p l a y i n c o n d i t i o n i n g the value of the l a n d . The next c o n t r i b u t i o n to land rent theory was made by R. M. Hurd who o u t l i n e d a theory of urban land values p a r a l l e l to Von Thunen's. "Since value depends upon economic r e n t , and rent on l o c a t i o n and l o c a t i o n on convenience, and convenience on nearness, we may e l i m i n a t e the i n t e r m e d i a t e steps and say that value depends upon nearness." 5  Co.,  A l f r e d M a r s h a l l , Principles 1917), e s p e c i a l l y Appendix G.  ^R. M. Hurd, Principles and Guide, 1903). ^Tbid., p. 11.  of Economics  (London:  of City Land Values  Macmillan and  (New York:  The Record  5  The next step i n the e v o l u t i o n of the theory was the f o r m a l i z a t i o n of Hurd's a n a l y s i s i n t o a canon. that there was a complementarity  R. M. Haig f i r s t advanced the p r o p o s i t i o n of r e n t and t r a n s p o r t a t i o n c o s t s .  In  other words, the r e n t on any s i t e was equal to the t r a n s p o r t costs not paid.  U n t i l v e r y r e c e n t l y , t h i s "law" formed the b a s i s of l a n d r e n t 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 c o n t r i b u t i o n s i s Transportation  and Urban Land by Lowdon Wingo, J?J  The assumption made by Wingo  i s common to most of the work I n the f i e l d s i n c e Hurd.  A featureless plain  w i t h 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 p e r f e c t competition i n the labour markets and the workers i n the c i t y have completely homogeneous t a s t e s and incomes. A l s o assumed i s the complementarity  of t r a n s p o r t costs and r e n t .  Transport  costs are composed of f i n a n c i a l costs which i n c l u d e the expenditure dependent upon the d i s t a n c e t r a v e l l e d and t e r m i n a l costs which are a f u n c t i o n of the congestion i n the c i t y .  In a d d i t i o n there are the o p p o r t u n i t y 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 o p p o r t u n i t y c o s t s a r e  thought to  Since the labour market  be an extension of the working day.  i s assumed to be p e r f e c t the worker can simply s h i f t them back to the emp l o y e r 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 .  Journal  R. M. Haig, "Toward An Understanding of M e t r o p o l i s , " of Economies, V o l . 35, No. 2 CMay, 1926).  Lowdon Wingo, J r . , Transportation and Urban Land (Washington, Resources f o r the Future, I n c . , 1961). 7  D.C.:  Quarterly  6  Three equations First,  f o r m t h e b a s i s o f Wingo's t h e o r e t i c a l  system.  t h e r e i s t h e 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 ) = k ( t ) o  (1)  m  where p i s the p r i c e p e r square f o o t o f l a n d ; q t h e q u a n t i t y o f l a n d sumed; k ( t ) t h e 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 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 m  con-  the p o i n t of settlement to the f r i n g e .  This  equa-  t i o n s t a t e s that the transport costs plus the rent f o r a s i t e i s equal f o r a l l w o r k e r s i n t h e c i t y and i s e q u a l t o t h e t o t a l centre  0  f the c i t y  commutting c o s t s from t h e  to the f r i n g e .  S e c o n d , t h e r e i s t h e demand f o r s p a c e w h i c h i s a s i m p l e  parametric  expression: q = (a/p)  (2)  b  where q and p a r e as b e f o r e and a and b a r e p a r a m e t e r s . t i o n equation  A t any g i v e n  (1) g i v e s t h e amount s p e n t on l a n d w h i l e e q u a t i o n  loca-  (2) i n d i -  c a t e s t h e amount o f l a n d consumed b y t h e w o r k e r . Third, i f the a v a i l a b i l i t y  of l a n d i s given by a s i m p l e  expres-  2 s i o n s u c h as S = i r t w h e r e TT i s t h e c o n v e n t i o n a l e x p r e s s i o n and t t h e r a d ius  of the c i t y  to  the f o l l o w i n g  , th_en Wingo c a l c u l a t e s t h e m a r g i n o f s e t t l e m e n t equation: t n = 2TT /  m  tq(t)dt  where n i s the p o p u l a t i o n o f t h e c i t y ; and  according  (3)  l / q ( t ) i s the d e n s i t y of settlement  t h e o t h e r v a r i a b l e s a r e as d e f i n e d a s 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 w h i c h i s f o u n d s i m u l t a n e o u s l y w i t h e q u a t i o n s  (1) and ( 2 ) .  7  One of the important conclusions Wingo a r r i v e s at i s that the t r a n s p o r t technology w i l l be r e f l e c t e d by the land p r i c e s i n the  city.  I f the cost of movement i s h i g h , 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 p r i c e s f o r the r e s i d e n t i a l s i t e s c l o s e to the job w i l l command a premium.  The a c t u a l l e v e l of land p r i c e i s a  f u n c t i o n of the numbers of workers i n the c i t y .  This i s elementary but  important. Recently, there has been c o n s i d e r a b l e e m p i r i c a l 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  location.  J . F. Kain i s perhaps the most known of researchers i n t h i s area.  His theo  retical  base i s s i m i l a r to Wingo and the land economists, i n that the compl  mentarity of t r a n s p o r t costs and r e n t i s assumed. The t r a n s p o r t costs of the household are broken i n t o three c a t e gories : 1. The costs of t r a v e l l i n g to and from s e r v i c e o b t a i n a b l e 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 s e r v i c e s only a v a i l a b l e o u t s i d e the r e s i d e n t i a l a r e a . K a i n i s vague on the d e f i n i t i o n of area and appears to use word interchangeably w i t h r i n g .  He presents some p r e l i m i n a r y  evidence to show the importance of the journey to work.  Location, 1962.  For  the  statistical example, 43.9  J . F. K a i n , The Journey to Work as a Determinant of Residential Papers and Proceedings of the Regional Science A s s o c i a t i o n , IX  8  per cent of a l l t r i p s undertaken by households sampled from 39 major c i t i e s are journeys to work, while 21.4 per cent are s o c i a l and recreational journeys.  Since many of the recreational and s o c i a l centers i n c i t i e s are close  to employment centers, the s o c i a l destination t r i p s 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 i s a normal good.  The assumption about the complementarity of the rent and transport costs i s 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.  I f 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 consumed as the household moves away from the job s i t e .  While the l o c a t i o n r e n t d e c l i n e s w i t h movement away from the employment s i t e , the t r a n s p o r t costs i n c r e a s e .  T(x) I s t k e t r a n s p o r t cost  f u n c t i o n and the t o t a l t r a n s p o r t costs p a i d 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 p l a c e of employment to the r e s i d e n t i a l site.  S i m i l a r l y , the t r a n s p o r t cost savings or l o c a t i o n r e n t i s the  area under the isospace  curve that a p p l i e s to the amount of land being  consumed by the household.  The minimum costs l o c a t i o n i s simply a t the  i n t e r s e c t i o n of the two curves. The l o c a t i o n s which minimize the l o c a t i o n costs of the households are now known.  The t o t a l l o c a t i o n cost d i v i d e d by the space consumed  i s the p r i c e the household must pay f o r 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 s e r v i c e s , 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 r e s i d e n -  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 f o r households of  d i f f e r e n t types, depending upon e t h n i c o r i g i n , income, age composition and f a m i l y s i z e , there w i l l be d i f f e r e n t p r o p e n s i t i e s to undertake a journey to  10  work of given length..  A l s o , r e s i d e n t i a l l o c a t i o n i s a f u n c t i o n of the  job s i t e . The e m p i r i c a l t e s t c o n s i s t s 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 rings.  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 p r o p o r t i o n  of land area increases as d i s t a n c e from the i n n e r r i n g i n c r e a s e .  The i n n e r  r i n g i s , of course, the prime employment area i n any urban area. I t was d i s c o v e r e d t h a t most of the journeys outer to inner r i n g s .  This supports  to work a r e from t h e  the theory i n that the only way t o  reduce the l o c a t i o n r e n t p a i d 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 c o n s t r a i n t on space consumption eases. Other f i n d i n g s of i n t e r e s t a r e that workers i n the CBD make cons i d e r a b l y longer journeys  to work.  to work i s a f u n c t i o n of income. for  I n a d d i t i o n , the l e n g t h of t h e journey The r i c h appear t o have a h i g h  space and can a f f o r d the t r a n s p o r t costs to get i t .  Very s m a l l house-  holds and l a r g e households tend t o make the s h o r t e s t journeys middle s i z e d households undertaking  preference  t o work w i t h  longer commuting journeys.  At f i r s t b l u s h , the empirics seem to s u b s t a n t i a t e the claims of the l a n d economists.  The household does appear t o s u b s t i t u t e savings i n  land costs f o r t r a n s p o r t expenditures.  Before c o n s i d e r i n g o b j e c t i o n s 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 t h e other land economists i n t h a t where Q  Wingo p o s t u l a t e s separate demands f o r space and l o c a t i o n , Alonso i n t e g r a t e s  W i l l i a m Alonso, Location vard U n i v e r s i t y P r e s s , 1961).  and Land Use (Cambridge, Mass.:  Har-  11  them i n t o a u t i l i t y maximizing framework. the  Both space and l o c a t i o n enter  u t i l i t y and budget equations of the household.  Previous work was  con-  tent w i t h 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 i n v e s t i g a t e d , and a r b i t r a r y demand f o r space equat i o n w i t h l i t t l e b a s i s i n r e a l i t y i s employed. The b a s i c assumptions used are common to most work i n land economics:  a f e a t u r e l e s s p l a i n w i t h no g e o g r a p h i c a l b a r r i e r s or f e a t u r e s  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  t r a n s f e r of p r o p e r t y between l a n d l o r d s are assumed to e x i s t .  Similarly,  t r a n s p o r t a t i o n i s u n i n h i b i t e d by n a t u r a l b a r r i e r s and t h e r e are no unusual costs to overcoming the f r i c t i o n of space. the  P e r f e c t knowledge abounds w i t h  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  f i r m pays f o r the r i g h t to use one u n i t of land.  Under these terms are  subsumed the costs of ownership, c o n t r a c t r e n t and s a l e s p r i c e i n the l o n g term, which given p e r f e c t c o m p e t i t i o n tend to e q u a l i t y ( i n terms of d i s counted present v a l u e ) . 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 p l a c e 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 v a r i o u s 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 f o r t h i s assumption i s l e n t some p l a u s i b i l i t y . Since the consumer i s asked to make the dual d e c i s i o n of how much to buy and where to s e t t l e t h i s must be i n c l u d e d i n the income and functions.  utility  The income equation c o n s i s t s 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 , p l u s the costs of a l l  12  other all  goods and  services.  To p r e v e n t t h e s t u d y  from becoming too  complex,  goods a s i d e f r o m l a n d a r e a g g r e g a t e d i n t o a c o m p o s i t e commodity — The  budget e q u a t i o n  y = P  z.  a p p e a r s as f o l l o w s :  C4)  z + p(t) q + k(t)  z  pCt)  where p^ i s t h e p r i c e of t h e c o m p o s i t e good; z i s t h e c o m p o s i t e good; is  the p r i c e o f l a n d a t d i s t a n c e t f r o m t h e c e n t e r  amount of l a n d consumed; and that p a r t i c u l a r The  o f th_e c i t y ; q  the  k ( t ) the c o s t s of commutting a s s o c i a t e d  site. utility  f u n c t i o n i s simple U = u ( z , q,  Subjecting  and  straightforward:  (5)  t)  these expressions  c a l c u l u s , Alonso obtains  to the u s u a l  t o o l s of  differential  the f o l l o w i n g s o l u t i o n s : u /u = p(t)/p a z z u  The [equation  C6)]  t  /u  z  =  (6)  (qdp/dt + dk/dt)/p  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 s t a t e s the marginal  a l l o t h e r goods i s e q u a l  [equation  The  (7)1  s t a t e s t h a t th_e  The  first  marginal equal  t o t h e t o t a l amount  the c o s t s of commutting to  numerator i n the second e x p r e s s i o n  and  In a  a l l o t h e r goods i s a g a i n  p r i c e of d i s t a n c e i s equal  s p e n t on l a n d a t p o i n t t C q d p / d t ) , and The  i s simple.  r a t e of s u b s t i t u t i o n b e t w e e n l a n d  r a t e o f s u b s t i t u t i o n b e t w e e n d i s t a n c e and to t h e i r p r i c e r a t i o s .  (7)  z  t o t h e 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 .  s i m i l a r v e i n the second e q u a t i o n  t(dk/dt).  with  [equation  (7)]  point i s impor-  13  tant s i n c e i t i n d i c a t e s t h a t cost of a marginal movement I s equal to the change i n the amount p a i d f o r land plus the change i n commuting  costs.  Since consumption of the composite good i s p l e a s u r a b l e , t h e r e f o r e p o s i t i v e and s i n c e commuting fr^s d i s u t i l i t y negative.  Commuting  is  attached to i t , the U  costs are r e l a t e d to the d i s t a n c e t r a v e l l e d  is and  even i f there were no d i r e c t costs to movement there would always be opportunity c o s t s , t h i s makes dk/dt negative.  the  Since q i s p o s i t i v e , t h i s  i m p l i e s that the p r i c e of land d e c l i n e s as one moves away from the  center  of the c i t y . The c o n c l u s i o n that Alonso d e r i v e s from t h i s a n a l y s i s i s that the consumer of urban housing s e t t l e s at the p o i n t where the costs of commutt i n g are j u s t greater than the savings from consuming cheaper l a n d .  In  other words, the p o i n t of e q u i l i b r i u m f o r the r e s i d e n t i s the p o i n t at which the costs of commutting i n c u r r e d by moving i n c r e m e n t a l l y from the centre are e x a c t l y balanced by the savings i n the p r i c e of land by such a move. Alonso proceeds to examine the a c t i o n s of commercial and t u r a l users of land.  agricul-  Instead of maximizing u t i l i t y , they maximize p r o f i t s .  A revenue and cost f u n c t i o n are s u b s t i t u t e d f o r the u t i l i t y and budget constraint.  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 s k i p  them and proceed to consider the nature o f . t h e b i d p r i c e f u n c t i o n which forms a c r u c i a l l i n k i n Alonson's wrork.  I t a l s o i s an important  step i n  the c o n s t r u c t i o n of the s i m u l a t i o n model and thus needs to be examined carefully.  14  THE RESIDENTIAL BID PRICE CURVE As was s t a t e d i n the i n t r o d u c t i o n , the b i d p r i c e curve f o r an i n d i v i d u a l i s d e f i n e d as "the s e t of land p r i c e s an i n d i v i d u a l would pay at v a r i o u s distances and s t i l l d e r i v e a constant l e v e l of s a t i s f a c t i o n . Several p o i n t s need to be s t r e s s e d .  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 v e r y d i f f e r e n t .  Secondly, a p a r t i c u -  l a r b i d p r i c e curves r e f e r s to a g i v e n 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 b i d p r i c e curves corresponding of s a t i s f a c t i o n .  to d i f f e r e n t l e v e l s  T h i r d , the b i d p r i c e curve i s i n no way r e l a t e d to th_e  p r i c e that i s e v e n t u a l l y p a i d .  There i s no c o n s i d e r a t i o n of supply f a c -  tors and t h e r e f o r e i n a sense t h i s i s a v e r y u n r e a l i s t i c concept. p o i n t w i l l a r i s e l a t e r i n the essay. for  This  Once the b i d p r i c e s a r e e s t a b l i s h e d  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 u s e r s , Alonso employs a game  t h e o r e t i c approach where the users compete f o r the land w i t h the s a l e going to the h i g h e s t b i d d e r .  Since commercial and a g r i c u l t u r a l users a r e e x t r a n -  eous to the essay I w i l l pass over t h i s p o i n t . The b i d p r i c e f u n c t i o n i s derived very simply. bid  By d e f i n i t i o n a  p r i c e i s the amount the household pays f o r 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 d i s t a n c e a r e t r e a t e d  i n i t i a l l y as given w i t h u t i l i t y s e t a t u o u t i l i t y f u n c t i o n now appears as f o l l o w s : U  q  = uCz, q, t )  Alonso, op. cit., p. 59.  Q  and l o c a t i o n set a t t . The o  (8)  15  and  t h e budget c o n s t r a i n t w i t h y^, p , k ( t ) a l l g i v e n : z  y  = p  o  z  z +  P  CO  q  C9)  + k t ) C  o  The s i g n s o f t h e p a r t i a l s r e m a i n t h e same as b e f o r e .  I f t h e problem i s  c a s t i n a Lagrangean framework t h e f o l l o w i n g s e t o f e q u a t i o n s  U = u ( z , q , t ) - X [ y - Cp  z  Q  Z + pCt ) Q  q + K O  result,^  ]  (10)  8u  •| = u  q  - Ap(d ) = 0 o  C  12)  8u and  f r o m (11) and ( 1 2 ) : _E u z  Equations  -  P < d  p-  o  }  (14)  2  ( 8 ) , ( 9 ) and (14) now f o r m a s y s t e m o f 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 s y s t e m o f l i n e a r e q u a t i o n s , i t c a n be made p a r a m e t r i c s i m p l y b y c h o o s i n g a " g i v e n " and d e n o t i n g i t a p a r a m e t e r .  I n t h i s case, i f t i s chosen  as t h e p a r a m e t e r , t h e n t h e p r i c e o f l a n d p ( t ) c a n 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 t h e 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 . Some I m p o r t a n t  c o r o l l a r i e s a r e proved by Alonso:  12  11„ Alonso uses t h e t o t a l d i f f e r e n t i a l i n s t e a d o f t h e Lagrangeanhowever, t h e r e s u l t s a r e t h e same. ' ' A l o n s o , op. ext., A p p e n d i x H.  16  1. The b i d p r i c e curve i s s i n g l e valued, i m p l y i n g that f o r any given u t i l i t y f u n c t i o n a t 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 o r the household. 2. Lower b i d p r i c e s imply g r e a t e r u t i l i t y s i n c e they s i g n i f y that the b i d s f o r land i n the community are lower. 3. B i d p r i c e curves f o r the same household do not c r o s s . The e q u i l i b r i u m 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 F i g u r e 1.  I n c i d e n t a l l y , i t can be shown t h a t the b i d p r i c e curves  slope  downward.  I n a d d i t i o n , Alonso shows that the p r i c e of land d e c l i n e s l e s s  r a p i d l y as the d i s t a n c e from the centre of the c i t y i n c r e a s e s as i s a l s o shown i n F i g u r e 2.  The p o i n t becomes c l e a r when the next author's work  i s considered and as w i l l be demonstrated has to do w i t h the t r a n s p o r t technology  of the c i t y .  $/sq. f t .  P(d) BPC 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 s u f f e r s from s e v e r a l s e r i o u s def e c t s that stem from the assumptions employed by some or a l l of the r e s e a r c h ers . The f i r s t i s s u e i n v o l v e s 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 p e r f e c t competition.  Wingo's  assumption of p e r f e c t 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 market i s very  restrictive. The requirements f o r competition  i n a market are w e l l known and  are f u l f i l l e d by the f o l l o w i n g c o n d i t i o n s . 1.  Buyers and s e l l e r s must be numerous.  2.  The t r a n s a c t i o n s of any one economic u n i t must be s m a l l enough not to have any e f f e c t on the p r i c e s or q u a n t i t i e s o f f e r e d 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 f r e e and unimpeded f o r 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 c o s t l e s s 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 t r a n s a c t i o n .  7.  The product i s homogeneous.  These p o i n t s can be summarized by three c o n d i t i o n s that there be homogeneous goods, many buyers and s e l l e r s and the costs of Information a c q u i s i t i o n are n i l .  and  Taking these p o i n t s i n t u r n , i t becomes apparent that  18  the housing market may by d e f i n i t i o n be i m p e r f e c t . are d i f f e r e n t i a t e d :  V i r t u a l l y a l l goods  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. i n g i s perhaps the most complex of consumer goods.  Hous-  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 a u t o m a t i c a l l y a c t s to g i v e 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 t h a t housing i s one economic good t h a t 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 e s t a t e companies act  to disseminate t h i s i n f o r m a t i o n i n an e f f i c i e n t manner.  Information i n a  market does not merely c o n s i s t of easy knowledge of what goods are p r e s e n t l y a v a i l a b l e but what goods w i l l be demanded and i n supply i n the f u t u r e . to the nature of housing, there tends to be a long p r o d u c t i o n p e r i o d . a d d i t i o n , housing i s d u r a b l e .  Due In  D u r a b i l i t y , long p r o d u c t i o n periods and  f i x e d l o c a t i o n are the common reasons that are g i v e n as to why  the housing  market should by d e f i n i t i o n be i m p e r f e c t . Surely the most important c o n s i d e r a t i o n when examining  the p e r -  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 sellers.  I f s i n g l e f a m i l y u n i t s 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 p e r f e c t c o m p e t i t i o n assumption i s used.  I f the mar-  ket c o n s i s t s of m u l t i p l e d w e l l i n g u n i t s , then i t i s l i k e l y t h a t the numbers of owners or s e l l e r s i s v e r y much, l e s s than the number of buyers. Since t h e r e i s l i t t l e e m p i r i c a l evidence on the s t r u c t u r e of the housing market, a l l that can be s a i d i s that a priori market.  housing i s an imperfect  19  A second objection to the l i t e r a t u r e 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 consumed. Some evidence i n the form of the frequency of the trips to various destinations about the c i t y 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 i s the sole determinant of r e s i d e n t i a l 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 establ i s h i n g other locational motivations to the r e s i d e n t i a l 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 s u f f i c i e n t 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 d i r e c t l y 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 resident i a l location that other motivations can be ignored. The proposition was tested s i m i l a r l y to Kain.  Vancouver was  divided into s i x rings and the labour force was c l a s s i f i e d into s i x occupat i o n a l classes. The percentage of each occupational group i n each ring was  (Vancouver;  J. Wolforth, Residential Tantalus Press, 1965).  Location and Place of Work in Vancouver,  20  compared to the mean income of the r i n g .  W o l f o r t h d i s c o v e r e d that lower  income workers tend to l o c a t e c l o s e r 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  areas assumed to have homogeneous housing c o s t s . i n g i n each census t r a c t was ular tract.  divided  into  The median v a l u e of hous-  assumed to be the c o s t of housing i n a p a r t i c -  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-  p a t i o n a l groups ranked according to percentage i n each t r a c t and groups ranked by income. The  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  occupation positive.  c o n c l u s i o n of Wolforth's study i s that there i s c o n s i d e r a b l y  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 p a t t e r n s than would be expected i f p r o x i m i t y to work was  the only m o t i v a t i o n to choosing a house.  Unfortunate-  l y , l i t t l e more can be s a i d from h i s study. The  d i f f e r e n c e between Wolforth's and Kain's study can  be a t t r i b u t e d to d i f f e r e n c e s been e s t a b l i s h e d a transport ium was  i n the c i t i e s s t u d i e d .  f o r a long period  technology that was  c i a l a c t i v i t y was  Those c i t i e s that have  of time (more than 100 y e a r s ) , grew u s i n g  c o s t l y to i n d i v i d u a l s .  placed upon c e n t r a l i t y and  largely  As a r e s u l t , a prem-  the placement of i n d u s t r i a l and  l o c a t e d at the core.  commer-  Thus these c i t i e s , such as New  York,  D e t r o i t , Montreal, e t c . , c o n d i t i o n e d the I n h a b i t a n t s i n t o a c c e p t i n g c e r t a i n l o c a t i o n patterns.  A d m i t t e d l y , these c o n s t r a i n t s  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 u s i n g more i n d i v i d u a l and  less costly transportation  (the car] they s t i l l  l i k e l y p l a c e more c o n s t r a i n t upon the consumption a l t e r n a t i v e s of the worker.  21  The p o i n t t h a t must be made here i s the household I n a l l l i k e l i h o o d i s responsive t o other t r i p s .  The p r o p e n s i t y to t r a v e l t o v a r i o u s  d e s t i n a t i o n s such as shopping c e n t r e s , s c h o o l s , r e c r e a t i o n a l c e n t r e s , n i g h t c l u b s , e t c . v a r i e s w i t h t h e s t r u c t u r e o f the household.  The number  of c h i l d r e n and t h e i r ages a r e important f a c t o r s i n where to l o 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, p r o x i m i t y to  n i g h t 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 c l o s e to such f a c i l i t i e s . The households responds t o many l o c a t i o n a l p u l l s .  The theory  surveyed w h i l e paying l i p s e r v i c e to t h i s has assumed that these  trips  were i n s i g n i f i c a n t and no damage t o the r e a l i s m 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 m o t i v a t i o n . 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 s t r u c t u r e and income v a l u e differently. lined.  I n the next chapter the theory u n d e r l y i n g the model i s out-  Two separate strands —  from modern demand theory —  one from operations r e s e a r c h , the other  a r e u n i t e d to form the base o f 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 decides on the b a s i s of only two c h a r a c t e r i s t i c s — work.  space and l o c a t i o n t o  The reason f o r t h i s i s that t r a d i t i o n a l economic theory i s q u i t e  constrained i n the a n a l y s i s of consumer goods.  I n t h i s chapter I out-  l i n e a theory o f demand f i r s t developed by K. J . L a n c a s t e r .  He employs a  l i n e a r a c t i v i t y a n a l y s i s to study the behaviour o f consumers when goods are considered t o 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 o u t l i n e d , i t becomes apparent that f o r con-  sumer durables such as housing, the a n a l y s i s g i v e n by Lancaster needs t o be amended to an i n t e g e r programming framework or assignment model. t h i s i s e s t a b l i s h e d , the work of W. F. Smith i s reviewed. an assignment approach t o housing s t u d i e s .  Once  Smith has used  From here i t i s a s h o r t step  to my model of housing demand.  THE NEW THEORY OF DEMAND T r 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, o f t e n t o t a l l y - u n r e l a t e d and pinned t o r e a l i t y only by a budget c o n s t r a i n t .  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 b u t t e r w i t h the t r a d e - o f f v a r y i n g ( i n two dimensions) from a s t r a i g h t l i n e f o r p e r f e c t s u b s t i t u t e s to a r i g h t angle f o r p e r f e c t complements. 22  An under-  23  c u r r e n t of economic theory has always argued that the c h o i c e i s more ordered. Cars are traded o f f w i t h commuter s e r v i c e of v a r i o u s types, b u t t e r w i t h marg a r i n e and shoes w i t h other c l o t h e s , Implying t h a t consumers choose among c h a r a c t e r i s t i c s r a t h e r than goods. K. J . Lancaster has f o r m a l i z e d t h i s view i n t o a f a i r l y r i g o r o u s theory and p o s t u l a t e s three assumptions as the departure from c o n v e n t i o n a l t h i n k i n g on the matter."^ 1. I t i s the c h a r a c t e r i s t i c s i n h e r e n t 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 g e n e r a l , goods possess more than one teristic.  charac-  3. Goods i n combination, or complements may g i v e 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 charact 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 a n a l y s i s to consumption theory i s merely the reverse of p r o d u c t i o n theory.  In p r o d u c t i o n  theory an a c t i v i t y i n v o l v e s the combination of s e v e r a l i n p u t s i n the c r e a t i o n of one product, w h i l e i n consumption theory the act of consuming i n v o l v e s one i n p u t or good and s e v e r a l j o i n t outputs or c h a r a c t e r i s t i c s . Therefore, a s s o c i a t e d w i t h each good there i s a v e c t o r of characteristics.  If  i s t h i s v e c t o r 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 v e c t o r s of c h a r a c t e r i s t i c s may  be  represented by a m a t r i x B which Lancaster r e f e r s to as the consumption  14  K. J . Lancaster, Mathematical Economics (New York: 1968); "The New Theory of Consumer Demand," Journal of Political V o l . 74, No. 4 ( J u l y , 1966).  Macmillan, Economy,  24  technology m a t r i x .  In g e n e r a l , 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 p o s t u l a t e s that f o r advanced economies the number of goods i s g r e a t e r 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 a r r a y 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 p o s s i b l e .  A l s o , i f the m a t r i x  B i s square, and can be decomposed i n t o a d i a g o n a l m a t r i x , t h i s new i s nothing more than a restatement  theory  of the o l d theory.  Since the consumer operates i n c h a r a c t e r i s t i c space and goods space, the u t i l i t y f u n c t i o n i s of the f o l l o w i n g  not  form:  U = U(z )  (16)  This i s maximized s u b j e c t to the f o l l o w i n g  constraints:  z = Bx  (17)  px <_ k  (18)  x  (19)  >o  where p i s a v e c t o r of goods p r i c e s and k i s the income of the household. The program as i t stands i s n o n - l i n e a r but can be transformed 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  simply by  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 nonnegative, then the " a t t a i n a b l e 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 r e p r e -  25  sented by rays which, i n d i c a t e the m i x t u r e of c h a r a c t e r i s t i c s each possesses. The a t t a i n a b l e c h a r a c t e r i s t i c s s e t i s shown by the shaded area.  The q u a n t i -  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 l e n g t h of the r a y i n q u e s t i o n .  Some goods Cas  good 3 i n F i g u r e 3) may not be considered a t a l l by any consumer.  The p e r -  s o n a l choice i s shown by the i n 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 t h e expenditure of the household w i t h i n the c o n s t r a i n t s  s e t by the a t t a i n a b l e  characteristics-set.  There are two s u b s t i t u t i o n e f f e c t s i n o p e r a t i o n 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 p r i c e of good two r i s e s .  This has the e f f e c t of mov-  i n g p o i n t X2 along the ray 2 (Figure 4 ) . As soon as #X1, X2, X3 form a  26  Figure 6  Centrality  27 s t r a i g h t l i n e the consumer w i l l maximize h i s w e l f a r e by s w i t c h i n g 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-  v e n t i o n a l s u b s t i t u t i o n occurs when the e n t i r e c o n s t r a i n t f u n c t i o n s h i f t s inwards and the i n d i f f e r e n c e map h e l d as i n F i g u r e  governs the s w i t c h i n p o r t f o l i o of goods  5.  APPLICATIONS OF THE THEORY TO HOUSING Consider a s t r a i g h t f o r w a r d approach to housing w i t h no m o d i f i c a t i o n of the theory. of c h a r a c t e r i s t i c s .  Assume a r e s i d e n t i a l bundle w i t h a w e l l d e f i n e d set These i n c l u d e such f a c t o r s as p r o x i m i t y to work, p l a y ,  the s c h o o l s , the n o i s e l e v e l of the area, the e t h n i c mix of the neighbourhood, e t c . 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 F i g u r e 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 i m p l i e s t h a t given  the type of u t i l i t y f u n c t i o n i n the f i g u r e , then the r a t i o n a l household would h o l d two types of houses.  However, the f o r m u l a t i o n ignores the prob-  lem of the costs of m u l t i p l e d w e l l i n g ownership. these are very h i g h .  Representation  I t i s very l i k e l y that  of t h i s s i t u a t i o n becomes very  diffi-  c u l t s i n c e 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 f o r c e the consumer to  h o l d only one house, however the economic meaning of such geometry i s dubious.  In e f f e c t , t h i s f o r m u l a t i o n i m p l i e s that there i s some smooth f u n c -  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 t r a n s a c t i o n c o s t s . A l l that can be r e a l l y s t a t e d unequivocably i s that the f e a s i b l e r e g i o n remains non-convex f o r most consumers of housing w i t h incomes below a c e r t a i n l e v e l simply because the h o l d i n g of m u l t i p l e d w e l l i n g u n i t s f o r pure consumption purposes i s very r a r e .  28  A second important  i s s u e r e v o l v e s around t h e a c t u a l r e p r e s e n t a -  t i o n o f houses i n t h i s framework.  By u s i n g a c o n t i n u o u s  v e c t o r , what i s  i m p l i e d i s t h a t t h e consumer c a n f r e e l y a d j u s t t h e amount o f h o u s i n g i s consumed.  that  I n o t h e r w o r d s , t h e p r i c e l i n e may f a l l anywhere a l o n g a  g i v e n r a y and t h e r e w o u l d b e a h o u s e t h a t w o u l d e x i s t w i t h t h e e x a c t m i x 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 .  In reality  this i sunlikely.  sumer goods s u c h as h o u s e s need t o be r e p r e s e n t e d 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  Large con-  by d i s c r e t e p o i n t s i n  t h e n t h e p r o b l e m becomes  an i n t e g e r programming p r o b l e m w i t h a l l t h e 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 . The  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 .  next step i n the e v o l u t i o n of the housing  l i n e an a s s i g n m e n t m o d e l o f h o u s i n g  demand d e v e l o p e d  model I s t o o u t -  b y W. F. S m i t h .  The  a s s i g n m e n t p r o b l e m i s one f o r m o f i n t e g e r programming p r o b l e m t h a t h a s r e ceived considerable refinement  i n recent  years.  W. F. SMITH AND A MATRIX ANALYSIS OF NEIGHBOURHOOD CHANGE The  l a s t c h i n k i n t h e t h e o r e t i c a l a t m o s p h e r e i s now t o b e f i l l e d .  To r e c a p i t u l a t e , h o u s i n g  has been c o n c e i v e d  o f as a b u n d l e o f 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 o f t h e h o u s e and t h o s e i n t r i n s i c t o t h e good i t s e l f .  Of c o u r s e , q u a l i t y i s n o t  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, t h e concept i o n of such interdependence  l e t a l o n e t h e measurement i s beyond me.  also view the urban landscape  as t h e r e s u l t o f a c o m p e t i t i v e b i d d i n g  I pro-  c e s s 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 o f l a n d compete w i t h one another  and t h e p r o p e r t y g o e s t  to thehighest bidder.  T h i s w i l l be f o r m a l -  i z e d s h o r t l y i n t h e context of t h e "optimal assignment model."  29  Smith's  theory or urban r e s i d e n t i a l s t r u c t u r e i s a d i r e c t 16  descendent of the s e c t o r theory formulated by Homer Hoyt  i n the 1930's.  At t h a t time the controversy was over the concept of " f i l t e r i n g . "  Basic-  a l l y , the p r o p o s i t i o n was t h a t f i l t e r i n g i s the process whereby the demand f o r 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  c o n s t r u c t i o n i s i n h a b i t e d by the r i c h who "bequest" t h e i r o l d homes f o r the next lower s t a t u s group.  As an armchair e m p i r i c a l f a c t there appears to be  l i t t l e to d i s p u t e , however, c o n t r o v e r s y e x i s t e d as t o whether e n l i g h t e n e d s o c i a l p o l i c y c o n s i s t e d of b u i l d i n g h i g h q u a l i t y housing t o Induce t h e r i c h to move and thereby i n c r e a s e the supply of housing f o r poor people, or whether i t was p r e f e r a b l e t o c o n s t r u c t low income housing p r o j e c t s . Even today housing p o l i c y i s very much d i v i d e d on t h i s matter. The s e c t o r theory formulated by Hoyt seemed t o i n d i c a t e t h a t the succession of houses from the r i c h to the poor was a n a t u r a l f a c t of urban ecology.  As the c i t y matures the r i c h areas were hypothesized t o  move outward i n rays resembling p i e s l i c e s .  The exodus of the r i c h leaves  behind housing that i s q u i c k l y 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 s e v e r a l r u l e s of m i g r a t i o n .  G e n e r a l l y the r i c h  move to the h i g h ground, along t r a n s p o r t a t i o n routes and tend t o avoid  W. F. Smith, Filtering and Neighborhood Center f o r Real E s t a t e and Urban Economics, 1965). 15  16  Hoyt, c i t e d i n Smith, op. cit., p. 9.  Change (Berkeley:  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 r e c o g n i t i o n that the r a t e of change i n p o p u l a t i o n i s an important determinant  of the success of t h e r i c h m i g r a t i n g 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 s l u g g i s h supply response t o new housing demand, the d i s t i n c t i o n s between the s e c t o r s may v e r y w e l l become b l u r r e d . The s e c t o r theory i s not s u f f i c i e n t to p r e d i c t or even adequatel y e x p l a i n the e x i s t i n g s p a t i a l s t r u c t u r e of c i t i e s .  I n p a r t i c u l a r , the  a c t i o n s of the middle income p o r t i o n of the p o p u l a t i o n were never accounted for i n d e t a i l .  C e r t a i n l y c o n s i d e r a b l e p o r t i o n s of the new housing  stock  was aimed a t the middle income groups simply because the r i c h d i d not form a s u f f i c i e n t p a r t of the p o p u l a t i o n to a l l o w the lower income groups to i n h e r i t enough houses.  Smith's model of housing i s v e r y simple and i s based  upon the optimal assignment model from the theory of i n t e g e r programming. I t i s a process whereby the e x i s t i n g p o p u l a t i o n i s assigned according to some predetermined  r u l e to the e x i s t i n g housing stock.  Households a r e  d i f f e r e n t i a t e d w i t h r e s p e c t to income w h i l e houses a r e c h a r a c t e r i z e d accordi n g to "value" or p r i c e . S e v e r a l questions are asked of the model: 1. What i s the p a t t e r n of urban r e s i d e n t i a l s t r u c t u r e produced by the p u r e l y c o m p e t i t i v e market? 2. What i s the impact w i t h i n the c o n s t r a i n t s 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 p o p u l a t i o n 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 c o n s t r u c t i o n ?  31  4. Given the a d d i t i o n of such new how w i l l the p a t t e r n of occupancy change?  stock,  5. I f there i s an i n c r e a s e i n the housing s t o c k and p o p u l a t i o n , what new housing w i l l be constructed? S e v e r a l 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 c o n s i d e r a b l y .  Only f i v e f a m i l i e s are assumed to i n -  h a b i t 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 available.  P r i c e i s d e f i n e d 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 p o i n t i n the c o n s t r u c t i o n of the model i s the c r e a t i o n of rent o f f e r s f o r v a r i o u s house types.  Each household makes an o f f e r on each-house:  an o f f e r which v a r i e s according to the d e s i r a b i l i t y of the house and income of the f a m i l y . The e f f e c t s of v a r i o u s income l e v e l s and house q u a l i t y on the r e n t o f f e r s of f a m i l i e s can be shown simply as a m a t r i x :  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 f o r a l l f a m i l i e s .  L i s the lowest r e n t o f f e r e d by any house-  h o l d f o r any house; i t represents the b a s i c demand f o r s h e l t e r . What i s important  i s not the amount L but the d i f f e r e n c e s from L that w i l l  o f f e r e d f o r v a r i o u s houses by v a r i o u s households.  be  32  Smith, then examined the i m p l i c a t i o n s of t h i s m a t r i x and concluded that i t i s not r e a l i s t i c t o 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 i n c r e a s e i n q u a l i t y would be worth  something to a h i g h e r 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 .  I n other words, each increment up i n b o t h  income and q u a l i t y r e s u l t s i n an i n c r e a s e 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 s i n c e the a n a l y s i s i s unchanged as long as the income and q u a l i t y , e f f e c t s a r e p o s i t i v e .  HOUSEHOLDS  HOUSES A  B  C  D  E  2  +1  +2  +3  +4  3  +2  +4  +6  +8  4  +3  +6  +9  +12  5  +4  +8  +12  +16  1  The two m a t r i c e s a r e now s i m p l y added together and r e s u l t i n a rent o f f e r which d e p i c t s the b i d made by each household f o r each house. The problem i s now to a s s i g n households to house according to some r u l e . I f the houses and households are ranked according to some o b j e c t i v e c r i t e r i o n such as s a l e s p r i c e and income, then theory from l i n e a r programming i n d i c a t e s that an o p t i m a l s o l u t i o n e x i s t s when houses a r e matched t o households along the main diagonal as i n the f o l l o w i n g :  33  HOUSEHOLDS  HOUSES A  1  B  O  D  E  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 rent o f f e r s of t h e community.  of the  Smith uses the r e s u l t s from the theory of  pure competition t o j u s t i f y t h e r e l a t i o n o f a Pareto o p t i m a l s o l u t i o n and r e n t maximizing p r o g r a m . ^  Note that l i k e the b i d p r i c e of Alonso,  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 p a i d .  these  Before examin-  i n g some of the experiments that Smith s u b j e c t s h i s model t o , I might j u s t p o i n t out that any other assignment other than the one shown r e s u l t s i n a lower aggregate r e n t o f f e r on the p a r t of the community. One  of the assumptions that was made a t 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 t o the costs of p r o d u c t i o n .  I f household f i v e leaves the  community and then a f a m i l y 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 r e n t o f f e r now dips  to L + 130 which r e f l e c t s the l o s s of h i g h income f a m i l y .  The f a c t that  the aggregate r e n t o f f e r has d e c l i n e d i n no way has any I m p l i c a t i o n upon the standard of housing i n t h i s simple model.  What i s notable i s that  f a m i l i e s 3, 4 and 5 now i n h a b i t " b e t t e r " housing.  "^See  Appendix 1 f o r a simple e x p l a n a t i o n .  34  Smith a l s o p o i n t s out t h a t p r e d i c t i o n s as to what type of housing  i s r e q u i r e d to meet a n t i c i p a t e d demand can be made u s i n g t h i s  work.  frame-  The b a s i c technique i s to c a l c u l a t e the "economic v a l u e " of a  p a r t i c u l a r type of house ( i . e . , i t s r e n t o f f e r i n the o p t i m a l a s s i g n ment) and then compare t h i s w i t h the c o n s t r u c t i o n c o s t .  This i s done  by c o n s i d e r i n g the o r i g i n a l m a t r i x and examining the change i n aggregate rent o f f e r s when d i f f e r e n t types of houses are added.  For example, the  change i n r e n t o f f e r s when houses of type A, B, C, D, or E area added are as f o l l o w s : 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 w i t h households 1 occupying house type B, household 2 occupying house type C, e t c .  I f the v a l u e curve Is p l o t t e d and compared  w i t h 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 c o n s t r u c t i o n would be j u s t i f i e d unless the economic value equals or exceeds the c o n s t r u c t i o n c o s t s .  Here house type C or  b e t t e r 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 s e v e r a l d i r e c t i o n s .  In the f i r s t p l a c e , the  35  concept  o f b i d f u n c t i o n i s a m p l i f i e d t o i n c l u d e b i d s by h o u s e h o l d s f o r n o t  j u s t " v a l u a b l e " houses. the concept  One  of the weaknesses i n Smith's t h e o r y i s t h a t  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  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 p e r s o n w i l l a l s o be by a n o t h e r .  Secondly,  and  ob-  valued  r e l a t e d to the f i r s t p o i n t , i s the assumption  t h a t t h e b a s i s f o r s u c h j u d g e m e n t s i s upon i n c o m e a l o n e . Once t h i s a s s u m p t i o n i s q u e s t i o n e d and r e j e c t e d on t h e g r o u n d s t h a t seem i n t u i t i v e l y a p p a r e n t ,  a n o t h e r more complex p r e m i s e i s r e q u i r e d .  I postulate that for different  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 t h e d e s i r a b i l i t y o f a p a r t i c u l a r  h o u s e w i l l v a r y w i t h n o t o n l y income b u t t h e h o u s e h o l d gin,  c l a s s o r i g i n and  In for to  other i d i o s y n c r a t i c  size, ethnic  ori-  details.  the next chapter I p r e s e n t a model w h i c h d i v i d e s the b i d  v a r i o u s h o u s e s a c c o r d i n g t o what d i f f e r e n t h o u s e h o l d s c a n b e o f f e r f o r various housing  attributes.  expected  At t h i s stage w i t h very  e m p i r i c a l work done i n t h e a r e a of l a n d v a l u e s and  little  the determination,  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 . 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  I  characteristics  a r e a d d i t i v e and f o r m a b i d - f u n c t i o n m a t r i x w h i c h r e f l e c t s t h e b i d b y 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 , t h e n  each  an  a l g o r i t h m b a s e d upon t h e o p t i m a l a s s i g n m e n t p r o b l e m i s u s e d t o 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  r e n t of t h e community.  the  aggregate  CHAPTER THREE AN ASSIGNMENT MODEL OF URBAN HOUSING DEMAND  In  t h i s s e c t i o n o f t h e essay I o u t l i n e an e x t e n s i o n o f t h e work  of S m i t h , w h i c h i n c o r p o r a t e s some o f t h e a s p e c t s As m e n t i o n e d i n t h e p r e v i o u s presented  c h a p t e r , one o f t h e w e a k n e s s e s o f t h e m o d e l  b y S m i t h was t h e vague u s e o f t h e n o t i o n o f q u a l i t y o f h o u s i n g .  The m o d e l I p r e s e n t housing  o f modern demand t h e o r y .  attempts  t o overcome t h i s d e f i c i e n c y b y d i s a g g r e g a t 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 m a k i n g t h e s e t h e b a s i s f o r t h e d e s i r -  a b i l i t y o f p a r t i c u l a r h o u s e s b y t h e h o u s e h o l d s i n t h e community. it  i s no l o n g e r p o s s i b l e t o a priori  have d i f f e r e n t  rank the housing  As a r e s u l t ,  s t o c k by c l a s s e s which  quality.  W i t h 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 t o h a v e 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 h a v e t h e o p t i m a l s o l u t i o n pop o u t a s does  Smith.  A complex a l g o r i t h m i s needed t o f i n d  t h e o p t i m a l s o l u t i o n . The  s e c o n d m o d i f i c a t i o n t h e n i s m e r e l y t o u s e s u c h a n a l g o r i t h m t o do t h e assignment. Before  examining t h e s t r u c t u r e of t h e b i d f u n c t i o n m a t r i x as I  employ i t , some e x a m i n a t i o n s h o u l d be made.  of t h e p r o p e r t i e s of t h e assignment problem  I t i s a s u b - c l a s s o f l i n e a r programming p r o b l e m s i n t h a t  t h e same a s s u m p t i o n a b o u t 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; h o w e v e r , i n t h i s c a s e , no f r a c t i o n a l answers a r e permitted.  I t was e v o l v e d b y o p e r a t i o n s r e s e a r c h e r s t o s o l v e t h e p r o b l e m  t h a t a r i s e s when a j o b h a s t o b e a s s i g n e d to  that  facility. 36  to a specific f a c i l i t y  and o n l y  37  THE ASSIGNMENT PROBLEM The assignment problem i n v o l v e s s e v e r a l f a c t o r s whose p r o d u c t i v i t y can be measured to s e v e r a l jobs i n such a manner as to maximize the 18 aggregate r e t u r n .  An example Is the matching of employees to t a s k s .  The  c r u c i a l requirement i s t h a t one and only one f a c t o r be matched t o one j o b . Mathematically,  the problem can be s t a t e d as f o l l o w s .  Given an n x n m a t r i x , A^j  (the r a t i n g m a t r i x ) w i t h a „ >_ 0  for a l l i , j , f i n d an n x n m a t r i x X.. such that IX.. i  =  1  EX..  =  1  1 J  1 3  J  I I a.. X.. 1  =  max  3  The f i r s t two c o n d i t i o n s ensure t h a t the v a l u e of X w i l l be 1 i f f a c i l i t y 1 i s assigned  to j o b j and w i l l be 0 otherwise.  Each column  and row c o n t a i n only one entry of u n i t v a l u e w i t h a l l the others zero.  The  t h i r d c o n d i t i o n s p e c i f i e s that the elements chosen from the r a t i n g m a t r i x  will  maximize the product. With a few amendments, the assignment problem can be transformed i n t o one which has a very simple and s t r a i g h t f o r w a r d s o l u t i o n .  See Appendix 2 f o r more d e t a i l s "on the a l g o r i t h m .  The housing  38  market t k a t Smith has s o l v e d by the assignment process can be d e s c r i b e d 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 o f f e r s a rent f o r each  3.  Rent o f f e r s i n c r e a s e w i t h income and  house. quality. 4. A premium i s o f f e r e d by each household for increases i n q u a l i t y . The s i t u a t i o n i s s t a t e d mathematically by Smitk as f o l l o w s : "Let a., be the r e n t o f f e r of the i t h f a m i l y f o r tke j t h house, tnen there i s a m a t r i x of r e n t o f f e r s i n which i + 1 i s a higher income l e v e l than i and j + 1 i s h i g h e r q u a l i t y d w e l l i n g than j suck t h a t , a(ij) < a(i+l, j) a(ij) < a(i,  j+1)  [ a ( i , j+1) - a ( i j ) ] < [aCi+1,  - a(i+l, j ) J  I f the r e n t o f f e r ( b i d f u n c t i o n m a t r i x ) i s s e t up w i t h households and houses ranked, then a s s i g n i n g the L t h to the j t h house w i t h j = I r e s u l t s i n a r e n t maximizing assignment." 20 With the simple s t r u c t u r e of Smith's model, i t i s simple to prove that any assignment other than that which a s s i g n s 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 r e n t of the - 21 matrix.  Smith, op. ait.. Appendix  1.  Ib£d. p. 68.  )  3  'Ibid., pp. 67-70 f o r p r o o f s .  39  I n f a c t , t h e r e i s no need t o r a n k t h e b i d f u n c t i o n m a t r i x . p u r p o s e o f 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  v a r i o u s c l a s s e s of housing.  Once t h e h o u s i n g s t o c k i s d i s a g g r e g a t e d  The the into  economic goods w i t h c h a r a c t e r i s t i c s w h i c h e a c h p o s s e s s i n d i f f e r e n t d e g r e e s and  the households are not  simply  c h a r a c t e r i z e d by  Income b u t  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 and  rows i s  t u r n and  columns  examine t h e b a s i s f o r a more e x t e n d e d and  plete bid function matrix.  For  t h i s I must r e p e a t ,  s t a t i n g , t h e b i d p r i c e c u r v e as f o r m u l a t e d  BID  to rank the  impossible. I now  THE  ethnicity,  by  at the r i s k of  comover-  Alonso.  FUNCTION MATRIX T h i s s e c t i o n d e v e l o p s t h e n o t i o n of b i d f u n c t i o n m a t r i x w h i c h i s  very of  c l o s e t o t h e b i d p r i c e c u r v e employed by A l o n s o .  To r e p e a t  the usage  Alonso: "A b i d p r i c e c u r v e o f a r e s i d e n t i s t h e s e t o f 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 constant 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 t h e manner d e s c r i b e d by t h e b i d p r i c e c u r v e , t h e n ^ t h e i n d i v i d u a l w o u l d 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. household.  The  b i d p r i c e curve r e f e r s to the  2.  The  curves v a r y from i n d i v i d u a l to  c u r v e and  individual  individual.  3. T h e r e i s no r e l a t i o n between t h e b i d p r i c e the p r i c e that i s a c t u a l l y paid f o r land.  'Alonso, op.  c i t . , p.  5,8.  40  The e x t e n s i o n of Alonso's work I wish to make i n v o l v e s  develop-  ing b i d p r i c e curves not only f o r land but some of the o b j e c t i v e c h a r a c t e r 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 d i s t a n c e t o work, shopping, schools and other d e s t i n a t i o n s .  I am assuming that the  household when i n the market f o r a house has a c l e a r idea what e x a c t l y i t wants and can s t a t e 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 a r e unimportant.  Thus not only  i s there a b i d p r i c e curve f o r land but b i d p r i c e c u r v e s . f o r each of the d i s t a n c e parameters mentioned, v a r i o u s amenities a s s o c i a t e d w i t h the n e i g h 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  t o c a s t 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.  F o r the  s i t u a t i o n that Alonso was c o n s i d e r i n g i t was f a i r l y reasonable  to p i 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 l a n d and d i s t a n c e was discovered; however, I demand  that the consumer not only f i n d an e q u i l i b r i u m between l a n d , and d i s t a n c e from the c i t y c e n t r e , but an e q u i l i b r i u m 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 v a r i a b l e s as used by Alonso a r e r e t a i n e d except f o r a second d i s t a n c e or v a r i a b l e - d i s t a n c e from shopping areas.  The problem now appears as f o l l o w s .  The urban housing consumer i s p i c t u r e d as s e p a r a t i n g each p r o s p e c t i v e d w e l l i n g i n t o i t s c o n s t i t u e n t c h a r a c t e r i s t i c s upon w h i c h i t p l a c e s a value.  The b i d that i s o f f e r e d upon the house i s the sum of the bids  o f f e r e d 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  m o d e l and i s c e r t a i n l y t h e most s u s p e c t a n d u n r e a l i s t i c . the  utility  Implied i s that  f u n c t i o n f o r t h e s e 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 w r o n g .  F o r example,  t h e e a s e o f d r i v i n g t o w o r k may b e  e n t i r e l y negated by t h e l a c k of p a r k i n g or t h e p r o x i m i t y t o shopping areas c o u l d n o t b e a f a c t o r s i m p l y b e c a u s e t h a t p a r t i c u l a r h o u s e h o l d does a l l its  shopping t o and from work.  The i n t e r d e p e n d e n c e o f 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 o f L a n c a s t e r ' s t h e o r y o f demand.  The assump-  t i o n o f a d d i v i t y seems t o me t o h e t h e o n l y w o r k a b l e e m p i r i c a l h y p o t h e s i s . In  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 o f s i f t i n g  t h e impacts of c h a r a c t e r i s t i c s  f r o m one a n o t h e r i s d e r i v e d , i t a p p e a r s t h a t t h i s a s s u m p t i o n needs t o b e retained.  A MODEL OF URBAN HOUSING DEMAND The m o d e l i s i n two p a r t s . t i c s o f h o u s e h o l d s and h o u s e 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 -  and f o 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 t h e  second p a r t o f t h e model i s an a l g o r i t h m w h i c h a l l o c a t e s t h e h o u s i n g s t o c k to  the households  i n s u c h a way as t o m a x i m i z e  t h e aggregate rent  offered  by t h e community and s u c h t h a t no h o u s e h o l d c a n b e r e a l l o c a t e d w i t h o u t making  any one o t h e r h o u s e h o l d w o r s e o f f . The f u n c t i o n s t h a t a r e employed h a v e no b a s i s i n e m p i r i c a l w o r k  s i n c e l i t t l e w o r k h a s b e e n done w h i c h c o u l d shed i n s i g h t i n t o t h e 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 housing. Arbitrarily, are  I c h o s e 15 h o u s e t y p e s and 15 h o u s e h o l d t y p e s .  The h o u s e h o l d s  c h a r a c t e r i z e d b y t h e number o f p e o p l e I n t h e h o u s e h o l d t o a maximum o f  42  t h r e e , and income o f w h i c h t h e r e a r e f i v e c l a s s e s , r a n g i n g f r o m $500 p e r y e a r t o $15,000.  A l l t h e numbers u s e d a r e 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 o f s p a c e and l o c a t i o n t o t h e downtown a r e w e i g h t e d o r measured  by an i n d e x number and how  w i l l be d i s c u s s e d l a t e r .  t h i s number i s o b t a i n e d  I make t h e a s s u m p t i o n t h a t g i v e n a l m o s t com-  p l e t e i g n o r a n c e o f t h e r e l a t i o n between  t h e v a l u a t i o n , t h e demand f o r  u r b a n s p a c e and f a m i l y s i z e and income, t h e r e l a t i o n i s l i n e a r and t a k e s the form:  Bidspace ( i j ) = f[Income i ,  Space j ,  Pers i ]  I n t h i s manner a m a t r i x w i t h h o u s e t y p e a l o n g t h e rows and h o u s e h o l d t y p e a l o n g t h e column i s c o n s t r u c t e d w h i c h shows t h e b i d by each h o u s e h o l d f o r e a c h h o u s e .  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.  assume t h a t t h e i m p o r t a n c e o f 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 t h e income;  I does  The f u n c t i o n t h a t i s b e i n g u s e d a t t h e moment t a k e s t h e  form:  Bidloc  ( i j ) = f[Income i ,  Loc j ]  where Space j i s t h e s p a c e i n house j ; Income i i s t h e income o f h o u s e h o l d  i;  P e r s i i s t h e number o f p e o p l e i n h o u s e h o l d  i;  L o c j i s t h e i n d e x o f 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 characteristics  i n the u t i l i t y f u n c t i o n , the b i d l o c a t i o n matrix i s simply added  to the b i d space m a t r i x to form the b i d f u n c t i o n m a t r i x which i s the same m a t r i x 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 d e r i v e d . At t h i s p o i n t the households are assigned to houses by an a l g o r ithm which i s e x p l a i n e d 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 m a t r i x w h i l e Table 2 i s the b i d space m a t r i x .  The f i n a l s o l u t i o n w i t h aggregate b i d o f f e r e d by the community  i s shown i n Tables 3 and 4.  The t a b l e s are presented i n Appendix 4.  The model as i t now stands i s v e r y 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 c o n t r a v e n t i o n 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 i n d i c a t e d t h a t low income f a m i l i e s l o c a t e i n the most remote housing w h i l e h i g h income f a m i l i e s are c l u s t e r e d about the core of the c i t y .  Every  study on r e s i d e n t i a l s t r u c t u r e and even casual e m p i r i c a l observations contradict this result.  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 f u n c t i o n m a t r i x .  A more complete a n a l y s i s would no doubt add  a l more v a r i a b l e s and have more s o p h i s t i c a t e d b e h a v i o u r a l f u n c t i o n s .  severMore  w i l l be s a i d about t h i s l a t e r . However, perhaps the most g l a r i n g p o i n t at t h i s moment i s the omission of any c o n s i d e r a t i o n of the budget c o n s t r a i n t s faced by the urban housing consumer.  I n a way  this, i s i n c l u d e d i n the b e h a v i o u r a l f u n c t i o n s ;  n e v e r t h e l e s s , some e x p l i c i t supply factors: need to be i n c o r p o r a t e d i f the model i s to approach some r e a l i s m .  44  The s t r e n g t h of t h i s s t y l e of t h i n k i n g i s t h a t 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  v a r i o u s 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.  Little  i s known about the importance of journeys other than t h a t of work-oriented journeys i n the demand f o r housing and t h e r e f o r e u n t i l the e m p i r i c a l work has been completed, n o t h i n g other than c o n j e c t u r e can be advanced. same must be s a i d about other c h a r a c t e r i s t i c s than space.  The  One c e r t a i n l y  could say t h a t elements such as design (ranch, apartment, townhouse or s p l i t l e v e l ) have f o r 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 t h e r e .  In the next s e c t i o n 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 d i s c o v e r e d . I n 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 i n t e g r a t i n g a supply s i d e to housing a l l o c a t i o n .  I conclude the essay w i t h a b r i e f  examination of the p o l i t i c a l b i a s of the model — vant f o r income l e v e l s ?  namely, i s the model r e l e -  45  THE MODEL:  AN EVALUATION As i t stands now, the model that I have presented i s very naive  and r e s t r i c t e d .  In the f i r s t p l a c e , the h i d f u n c t i o n s and b e h a v i o u r a l  e q u a t i o n s that I have used are very s i m p l i s t i c and have no b a s i s i n r e a l i t y . Secondly, the model i s not dynamic.  SOME STATISTICAL CONSIDERATIONS The problem q u i t e simply i s to discover how the b i d f o r houses of  d i f f e r e n t types v a r i e s w i t h the s t r u c t u r e of the household.  A priori,  the premium placed on space and p r i v a c y w i l l be a f u n c t i o n of the number of persons i n the household and the age s t r u c t u r e of the f a m i l y .  Similarly,  the e f f e c t of i n d u s t r i a l centres w i l l be l e s s f o r those households whose members possess s k i l l s normally used i n the CBD, such as p r o f e s s i o n a l s . 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 a t t r i b u t e s of t h e i r present house they l i k e or d i s l i k e .  This type of information would then be c o r r e -  l a t e d w i t h data on the family s t r u c t u r e .  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 a l t e r n a t e method would be to use land values as a trap f o r the advantages  and disadvantages of various l o c a t i o n s and types of house.  The r e l a t i o n between the sales value of a house and the c h a r a c t e r i s t i c s of a house such as space, i n d i c e s of p r i v a c y , i n d i c e s of environmental q u a l i t y , and i n d i c e s of proximity to work, shopping, schools, e t c . could be measured using r e g r e s s i o n 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 b i d s v a r y w i t h the s t r u c t u r e of the household i s p o s s i b l e .  In a l l l i k e -  l i h o o d , the market i s too s l u g g i s h t o be v e r y s e n s i t i v e t o changes of f a m i l y s t r u c t u r e 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 ( e x a c t l y the same a t t r i b u t e s ) , the s a l e s p r i c e l i k e l y would not r e f l e c t any d i f f e r e n c e i n the s t r u c t u r e of the two f a m i l i e s l i v i n g i n these houses.  Other c o n s i d e r a t i o n s such as b a r g a i n i n g s k i l l s of the v a r -  ious buyers and s e l l e r s would be important i f home ownership were the case. The only way to r e s o l v e t h i s d i f f i c u l t y appears to be t o use both methods to g a i n some i d e a of how consumer demand v a r i e s , then t o cons t r u c t b i d f u n c t i o n 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 t e s t s but a r e not d i r e c t l y r e l a t e d to the parameters discovered. combination  Thus a  of q u e s t i o n n a i r e and r e g r e s s i o n should i n d i c a t e which v a r i a b l e s  or a t t r i b u t e s of a house and household are t h e 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 o f households of houses i n a p a r t i c u l a r c l a s s a r e g r e a t e r 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 c o r r e c t e d . At the moment, only s i m p l i s t i c s o l u t i o n s come to mind. These c l a s s e s of household or houses which are i n excess supply could simply be ignored. A b e t t e r s o l u t i o n would be to have the model a s s i g n the v a r i o u s 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 c l a s s down.  I f one were t o s t a r t at the  47  t o p and move t h r o u g h t h e 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 o f  households, the households a t the bottom o f the s c a l e would be  forced  out  nothing  of the market.  conclusive  At t h i s p o i n t , t h i s i s pure conjecture  can be s a i d u n t i l a f i r m e r ground i s  and  constructed.  B I E  L I 0 G E A P  Alonso, William.. Location and Land Use. U n i v e r s i t y P r e s s , 1961.  EI  Cambridge, Mass.:  Ford, L.. R.. and D. R. F u l k e r s o n . " S o l v i n g the Assignment Management Science ( J u l y , 1956]. Gale, D..  Theory of Linear  Economic Models.  New York:  Harvard  Problem,"  McGraw-Hill, 1961.,  Haig, R. M. "Toward An Understanding of M e t r o p o l i s , " Quarterly 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 New York, 1963.  Journal  Guide.  K a i n , J . F . The Journey to Work as A- Determinant of Residential Location, Papers and Proceedings of the R e g i o n a l Science A s s o c i a t i o n , IX (1962). Kuhn, H. W. Logistics  "The Hungarian Method f o r the Assignment Problem," Quarterly, V o l . IV, No. 1 (1955).  L a n c a s t e r , K. J .  Mathematical  Economics.  New York:  Macmillan, 1968.,  . "The New Theory of Consumer Demand," Journal , Economy, Vol., 74, No. 4 ( J u l y , 19661. Marshall, Alfred. R i c a r d o , David.  Principles  of Economics., London:  On the Principles  of Political  of  Political  M a c m i l l a n .d Co., an  Economy and Taxation.  Smith, W.. F. Filtering and Neighborhood Change. f o r R e a l E s t a t e and Urban. Economics, 1265. 48  Naval  Berkeley, Calif.,:  1917. 1817..  Center  49  Von Thunen, Johann H_.  The Isolated  Wingo, Lowdon.. Transportation f o r t h e F u t u r e , 1961..  State..  1826.  and Urban Land.  Washington, D..C.,: Resources  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 c o m p e t i t i v e markets and resource a l l o c a t i o n . Gale gives an example of p r i c e e q u i l i b r i u m u s i n g the housing market.  The  b a s i c d i f f e r e n c e between the housing market d e s c r i b e d by Gale and the market and the one used by Smith and myself i s that s u p p l i e r s are used by Gale.  The use of assignment techniques i n the a n a l y s i s i s v a l i d .  The  housing s t o c k i s v a r i e d and households d i f f e r i n b o t h t h e i r a b i l i t y to b i d f o r v a r i o u s houses and t h e i r t a s t e s .  Because of long p r o d u c t i o n lags  the s o l u t i o n that flows from such a n a l y s i s can be used f o r f a i r l y long periods.  Changes i n the s t o c k of houses i s a l s o i n f l u e n c e d by the c o n d i -  t i o n and numbers of the present housing s t o c k . The market used by Gale has n i n d i v i d u a l s i n t e r e s t e d i n buying n houses.  A v a l u e m a t r i x a., i s c o n s t r u c t e d which shows the worth of each  house t o each household. each house.  A l s o , the s u p p l i e r s have s e t s a l e s p r i c e s p^ on  N a t u r a l l y a household would not be i n t e r e s t e d i n purchasing  unless i t s v a l u a t i o n of that house were h i g h e r than the s a l e s p r i c e .  Gale  uses programming theory to show that such a market r e s u l t s I n that i m p o s s i b l e dream of t h e " g r e a t e s t good f o r the g r e a t e s t 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 v a l u e maximizing arrangement of house s e t t l e ment . 51  52  From the theory of competitive markets i t can be s a i d that these markets r e s u l t i n a v a l u e maximizing  arrangement of the stock.  The  market that both Smith and I use are p u r e l y c o m p e t i t i v e i n nature s i n c e the housing stock and s i n c e the household b i d s 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 p o s s i b l e and i s c o n s i s t e n t w i t h the theory of pure competition.  See D. Gale, Theory of Linear 'Economic Models (Toronto: H i l l Co. I n c . , 1961).  McGraw  APPENDIX 2 AN ALGORITH TO SOLVE THE TRANSPORTATION PROBLEM  The a l g o r i t h m that i s used i n my housing model i s a v a r i a t i o n of a r o u t i n e designed by L. R. Ford and D. R. F u l k e r s o n Hithcock t r a n s p o r t a t i o n problem.  1  to s o l v e the  The problem can be s t a t e d  mathematically  as f o l l o w s : F i n d an m x <n a r r a y of numbers x = ( x „ ) , i = 1, 2, . . ., m and j = 1, 2,. . ., n that minimizes  T. c..x.. s u b j e c t ij  1 3  1 3  to the c o n s t r a i n t s X x. . - a. £ x. . = b . T-3 3 3 > 0 x. . — where a., b., c.. are non-negative i n t e g e r s and the sum of the i 3 1J v e c t o r a = the sum of the v e c t o r b.  I f m = n and a and b a r e  equal t o 1 t h i s becomes the optimal assignment problem. U s u a l l y the c.^ matrix i s a tableau of u n i t shipping costs from p o i n t i t o p o i n t j ; a. i s a v e c t o r that i n d i c a t e s the supply o f goods a t p o i n t i and b. r e f l e c t s the demand a t p o i n t j . The purpose of t h e a l g o r 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 p o i n t s so that t r a n s p o r t a t i o n costs a r e at a minimum.  L. R. Ford and D. R. Fulkerson, " S o l v i n g the Assignment Problem," Management Science ( J u l y , 1956). 53  54  The a l g o r i t h m i s m o d i f i e d to search f o r a maximum v a l u e merely by scanning the cost m a t r i x , f i n d i n g the maximum value i n the e n t i r e matrix and s u b t r a c t i n g t h i s v a l u e from each element of the m a t r i x .  Using t h i s aug-  mented m a t r i x w i t h i n the cost m i n i m i z i n g framework produces the s o l u t i o n f o r a cost maximum. A d i s c u s s i o n of the a l g o r i t h m r e q u i r e s d e t a i l and development that would be o u t s i d e the scope of t h i s paper.  A l l that r e a l l y need be  s a i d 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 w i d e l y used i n l i n e a r programming.  Because of the unique f e a t u r e o f t h i s  problem, such as i n t e g e r v a l u e s , square value m a t r i x and no surpluses or shortages f o r any of the row o r column e n t i r e s of the v a l u e m a t r i x , s e v e r a l s h o r t c u t s can be used to a r r i v e a t the s o l u t i o n f a s t e r . Proofs and a d e s c r i p t i o n of the method can be found i n Ford 2 and F u l k e r s o n , and i n H. W. Kuhn..  Kuhn, H. W. "The Hungarian Method f o r the Assignment Problem," Naval Logistics Quarterly, V o l . IV, No. 1 (1955).  A P P E N D I X  3  FORTRAN IV  Of  G COMPILER  MAIN  03-02-72  15:19:17  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 ,  0001 0002 000 3 000 4 0005 000 6 000 7 0008 0009 0010 0011 • 0012 ;  PAGE 0 0 0 1  U  U  l  h  U  D  ******************** * * * * THIS PROGRAM IS INPUT FOR THE ASSIGNMENT MODEL IT READS IN DATA ON HOUSEHOLD AND HOUSE C H A R A C T E R I S T I C S AND CREATES COMPLETELY IMAGINARY BEHAVIOURAL FUNCTIONS TO BE ;ED IN THE CREATION OF THE BIDFUNC TION MATRIX R E A L * 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) 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 **,6H DIMENSION K ( 1 0 0 , 1 0 0 ) , L ( 1 0 0 , 1 0 0 ) DIMENSION IA(IOO) ,IW(100),IS( 1 0 0 ) , I C ( 1 0 0 ) , J R ( 1 0 0 ) , K R ( ] 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 ) 202 FORMA T( ' 'HOUSEHOLDS * * < , 1 7 ( 1 4 , 2 X , 1 H * ) / ( 1 1 X , ' 14,2X, '*' ) ) 203 FORMA T ( 1 X , 2 A 6 , 1 7 ( A 1 , A 6 ) ) 204 FORMA T 2041 FORMA T ^ x ; A 6 * J ; ^ x : ; 6 n ' ' ' "' ^"''»'' »' 30 6 FORMA T ( 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 { 15, i , , » | ) ) C READ(5,701)M,N,AA,BB,CC,DD 701 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 ^ 1, M ) 123 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 = 1,M) 77 F O R M A T ( F 6 . 0 , F 3 . 0 ) WRITE(6,1112)M,N,AA,BB,CC,DD 1112 F O R M A T ! • « , « 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 ) * S P A ( J ) + I N C O M ( I ) * P E R S ( I ) / 1 0 . * 1 . 0 / B B BIDLOC(I,J)=AA/INCOM( I )*LOC(J)*INCOM(I )/3.0 K(I,J)=BIDSPA(I,J)+BIDLOC( I ,J) 534 CONTINUE 533 CONTINUE LA=1 900 W R I T E ( 6 , 4 0 0 ) ( (I ) , I = 1 , M ) 400 FORMAT! '1 ' , ' HOUSETYPE',15!5X,I2) ) 40 2 FORMAT! • ' , ' L O C A T I O N ' , 10X , F 3 . 2 , 14 ( 4 X , F 3 . 2 ) ) WRITE(6,4 0 1 ) ( S P A ( J ) , J = 1 , N ) 401 F O R M A T ! ' ' , ' 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 ) WRITE(6,405 ) 40 5 <MA T( ' i * * * * * •V- 'r- * * * * Jji * * * ******** * ** * * ** * * *• ) * * * 40 6 W R I T E ! 6 , 4 0 6 ) FORMAT! • , 'HOUSEHOLD' ) WRITE(6,407) 40 7 F O R M A T ! ' ' , ' INCOME PERSONS.') 1=1 IF(LA-2) 500,501,502 500 W R I T E ( 6 , 4 0 8 ) I 40 8 FORMA T( ,7X,12) I  1  2  x  , I 4  2 x  1 H , , , / , l l x  2 H  1 7 , i  i  1  H  0013 ' 0014 0015 0016 J017 0018 0019 " 0020 0021 0022 ' 00-2 3 ;0024 0025 0026 002 7 •0028 0029 -00 30 0031 'OO 32 0033 .00 34 0035 0036 ,0037 ,0038 00 39 0040 , 041 1-1042 3043 )044  :  1  ;gRTRAN 0045 0046 0047 0048 049 •0O5O 0051 00 52 0053 00 5 4 00 55 0056 00 57 0O58 0059 0060 00 61 0062 0063 0064 0065 00 66 0067 0068  0069 .0070 .0071 )0 7 2 0073 00 7 4 0075 0076 00 7 7 00 7 8 0079 0080 10081 0082 00 8 3 00 8 4 00 8 5 0086 00 8 7 00 88 0089 "0090 *00 91 0092 0093 00 9 4 0095 00 9 6  MAIN  IV G COMPILER  40 9  501  502 740  410  700 102 103 105  03-02-72  15:19:17  W R I T E ( 6 , 4 0 9 ) I N C O M ( I ) , P E R S ( I ) , ( B I D S P A ( I ,,JJ = 1 , N ) 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 4 1 0 GO TO 5 0 0 WRITE(6,408) 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 ,J=1,N) 1=1+1 I F ( I . G T . M ) GO TO 4 1 0 GO TO 5 0 1 WRITE(6,408) I W R I T E ( 6 , 7 4 0 ) I NC OM ( I ) , PER S ( I ) , ( K ( I1,N) , 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 5 0 2 LA =LA +1 I F ( L A . G T . 3 ) GO TO 7 0 0 GO TO 9 0 0 DO 1 0 2 1=1,M IA ( I ) = 1 DO 1 0 3 J = 1 , N J B ( J ) =J MN=M*N  DATA P R I N T OUT WRITE(6,200) FUNCTION MATRIX') 2 0 0 FORMA T( WRITE(6,201) " , 9 X , 'HOUSES' , 6 X , A 2 , 1 4 ( 5 X , A 2 ) ) 2 0 1 FORMA T{ WRITE(6,202)(JB(J),J=1,N) DO 1 7 0 J J = 1,2 I F (N .GT. 1 7 ) GO TO 1 5 1 WRITE ( 6 , 2 0 3 ) D , D , ( E , D , I I 1,N) I F ( J J .EQ. 2 ) GO TO 1 6 0 GO TO 1 7 0 1, 1 7 ) WRITE ( 6 , 2 0 3 ) D , D , ( E , D , I I 151 CONTINUE 170 DO 30 6 0 I = 1 , M 160 W R I T E ( 6 , 2 0 4 ) I A ( I ) , { K ( I , J ) , =J 1= ,N) I F (N .GT. 1 7 ) GO TO 1 5 0 ( C, I I 1,N) W R I T E ( 6 , 2 0 4 1 ) AGO TO 3 0 6 0 150 WRITE ( 6 , 2 0 4 1 ) A , B , ( C , I I = 1,17) 30 60 CONTINUE DO 1 7 1 J J = 1,2 I F (N .GT. 1 7 ) GO TO 1 5 3 W R I T E ( 6 , 2 0 3 ) D , D , ( E , D , I I = 1,N) I F ( J J .EQ.2 ) GO TO 1 6 2 GO TO 1 7 1 153 W R I T E ( 6 , 2 0 3 ) D , D , ( E , D , I I.= 1 , 1 7 ) 171 CONTINUE C C O N V E R S I O N TO MAX P R O B L E M 1 6 2 M I N=K ( 1 , 1 ) DO 6 0 0 1=1,M 1  PAGE  0002  FORTRAN 0097 %09 8 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0111 0112 0113 0114 0115 0116  0117 0118 0119 ')l20 0121 :0122 012 3 :0124 :012 5 0126 0127 0128 0129 0130 0131 0132 0133 0134 3135  3136 0137 0138 3139 0140 3141 0142 0143 0144 |145 1146 )147 )148  IV G COMPILER  MAIN  03-02-72  15:19:17  DO 6 4 0 J = 1 , N I F ( K ( I , J ) .GT.MIN) M I N = K ( I , J ) 640 CONTINUE 600 CONTINUE W R I T E ( 6 , 7 4 1 ) MIN 7 4 1 FORMA T( • «, 1 7 ) DO 6 0 2 1=1,M DO 6 0 3 J = 1 , N K ( I , J ) = ( K ( I , J ) - M I N ) * ( '1) 603 CONTINUE 602 CONTINUE WRITE(6,702)((K(I • J),J=l,N),I=1,M) 7 0 2 FORMA T( • « ,15 1 5 ) DO 6 0 8 J = 1 , N 60 8 J B ( J ) = 1 DO 6 0 7 I = 1 , M 6 0 7 I A ( I ) =1 WRITE(6,1111)(JB(J1,J=1,N) W R I T E ( 6 , 1 1 1 1 ) { I A ( I ) , 1 = 1,M) 1 1 1 1 FORMAT! ' ' , 1 5 1 5 ) C T H I S C O M P L E T E S THE P R O B E L M C O N V E R S I O N TO MAXIMUM C * * * * * * * * * * * * * * * * * >; * * * > C GETTING STARTED 705 DO 1 1=1 ,M IS { I ) = I A ( I ) J I G = K { I ,1 ) DO 2 J = 1 , N JS(J)=JB( J) L (I ,J )=-l IF ( J I G - K ( I , J ) ) 2 , 2 , 3 3 JIG=KU,J) 2 CONTINUE IW(I)=-JIG DO 4 J = l ,N I F ( J I G - 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 JW(J)=0 GO TO 6 7 CONTINUE JIG=K(1 ,J)+IW(1) DO 11 1=1 ,M KR(I)=K(I,J)+IW(I) IF ( J I G - K R ( I ) ) 11,11,10 10 J IG =K R( I ) II CONTINUE J W(J)=-JIG DO 4 6 I =1 ,M IF (JIG-KR(I))46,95,46 95 L (I , J )=0 46 CONTINUE  p A G  E  0003  6 C C "^.0150 151 (0152 ! 0153  0156 ! 0157 \ 0158 [0159 i 0160 0161 10162 0163  14 C 15  C 16  13 12 C C  0164 0165 0166 0167 - 0168 10169 ' 3170 0171 0172 0173 - 0174  c c 57 19  21 20 36 C  0175 I 0176  0177 90178 -0179 0180 0181 I 0182 '0183 ; 0184 - 01 85  23 25 26  24 22 C  v 018 6  i' 01 8 7 01 88  29  i 0189  -ij 0 1 9 0 ' 0191 i 0192 ? 0193  31 32 33  03-02~72  MAIN  .OUTRAN I V G C O M P I L E R CONTINUE  * *  * *  * *  D E T E R M I N A T I O N OF I N I T I A L A L L O C A T I O N S DO 1 2 1=1,M DO 1 3 J = l , N IF { L ( I , J ) ) 1 3 , 1 4 , 1 3 IF ( I S ( I ) - J S ( J ) ) 1 6 , 1 5 , 1 5 J S L E S S THAN I S L ( I , J )=JS ( J ) IS(I ) = I S ( I ) - J S ( J ) J S ( J ) =0 GO TO 1 3 I S L E S S THAN J S L (I ,J)=IS (I ) JS(J)=JS(J)-ISU) IS(I)=0 CONTINUE CONTINUE GO TO 5 1  * * * * *  ITERATIVE  PROCEDURE  * *  15:19:17  * *  L A B E L I N G PROCEDURE DO 1 9 J = l , N JC(J)=-l IR ( J ) = -1 DO 2 0 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 L A B E L ROWS DO 2 2 1=1,M I F ( I C ( I ) ) 2 2 ,22 ,23 DO 2 4 J = l ,N IF (L ( I , J ) 1 2 4 , 2 5 , 2 5 IF ( I R ( J ) )26 ,24,24 IR(J)=IC (I) JC ( J ) = 1 IND=1 IF ( J S ( J ) ) 2 4 , 2 4 , 2 7 CONTINUE CONTINUE L A B E L COLUMNS DO 2 8 J = l ,N IF ( I R ( J ) )28,28,29 DO 30 1=1 ,M I F ( L ( I , J) ) 3 0 , 3 0 , 3 1 I F ( I C ( I )) 3 2 , 3 0 , 3 0 J R ( I ) =J I F ( L ( I , J) - I R ( J ) ) 3 3 , 3 4 , 3 4 IC ( I )=L ( ,I J )  PAGE 0 0 0 4  * * *  sjc  ?p  ?js  i£  ^js  J^C  * * * * * *  *  * * *  FORTRAN P194 01u l 9 5 0196 0197 0198 0199 0200  IV G COMPILER  34 30 28  02O2 02O3 0204 0205 0206 0207  0208 0209  0210 0211 0212 0213 0214 0215 0216 $217 0218 1  0219  1 0221 0220  0223 1 02 22 S 0224 0225 022 6 0227 .0228 0229 0230 0231 0232 0233 0234 0235 0236 0237 0238 0239 2 40 0 2 41 0242 3243  c 27 38 37 39  42 18  50 C  51  80 C C  35  60 61 62 59 590 63 72 64 58 5 80  f  66 70 91  03-02-72  IMD=-1 GO TO 3 0 I C ( I 1=IR(J 1 IND=-1 CONTINUE CONTINUE IF ( I N D ) 3 6 ,35 ,36  C  0201  MAIN  ^<  =jc  ifc  >',z >',< i|<  it<  if.  &  if  if  if  if  if  if  if  if  BREAKTHROUGH PROCEDURE IF (JS(Jl-IR(J)137,38,38 L H=IR(J) GO TO 3 9 LH=JS ( J ) J S ( J )=JS( J l - L H L(I,J)=L(I,J)+LH 11=1 I F ( J R ( I I ) ) 18 , 5 0 , 1 8 J1=JR ( I I ) L ( 11 , J 1 ) =L ( I I , J D - L H I1=JC ( J l ) L(11,J1)=L(I 1,Jll+LH GO TO 4 2 IS(I1) = IS(I1)-LH ARE A L L SHORTAGES S A T I S F I E D DO 8 0 J = 1 , N IF (JS(J1180,80,57 CONTINUE GO TO 4 4 *  *  *  *  *  *  *  JJ5  *  3jC  *  *  *  *  *  jjc  *  NONBREAKTHROUGH PROCEDURE LK=99999 DO 5 9 0 1 = 1 , M IF ( I C ( I ) 1 5 9 0 , 5 9 0 , 6 0 DO 5 9 J = l ,N I F ( I R ( J ) 1 6 1 , 5 9 ,59 LY=K ( I , J 1 +IW ( I 1 + J W ( J 1 IF (LY-LK)62,59,59 LK=LY CONTINUE CONTINUE DO 5 8 0 1 = 1 , M IF ( I C ( I ) 1 5 8 0 , 5 8 0 , 6 3 DO 5 8 J = 1 , N I F ( I R ( J 1 172 ,58 ,58 IF (K(I,J)+IW(I)+JW(J)-LK)58,64,58 L(I,J)=0 CONTINUE CONTINUE DO 65 J = 1 , N I F ( I R ( J ) 165 , 6 6 , 6 6 J W ( J 1 = J W ( J 1 +L K DO 9 0 1=1 ,M IF (L ( I , J 1 1 9 0 , 7 0 , 9 0 IF ( I C ( I ) 1 9 1 , 9 0 , 9 0 L ( I , J )=-l  * * *  15:19:17  PAGE  0005  FORTRAN  | 0244 0245 i: 0246 ~H 0247 | ^48 I 0249 ; 02 50  IV  G  90  CONTINUE  65  CONTINUE )  CONTINUE -v*f  TO -sV  J  >i<  ^  >yV  DO  205  KR(J  206  0257  *  J=l,N  )=0  DO  30 5  IF  ( L ( I , J  I=1,M  LY =L ( I , J  ) )2  1 0 , 3 0 5 , 2 0 6  )*K{I,J)  LC=LC+LY  0258  KR ( J ) =KR ( J ) + L Y  0259  GO  TO  L(I  , J ) =0  305  026O  210  0261  30 5  CONTINUE  0262  205  CONTINUE  0263  DO 611  0265  611  IA ( I DO  612  0267  1=1,M  )=1  612  J=1,N  J B (J ) = J WRITE ( 6 ,207 )  207  0269 0270  >;<:  LC=0  0254  0268  j(e #  PROCEDURE  TERMINATION  44  0255  0266  )-LK  5 7  N  >fV  0253  0264  ) 6 7 , 6 9 , 6 9  67  0252  02 5 6  (IC(I  IW ( I ) =IW ( I  C 0251  67  IF  GO  FORMAT  (1H1 ,30X , 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 208  J271  F O R M A T( 1 H  , 1 2 H TO TA L  COST=  ,110)  W R I T E ( 6 , 2 0 1 )  6272  W R I T E ( 6 , 2 0 2 ) ( J B ( J ) , J = 1 , N )  0273  DO  172  J J  0274  IF  (N  . G T .  0275  WRITE  0276  IF  ( J J  GO  TO  0277  =  1,2 17)  ( 6 , 2 0 3 ) . E Q .  2)  152 G O TO D,D, ( E , D , I I 161 G O TO  =  1,N)  172  0278  1 52  WRITE  0279  172  CONTINUE  0280  161  D,D, ( E , D , I  ( 6 , 2 0 3 )  I  1,  17)  0285  1 55  DO 3 0 7 I=1,M WRITE(6,204) I A ( I ) , ( L ( I ), , JJ = I , N : IF (N . G T . 155 17) G O TO WRI TE ( 6 , 2 0 4 1 ) A , B ,, (( C C ,, II II == 1,N) GO TO 3 0 7 A , B , ( C , I I = 1,17) WRITE ( 6 , 2 0 4 1 )  0286  307  CONTINUE  0281 0282 0283 0284  0287  DO  173  J J  0288  IF  (N  . G T .  0289  WRITE  0290  IF  ( J J  GO  TO  0291  =  1,2 17)  ( 6 , 2 0 3 ) . E Q .  2)  G O TO D,D, ( E G O TO  WRITE  0293  173  CONTINUE  1 63  W R I T E ( 6 , 3 0 6 ) ( K R ( J )  1  2 9 6  ( 6 , 2 0 3 )  WRITE(6,959)  0295 959  »Dr I I  1,N)  163  D,D, ( E , D , I 1  1 54  F  154  173  0292 0294  1 5 : 1 9 : 1 7  1=1,M  DO 69  C  0 3 - 0 2 - 7 2  MAIN  COMPILER  FORMAT* 1H1 )  ,  J=1,N)  =  1,17)  /)  PAGE 0 0 0 6  FORTRAN  IV G C O M P I L E R  0297 0298 0299 0300 0 301 0 30 2 0 30 3 0304  743 744 745 746  0305 0 30 6 0307 TOTAL  747  MEMORY  COMPILE  M A I N  15:19:17  DO 7 4 7 J = 1 , N DO 7 4 3 1=1,M I F ( L ( I , J ) . E Q . l ) GO TO 7 4 4 CONTINUE WRITE(6,745) I , J FORMA T( ', 'HOUSEHOLD' , 1 2 , ' L O C A T E D I N H O U S E ' , 1 2 ) W R I T E ( 6 , 7 4 6 ) I N C O M ( I ) , P E R S ( I ),S P A ( J ) , L O C ( J ) FORMAT( ,'INCOME , F 6 . 0 , 2 X , ' F A M SI Z E= * , F 3 . 0 , 5 X , ' •-2 , ' L O C A T I O N COEF , F 3 . 2 ) CONTINUE STOP END  PAGE  0 0 0 7  1  1  REQUIREMENTS  TIME =  03-02-72  17.4  1  017EC2 BYTES SECONDS  S P A C E  C0EF=',F3  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 SPACE LOCATION  1 .48 .48  2 .49 .49  3 .50 .50  4 .51 .51  5 .52 .52  6 .53 .53  7 .54 .54  8 .55 .55  9 .56 .56  10 .57 .57  11 .58 .58  .12 .59 .59  13 .60 .60  14 .61 .61  15 .62 .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  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  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 e r r o r t h i s table may not be the exact sum of Tables 1 and 2.  Ln 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  107  553  4528  4507  4433  4393  4353  3227  3373  3520  2383  2013  1643  1000  VO  

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