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Estimates of wasteful commuting in a sample of Canadian cities: a test of the monocentric model Pratt, Reagan A. 1993

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ESTIMATES OF WASTEFUL COMMUTING IN A SAMPLE OF CANADIAN CITIES:A TEST OF THE MONOCENTRIC MODELbyREAGAN ADAM PRATTHonours B.A., The University of Guelph, 1986A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE (BUSINESS ADMINISTRATION)inTHE FACULTY OF GRADUATE STUDIES(Department of Commerce)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1993© Reagan Adam Pratt, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureDepartment of ^CommerceThe University of British ColumbiaVancouver, CanadaDate^April 19. 1993DE-6 (2/88)ABSTRACTThis thesis is based on an article by Bruce Hamilton published inthe Journal of Political Economy 90(5) in 1982, titled "WastefulCommuting". The analysis compares "optimal" and "observed"commuting behaviour in 23 Canadian cities in order to provide atest of the monocentric model's ability to predict commutingbehaviour. Monocentric models are widely used in urban economicsdue to their simple structure but any model purporting to explainresidential and job choice location should be able to explainobserved commuting behaviour. In order to operationalize the modelit was necessary to estimate employment and population densitygradients for 23 Canadian cities using 1981 census data. Densitygradients were estimated using a two-point estimation techniquepioneered by Edwin Mills. The employment gradient estimatespresented in Chapter 4 are the only existing Canadian estimates fora large set of cities. The density gradient estimates were used tocalculate the minimum average commuting distance in each city. Theminimum average commute was compared with observed commutingbehaviour. Data on observed commuting was obtained from a 1977household survey. The results indicate that observed commuting is,on average, eight times the minimum necessitated by the separationof homes and jobs. Randomly assigning residents to homes and jobsexplains observed commuting better than the monocentric model.Like Hamilton's results, the results of this thesis draw intoquestion the validity of the basic monocentric model.Alspri.racfList of TablesTABLE OF CONTENTS List of Figures ^Acknowledgement CHAPTER ONE - INTRODUCTION ^CHAPTER TWO - A REVIEW OF THE MONOCENTRIC MODEL . .vivii142.1 Transportation Costs, Land Values & Urban Economics . 52.2 The Basic Monocentric Model ^ 72.2.1 Derivation of the Locational Equilibrium Condition 92.2.2 The Supply of Housing Services ^ 122.2.3 The Demand for Housing Services 142.2.4 Solution to the Model ^ 152.3 The Negative Exponential Population Density Gradient 192.4 A Digression on Employment Location ^ 202.5 Criticisms of the Monocentric Model 242.5.1 Assumptions of the Monocentric Model ^ 242.5.2 Predictions of the Monocentric Model 262.6 Summary ^ 28CHAPTER THREE - HAMILTON'S MODEL FOR ESTIMATINGWASTEFUL COMMUTING ^ 313.1 Hamilton's Model For Estimating Wasteful Commuting 313.1.1 The Theoretical Basis of Hamilton's Model^. . . 323.1.2 Calculating the Optimal Average Commute 363.1.3 Calculating the Average Random Commute 423.2 Two-Point Estimates of Employmentand Population Density Gradients ^ 443.2.1 The Mills Estimation Technique 483.2.2 The Macauley Estimation Technique 503.2.3 The Edmonston Estimation Technique ^ 523.2.4 Two-Point Versus OLS Density Gradient Estimation 553.3 Data Sources ^ 583.3.1 The Urban Concerns Survey (UCS) 653.3.2 Vehicle Survey (VS) ^ 673.3.3 Estimating the Zeros of Non-Linear Equations ^• • 683.4 Summary ^ 69CHAPTER FOUR - DENSITY GRADIENT ESTIMATES ^ 704.1 Previous Two-Point Density Gradient Estimates . . . 704.2 Population Gradients: New Canadian Estimates ^. . . 754.2.1 Macauley Estimates ^ 764.2.2 Edmonston Estimates 894.2.3 Mills Estimates 954.3 Employment Gradients: Canadian Estimates ^ 974.3.1 Macauley Estimates ^ 984.3.2 Edmonston Estimates 1054.3.3 Mills Estimates 1104.4^Summary  ^111CHAPTER 5 - ESTIMATES OF WASTEFUL COMMUTING IN 23CANADIAN CMAs  ^1135.1 Measuring Waste Using Macauley GradientEstimates: Base Case  ^1155.1.1 Assessing the Monocentric Model  ^1225.1.2 Comparing Canadian and American Cities  ^1295.2 Adjusting for the Areal Extent of the CBD  ^1345.3 Sensitivity to the Choice of Urban Boundary^     ^1415.4 Using Edmonston Density Gradients^ 1455.5 Summary  ^149CHAPTER 6 - SUMMARY AND DIRECTIONS FOR FURTHERRESEARCH  ^1526.1 Density Gradient Estimates  ^1526.2 Wasteful Commuting - A Critique of theMonocentric Model  ^1546.3 International Comparison  ^1556.4 Criticisms of Hamilton's Methodology  ^1566.5 Directions for Future Research  ^157BIBLIOGRAPHY  ^161APPENDIX 1 - INTEGRATION OF EQUATIONS 3.8 AND 3.15 ^166APPENDIX 2 - COMPARISON OF CMA EMPLOYMENT AND CENTRALCITY EMPLOYMENT BASED ON DATA FROM THE 1981 CENSUS ^169APPENDIX 3 - ESTIMATES OF WASTEFUL COMMUTING DERIVEDUSING EDMONSTON DENSITY GRADIENT ESTIMATES ^ 171LIST OF TABLES3.1 - Radius Calculation Based on Eight City Shapes . . ^543.2 - Population and Employment Data Used to EsitimateDensity Gradients  ^623.3 - Estimation of CMA Radius and the Parameter Phi . . ^643.4 - Summary of Observed Commuting Data  664.1 - Average Density Gradients, Mills and Macauley . ^714.2 - Average Values for Gamma and D, EdmonstonGoldberg and Mercer  ^754.3 - 1981 Population Density Gradient ParameterEstimates  ^774.4 - Cities With Small Suburban Population and LargeValues For D,  ^874.5 - 1981 Employment Density Gradient ParameterEstimates  ^994.6 - Summary of Macauley and Edmonston GradientEstimates  ^1125.1 - Optimal and Actual Commute Characteristics Using1981 Macauley Gradient Estimates^ 1165.2 - Alternative Measures of Wasteful Commuting DerivedFrom Estimates in Table 5.1  ^1255.3 - Summary Statistics Comparing Canadian and AmericanEstimates of Urban Structure and CommutingBehaviour  ^1325.4 - Alternative Measures of Wasteful Commuting Assuming^a CBD Radius of 1.0 Kilometres   1365.5 - Regression Estimates of the Relationship Between CMAPopulation and Wasteful Commuting ^ 1405.6 - Alternative Measures of Wasteful Commuting AssumingDifferent CBD Boundaries^ 1445.7 - Summary Statistics Comparing Urban Structure andCommuting Behaviour Using Edmonston Versus MacauleyDensity Gradient Estimates 148A3.1 - Optimal and Actual Commute Characteristics Using1981 Edmonston Gradient Estimates  ^172A3.2 - Alternative Measures of Wasteful Commuting DerivedFrom Estimates in Table A3.1  ^173LIST OF FIGURES3.1 - City With Dispersed Employment  ^353.2 - Location of CMAs Ranked By 1981 Population . • • •^603.3 - CMA Abbreviations  ^614.1 - 1981 Population Gradient Estimates MacauleyEstimation Technique  ^784.2 - Gradient Slope Rank 1981, Macauley EstimationTechnique^ 814.3 - Central Population Density Estimate 1981, MacauleyEstimation Technique  ^844.4 - Central Density Rank 1981, Macauley EstimationTechnique^ 854.5 - Measured Versus Calculated CMA Radius ^ 884.6 - Population Gradient Estimates 1981, EdmonstonEstimation Technique  ^904.7 - Comparison of Edmonston and Macauley PopulationGradient Estimate Rankings  ^944.8 - Central Population Density Estimate 1981, EdmonstonEstimation Technique  ^964.9 - Comparison of Edmonston and Macauley CentralPopulation Gradient Density Rankings  ^1004.10 - Employment Gradient Estimates 1981, MacauleyEstimation Technique  ^1024.11 - Central Employment Density Estimate 1981, MacauleyEstimation Technique  ^1044.12 - Employment Gradient Estimates 1981, EdmonstonEstimation Technique  ^1064.13 - Comparison of Edmonston and Macauley EmploymentGradient Estimate Rankings  ^1084.14 - Comparison of Edmonston and Macauley CentralEmployment Gradient Density Rankings  ^1095.1 - Ranking of Average Household and Job Distance Fromthe CBD Based on 1981 Macauley Density Gradient^Estimates   1195.2 - Ranking of Optimal and Observed Commute 1981Macauley Density Gradient Estimates   1215.3 - Optimum Versus Observed Commute 1981 MacauleyDensity Gradient Estimates  ^1245.4 - Average One Way Wasteful Commute Using 1981Macauley Density Gradient Estimates^ 1275.5 - Optimal As A Share of Observed Commute 1981Macauley Density Gradient Estimates 1285.6 - Observed Versus Random Commute 1981 MacauleyDensity Gradient Estimates  ^1305.7 - Ranking of Wasteful Commuting (C/D) Versus CMAPopulation  ^1395.8 - Measured Versus Calculated CMA Radius ^1435.9 - Optimal As A Share of Observed Commute AllowingThe Urban Boundary to Vary  ^146- vi -ACKNOWLEDGEMENTMy old man worked 50 years in a steel mill so his children couldhave the opportunity to do what would make them happy. My oldman is also one of the most naturally curious people I have everknown. Thanks Dad - for instilling in me your natural curiosityand at least a fragment of your perseverance. I would be remissif I did not thank Mike Goldberg who has been part mentor andpart friend during my time at the University of BritishColumbia. Although I know it is arrogant to hope for, when Igrow up I want to be just like Michael. I would also like tothank my thesis advisor Bob Helsley, a naturally giftededucator, brilliant economist and another who helped to feed mycuriosity. Anyone with a compilation of Bob Dylan's completelyrics on their desk must truly be a kindred spirit. Finally Iwould like to express my appreciation to the members of mycommittee for their interest in this project.CHAPTER ONEINTRODUCTIONThe monocentric urban model is virtually synonymous with the urbaneconomics sub-discipline and particularly with the paradigm ofinquiry referred to as the "new urban economics". Since the workof Muth [45] and Mills [43] in the 1960s monocentric models havebeen widely employed in urban economic analysis.The monocentric model yields a number of testable predictions aboutthe distribution of phenomena, particularly housing and population,in urban areas. Many of the model's predictions have beenempirically tested for numerous sets of U.S. cities. There hasbeen a dearth of similar testing of the model in countries otherthan the United States. One area where the monocentric model hasfailed is in predicting observed commuting behaviour. Because thisis a fundamental shortcoming in a spatial model that purports toexplain the location of households and employment based on thetrade-off between housing prices and commuting costs it isconsidered an area worthy of further analysis.The purpose of this thesis is to test a "strong form" of themonocentric model. The test focuses on the monocentric model'sability to predict observed commuting behaviour. The test wasoriginally developed and employed by Hamilton [26] for a sample of14 U.S. cities. Specifically, this thesis attempts to answer twoquestions:1. Given the existing distribution of homes and jobs in a- 1 -sample of Canadian cities is the monocentric model ableto accurately predict observed commuting behaviour?2.^Is the model's performance significantly different inCanada than in the United States? Or put differently, inthe aggregate, do Canadian and American commuters behavedifferently after controlling for the existing urbanstructure?In order to operationalize the model used to estimate expectedcommuting it is first necessary to estimate population andemployment density gradients for each city included in the sampleof Canadian cities. Although this is not specifically listed amongthe goals of the thesis, the population and employment gradientestimates should be of interest to urban researchers.^Theemployment gradient estimates are the first for any set of Canadiancities and the population gradient estimates are for the largestsample of Canadian cities covered by any research to date.Chapter 2 reviews the importance of the monocentric model withinthe urban economics sub-discipline. A simple mathematical versionof the model based on Hamilton and Mills [42] is presented in orderto highlight the predictions of the monocentric model and the keyrole played by commuting.^The spatial location equilibriumcondition, one of the fundamental results in urban economics, isalso presented. Finally criticisms of the monocentric model arereviewed.Chapter 3 reviews a model that can be used to estimate aggregatecommuting behaviour in monocentric cities. Originally, the modelwas developed and employed by Hamilton [26] for a sample of citiesin the United States. The logic of the model is straightforwdid.- 2 -The average distance of homes and the average distance of jobs,from the CBD is estimated. The difference between the two isreferred to as the optimum average commute. The optimum averagecommuting distance is then compared to observed commuting behaviourin a sample of cities.Chapter 4 presents estimates of population and employment densitygradients for a sample of 23 Canadian cities. The parameterestimates from the density gradients are necessary to estimateoptimal commuting distance in each city. Considerable attention isgiven to the estimation technique and discussion of the gradientestimates.Chapter 5 employs the model developed in Chapter 3 and thegradients estimated in Chapter 4, to test the ability of themonocentric model to predict observed commuting behaviour in 23Canadian cities. The results for Canada are compared withHamilton's results [26] for his sample of U.S. cities.Chapter 6 concludes the study and provides suggestions for furtherresearch into the determinants of commuting behaviour.CHAPTER TWOA REVIEW OF THE MONOCENTRIC MODELThis chapter reviews the monocentric city model. The importance ofthe monocentric model in urban economics and the evolution of themonocentric model is discussed. Monocentric models have beenwidely employed in urban economics due largely to the simplestructure of the model and the associated mathematical tractability[13]. There are at least three essential but unrealisticassumptions common to most versions of the monocentric model:1. There is a predetermined centre to which all householdscommute and to which all products are shipped;2. Individuals are homogeneous in preferences and the urbanarea is homogeneous in land and neighbourhoodcharacteristics; and3. Cities are instantaneously developed and infinitelymalleable (sometimes referred to as the putty-puttyassumption).The first two assumptions imply that distance to the centralbusiness district (CBD) fully characterizes the desirability of anylocation within the city: direction is irrelevant. Households areindifferent to a particular home or job except insofar as thetransportation (commuting) costs associated with particular sitesdiffer.' The third assumption implies that the spatial economy ischaracterized by a series of long run equilibria with capitalThe assumption of indifference among jobs is particularlyunrealistic in monocentric models that attempt to incorporatemultiple income groups. In reality an important factor indetermining income differentials is probably employment income.The monocentric model assumes that all jobs are equally desirableso income differentials must arise due to differences in initialendowments or differences in non-employment income.-4-investment adjusted every period.This chapter is divided into six sections. Section 2.1 discussesthe central role played by transportation costs in urban economicanalysis. Section 2.2 presents a simple version of the monocentricmodel. The model presentation relies heavily on Mills and Hamilton[42], (particularly Mills and Hamilton's Appendix A). Section 2.3briefly touches upon the issue of functional form and theappropriateness of the negative exponential population densityfunction. Section 2.4 examines employment location theory in thecontext of the simple urban model. The next section discusses somegeneral criticisms of the monocentric model, including a criticaltest of the model developed by Hamilton [26]. Hamilton'smethodology is subsequently built upon in the remaining chapters ofthis thesis. Finally section 2.6 concludes the chapter.2.1 Transportation Costs, Land Values & Urban EconomicsTransportation costs have long been recognized as a crucialdeterminant of both the formation of cities and the distribution ofeconomic activity within cities.2 In the absence of transportationcosts geographic proximity is not necessary in order to preserveeconomic linkages. Once transportation is costly all interactioninvolves the cost of overcoming distance.2 Transportation costs and scale economies in production (and toa lesser degree in consumption), are sufficient conditions for theexistence of cities.5Traditional neoclassical economic theory is presented in aspaceless economy which is, quite clearly, artificial becausespatial relationships and land use patterns affect every economicactivity. In a spatial economy the demand for land arises from theconsumption and production activity of individuals. Individualactivities are brought together by spatial agglomeration economieswhich arise from interaction and it is this interaction thatinvolves the cost of overcoming distance.Urban economics developed as an explicit attempt to incorporatespace into consumer and producer theory and this basic concern forthe geographic distribution of phenomena (particularly population)in cities still defines urban economics [57]. The majordifficulty inherent to incorporating space into neoclassical theorylies in the indivisibility of the land-location consumptiondecision. Locational fixity suggests that dwelling units differgreatly in their accessibility to production and consumptionactivities making it difficult to isolate the consumption of a goodsuch as housing from the consumption of "accessibility" [49].Many important insights into the operation of housing markets werederived during the 1960s from the realization that employmentaccessibility and housing are jointly purchased. However interestin the nexus between transportation costs and the value of landstretches back into the 19th century and the work of Johan vonThUnen. Von Thfinen postulated an agricultural land market wherethe price of land was determined by productivity (fertility) and6the distance of land to the nearest market town. 3 William Alonso[3] built on the work of von Thanen and another early twentiethcentury economist, Robert Haig. Alonso's work made explicit thecentral role played by transportation costs within cities throughthe development of bid-rent functions. In equilibrium, sitesdiffer by the transportation costs associated with distance fromthe CBD. The sum of land rent plus transportation costs must be a"constant" throughout the city for any particular land use, such ashousing, in order to establish long run equilibrium.From the work of Alonso urban economic analysis quickly evolvedinto spatial models focusing on equilibria and social optima. Ofcourse, the purpose of such models is to abstract from realityusing a few basic theoretical concepts in order to explain a largenumber of observed phenomena. The most important of the spatialmodels, the monocentric model, has become synonymous with urbaneconomics. But as Wheaton cautioned, "If the monocentric modelscontain a lesson it is that spatial relationships substantiallycomplicate microeconomics' simple analysis" [57].2.2 The Basic Monocentric ModelAccording to Wheaton [57] the family of monocentric modelsrepresents a distinct branch of microeconomics. While Alonso [3]is generally credited with incorporating bid-rent functions into3 Throughout the 19th century and for the early part of the 20thcentury it was geographers rather than economists who were in thevanguard of locational and spatial research.-7-urban analysis, Mills [43] and Muth [45] are often cited among thefirst to have formalized mathematical versions of the monocentricmodel. In the Alonso model households have a direct preference forland which determines residential density. In the model developedby Muth and Mills consumers do not have a direct preference for thegood land. Instead, residential density is determined on thesupply side by the ability to substitute capital for land in theproduction of "housing services".Monocentric models differ in sophistication and can incorporate anumber of important complications that are either internal orexternal to the city. Internal complications include: Incorporating dispersed employment as well as dispersedhousing;4 The addition of public goods; Allowing for transportation congestion; Including externalities such as race, pollution or crime;and Incorporating multiple (usually two) income groups.There are basically two variations in the outer structure of thespatial economy in which a city operates that have been examinedwithin the framework of the monocentric model: Open versus closed cities - open cities allow migrationinto and out of the city, which implies that the level ofutility is exogenously determined while closed cities donot allow migration implying that utility is determinedwithin the city; and4 Employment decentralization is discussed in Section 2.4 of thischapter and again, briefly, in Chapter 3.-8-• Distribution of land rents - landlords can be assumed toreside in the city or it can be assumed that all rentsare paid to absentee landlords. The former results inmultiple income groups.It is beyond the scope of this chapter to review each incarnationof the monocentric model as listed above. Only a simple version ofthe model is presented below, focusing on residential locationtheory. The model presented is, however, sufficient to highlightthe central role that commuting plays in the model. And it is theinability of the monocentric model to predict commuting behaviourthat was central both to Hamilton's [26] criticism of themonocentric model and to this thesis.2.2.1 Derivation of the Locational Equilibrium ConditionBegin by assuming that all individual households are utilitymaximizers and that there are only two normal goods: housing (h)and a composite non-housing good (g).^Household maximize:U s u(h, g)subject to the budget constraint:Pgg(x) + Ph (x) h (x) + tx ywhere:• y m household income;• t m is the cost of a round trip kilometre;• x m distance from the CBD; Ph '74 the price of one unit of the housing good;- 9(2 .1)(2.2) ^Pg Es.-- the price of one unit of the composite good.In equilibrium, each household will maximize utility at a pointwhere the slope of the indifference curve is tangent to the slopeof the budget constraint: 5Ah(x)^Pg A g(x)^Ph(x) (2.3)Now, in order to derive an important result, imagine the impact ofa small change in location (Ax) on the budget constraint given inequation 2.2. To maintain equilibrium the following must obtain:PgAg(x) + APh (x) h(x) + Ph (x)Ah(x) + tAx - 0 .^(2.4)Rearranging equation 2.3 yields:PgAg(x) + Ph (x) Ah(x) - 0.^ (2.5)Subtracting equation 2.5 from both sides of 2.4 and rearranginggives:APh(x)h(x)^- t.Ax (2.6)An indifference curve maps out all the possible combinationsof (h) and (g) that will yield a given level of utility for ahousehold. Households are indifferent among various combinationsof (h) and (g) that yield the same level of satisfaction.- 10-Equation 2.6 is an important result, often referred to as thelocational equilibrium condition. The locational equilibriumcondition implies that as distance to the CBD increases, thereduction in housing expenditure (the numerator on the LHS ofequation 2.6) is exactly offset by an increase in commuting costs,-t (the RHS of equation 2.6). The importance of commutingbehaviour in the model is illustrated by equation 2.6.Rearranging equation 2.6 yields an expression for the slope of thehousing price function:(2.7)Ax^h(x)Interesting implications arising from equation 2.7 include:1. The minus sign on the RHS implies that the housing pricefunction has a negative slope - i.e. prices fall withdistance from the CBD;2. The presence of h(x) in the denominator of the RHSimplies that when housing consumption is small the houseprice function is steeper than when housing consumptionis not small - i.e. Ph is convex with substitution inconsumption; and3. If we assume that non-land input prices do not vary withdistance from the CBD, housing prices can be steep onlywhere the land rent function is steep, near the centre ofthe city.Two realistic implications of the locational equilibrium are:1. Suburbanites consume more housing than central cityresidents in order to maintain spatial equilibrium; and2. Suburban houses have lower capital land ratios because inthe suburbs land is cheaper relative to other inputs inthe production of housing.Both implications entail lower population density in suburbanlocations. 62.2.2 The Supply of Housing ServicesTo understand how the urban economy works is to understand howmarkets combine land with other inputs in varying proportions atdifferent places.^The previous derivation of the locationalequilibrium condition ignored any formal inclusion of the supply ofhousing. This section will more formally examine the conditionswhich must obtain for equilibrium in the residential sector.Muth/Mills models are able to account for the substitution betweenlabour, land and capital in housing markets. In addition to thethree assumptions cited in the introduction to this chapter thecurrent exposition assumes:• A Cobb-Douglas housing production function;• The rental rate on capital (r) is not related to intra-urban location (x); and• Housing input and output markets are competitive.Using Cobb-Douglas notation assume the following housing productionfunction:6 It is possible to construct a model with fixed lot sizes inwhich suburban residents do not consume more housing. With fixedlot sizes suburban housing is cheaper than in more centrallocations. This enables suburban households to achieve theequilibrium level of utility by increasing non-housing consumptionby an amount exactly equal to the cost of commuting to the suburbs.- 12 -HS (x)= AL (x) a IC(x) 1 - a^(2.8)where:•^li.(x) m housing supply;•^A, a m constants with 0 < a < 1;• L(x) m land used in housing production; and• K(x) m capital used in housing production.Differentiating equation 2.8 yields the marginal product for land:a 14(x)MPL(x)^L (x) (2.9)and the marginal product for capital:(1 - a) 1-1,(x)MPI(06^K(x) (2.10)Multiplying the marginal products of land and capital by the priceof housing (Ph(x)) yields the respective value of marginal productexpression for land and capital: 7R(x)^a Ph (X) Hs (x)L(x) (2.11)R(x) is the rental rate on land at distance x from the CBD andr is the spatially invariant rental rate on capital. Given theassumption that input markets are competitive, in equilibrium thevalue of the marginal product of each factor of production mustequal its rental rate.- 13 -(1 - a) Ph (x) 1-1,(x) (2.12)K(x)2.2.3 The Demand for Housing ServicesAssume that all workers in the city have the same income (y)(determined exogenously), the same preferences and the sameindividual demand for housing services:ah (x)^Hyel Ph (x) e2^ (2.13)where: 01 m the income elasticity of demand for housing; 02 m the price elasticity of demand for housing; and B m is a constant.Because it was assumed that housing is a non-inferior good we knowthat ei > 0. Therefore, a downward sloping housing demand functionimplies that 02 must be less than zero. Total housing demand isthe product of individual housing demand times the number ofindividuals, N(x):HD(x) h(x) N(x)^ (2.14)s According to Mills and Hamilton [42], equation 2.13 is a demandfunction that has been widely employed in many demand studies. Itassumes that the income and price elasticities are constant.- 14 -2.2.4 Solution to the ModelTo complete the model and solve it assume that 0 radians of acircle are available for development at every distance from thepredetermined city centre with 0 5 27. 9 Thus 2r - 0 radians areunavailable for development.The model has five equilibrium conditions which must be satisfied:Hd(x) - H5(x)Pih (x) h(x) + t 0LOO^Itt•xR(/) =N(x) dx N.(2.15)(2.16)(2.17)(2.18)(2.19)Equation 2.15 simply says that in equilibrium total housing supplymust equal total housing demand. Equation 2.16 is the locationalequilibrium expression (derived in Section 2.2.1), equation 2.17implies that, in equilibrium, the amount of land used for housingcannot exceed the total land available and no land can be leftvacant. Equation 2.18 equates land rent at the urban boundary torent in non-urban uses and 2.19 specifies that the total number ofworkers in the urban area, N, is equivalent to the number ofworkers at any distance, N(x), for all x.9 0 cannot exceed 27r because a complete circle has 27 radians.- 15 -In order to solve the model the VMP equations 2.11 and 2.12 arerearranged isolating L(x) and K(x). L(x) and K(x) are thensubstituted into the housing supply expression, equation 2.8yielding:rl-aRwaPh (X) Aaa(1 - a) 1- a (2.20)Equation 2.20 indicates that the price of housing is proportionateto land rent, R(x), raised to the power of a. Because we know that0 < a < 1, we know that house prices are high when land rents arehigh but house prices rise less than land prices due to factorsubstitution in the housing production function. Taking thederivative of equation 2.20 with respect to x yields:P12 (x) if (1 -r77 a) )1 - 4 R (x) - 1 RI (x) (2.21)where R'(x) is the slope of the rent function R(x). Equation 2.21is a differential equation for housing prices but, because there isno initial condition for house prices, it is necessary to solve themodel for R(x) rather than house prices. Equation 2.18 providesthe initial condition necessary to solve the differential equationexpressed in terms of land rents.Substituting equation 2.13 into the locational equilibriumcondition, equation 2.16, for h(x), equation 2.20 into 2.16 forPh(x) and equation 2.21 into 2.16 for Ph'(x) yields:- 16 -E-1R(x) 13-1Ri (x) + t = 0^(2.22)where E and B are collections of constants:E-1^apyelAaa (1 - a )i - ar c1 +^r t1 - a) (I- 4- Oa)(3 - a (1 + 02 ) .Utilizing the initial condition provided by equation 2.18 resultsin a solution for R(x) from the differential equation, 2.22:R(x) =^+ /3 tE(I- x)]1 .^ (2.23)Equation 2.23 is a general expression for land rent, R(x). If Bequals zero land rent can be expressed as a negative exponentialfunction of x:1°R(x)^Re -tEci -^ (2.24)From the definition of B given above it can be seen that a zerovalue for B implies that 02 = -1: i.e. the price elasticity ofdemand for housing is -1.The final step in this simple derivation of the model involveslinking the rent gradient to the population density gradient.10 e is the base of the natural logarithm.- 17 -Using the equilibrium condition given by equation 2.15 theexpression for total housing demand, equation 2.14, can berewritten:N(x) - (x) h (x) (2.25)Taking the ratio of the value of the marginal product of land andthe value of the marginal product of capital (equation 2.11 and2.12) results in the following expression for K(x):K(x) =^a R(x)L(x).^(2.26)cc rSubstituting equation 2.26 into the Cobb-Douglas productionfunction, equation 2.8, yields:1 - a  11- - a(x) = A [  a r^R (x) 1 - a L (x) . (2.27)Substituting the expression obtained for Ph(x) in equation 2.20,into the individual housing demand expression given by equation2.13 and then substituting the entire expression into 2.25 for h(x)as well as substituting equation 2.27 into 2.25 for Hs(x) yields:N(x) - ER (x) 1 - P.L (x) (2.28)Equation 2.28 indicates that resident workers per unit area(population density) is proportionate to land rent raised to thepower (1 - B). If B is once again set to zero ( i.e. assume that- 18 -the price elasticity of demand is -1) population density isstrictly proportionate to land rent and, therefore, populationdensity declines exponentially with distance from the CBD, likeland rent in equation 2.24.2.3 The Negative Exponential Population Density GradientThe empirical regularity of a negative exponential populationdensity gradient was noted by Colin Clark almost 20 years beforeMuth and Mills (independently) constructed a formal theory toexplain the empirical regularity. Because the negative exponentialpopulation density gradient is generated by a strong form of themonocentric model, it has been subject a great deal of scrutiny andcriticism.There have been numerous empirical tests of the negativeexponential population density function using the Box-Cox methodwhich can statistically test for functional form. 11 Employing Box-Cox tests Kau and Lee [32] found that only 50 percent of the U.S.cities they tested were well characterized by a negativeexponential population density gradient. Anderson's [6] resultswere even less favourable with 22 of 30 cities poorly characterizedby a negative exponential population density gradient. Kau Lee andChen [31] found the negative exponential gradient did not apply in50 percent of the cities they tested. McDonald and Bowman [40]" The Box-Cox method is described in Johnston [30] and Meyer[46].- 19 -were able to identify several alternative functional forms thatperformed as well as the negative exponential density function insome cases, including generalized normal, gamma and standardnormal.Despite frequent criticism that the price elasticity of housingdemand is not equal to -1 and the empirical evidence cited above,the negative exponential population density function has been usedin numerous applied studies (e.g. [1], [4], [5], [14], [17], [18],[26], [38], [43], [45], [61] and [63]) since Mills formalized themonocentric model. This is largely because there are very few moremathematically complex models that are tractable [13].It is important to understand the issues surrounding the correctfunctional specification because the test of the monocentric modeldeveloped by Hamilton [26] simply assumes that the negativeexponential population (and employment function) obtains.' 2Although some would accuse Hamilton of constructing a "strawman" byusing the negative exponential form, the accusation rings hollowconsidering the amount of academic research that has been based onthe strong version of the monocentric model.2.4 A Digression on Employment LocationHamilton's [26] assumption of a negative exponential employment" Hamilton's [26] model is described in detail in Chapter 3 ofthis thesis.- 20 -gradient raises employment location as an important issue. 13 Infact, one of the strongest criticisms of Hamilton's test of themonocentric model [62] focused on Hamilton's assumption that whenemployment decentralizes from the CBD it does not cluster butdisperses in a uniform fashion.In the monocentric model described in section 2.2 of this chapterit was assumed that all employment was located in the CBD. This isclearly an unrealistic assumption but it can be easily relaxedwithout disturbing the locational equilibrium condition. If a firmmoves out of the CBD to a point 5 kilometres distant, any workerswho choose to work at the firm would be better off than workers whowork in the CBD, if both locations paid the same wage rate. Thus,a profit maximizing firm would offer a wage in the suburbs that islower than the CBD wage by the amount of the potential commutingsavings (in this case 5t). There is, therefore, a wage gradientwith slope -t which leaves the residential location equilibriumcondition, equation 2.6, undisturbed." The somewhatcounterintuitive result is that R(x) is unaltered by employmentdecentralization: workers still commute up the rent gradient, someall the way to the CBD and some only as far as the suburban13 For the specifics regarding Hamilton's use of the negativeexponential employment gradient see Chapter 3 of this thesis.14 This is true only if the number of workers located on thesuburban side of a ray passing from the CBD and through thesuburban firm location exceeds the demand for workers at thesuburban employment location. If it does not then the firm wouldhave to offer a higher wage in order to induce circumferential orbackward commuting. See Chapter 3 of this thesis or White [62] fordetails.- 21 -employer.Madden [39] provided one of the few studies to empirically test fora negatively sloped wage gradient. To test for a negative wagegradient Madden developed a regression model but had to assume thatthe negative exponential population gradient was a correctspecification. Madden's results were consistent with a negativelysloped wage gradient for a sample of U.S. cities, although noconclusion regarding functional form was possible. Similarly,Leigh [36] found evidence of a wage gradient in several U.S.cities, but only for caucasians.The assumption of negative exponential population and employmentgradients requires an identical job/housing pattern along every rayemanating from the CBD [62]. This leads to the key question ofwhether firms are likely to cluster when they suburbanize orwhether they are likely to disperse and what the decisions arebased upon.In general, employment location is poorly understood and much lessresearched than population location. Much of the employmentlocation research has focused on the location of manufacturing(e.g. [10],[27] [51]), which is a very small portion of totalemployment in most North American cities. Work examining thelocation of non-industrial employment frequently examines thenotion of agglomeration economies as a determinant of firm location(e.g. [11], [56]). Although the dictionary would treat- 22 -agglomeration economies as a virtual synonym for clustering, theterm carries a heavier, if poorly defined, meaning in economicwriting on the location of industries.It is important to note that even though firms can be allowed todecentralize in the monocentric model access to the CBD is stillvaluable or a firm would not choose to locate anywhere on the urbanrent surface. Mills and Hamilton [42] suggest that firms withagglomeration economies that decline rapidly with distance from theCBD will have a steeper bid-rent function than firms withagglomeration economies that are less sensitive to distance.There are two interesting implications of the Mills-Hamilton lineof reasoning. First, improvements in communication technologymight be expected to lessen the decline in agglomeration economiesfor a firm located at any distance from the CBD, with the resultthat more and more firms will choose suburban locations. 15 Second,by clustering in suburban locations firms may be able to generatesome positive externalities (i.e. agglomeration economies) in thesuburban location. The degree to which this is possible dependsupon he micro-foundations of the agglomeration economies. 16 Of15 This argument is often used to explain the fairly recentsuburbanization of office occupations which were typically thoughtto locate in the central city due to the need for face to facecommunications.16 The micro-foundations of agglomeration economies are poorlyunderstood. Possible explanations include: the city performs arole as a warehouse, incubator effects such that new firmsexperience lower information costs due to proximity to other firmsin the same industry, higher salvage values for capital assets,lower search costs for skilled labour, qualitative differences in- 23 -course, when suburbanized firms cluster they also offset some ofthe benefits of suburbanization by increasing both land rent andwages relative to a dispersed suburbanization scenario. Ultimatelythe choice to cluster or suburbanize will be dependant upon atrade-off between agglomeration benefits, land rent and wage costs.Unfortunately, there is no way to extend the monocentric model tocover the formation of sub-centres because the monocentric modelcontains only one descriptor of location (distance to the CBD).Clustered suburban employment necessitates circumferential and/orbackwards commuting making both distance and direction from the CBDimportant.'' And two dimensional models are extremely complex andyield few analytic results.2.5 Criticisms of the Monocentric ModelCriticisms of the monocentric model fall into two broad classes:1. Criticism of the model's central assumptions (e.g. [49][57]); and2. Criticism of the predictive powers of the monocentricmodel (e.g. [7], [15], [26], [32], [36], [39]).2.5.1 Assumptions of the Monocentric ModelAll economic modelling involves abstraction which usually takes thethe types of information exchanged etc." Dubin and Sung examined the impact of direction on the densitygradient for Baltimore [16].- 24 -form of simplifying assumptions in the model and the monocentricmodel is no exception. The monocentric model has receivedcriticism for a lack of realism in a number of areas including:• The assumption that all housing is putty-putty;• The lack of specificity regarding the reasons for theexistence of the CBD;• The assumption of homogeneous households;• The assumption that there exists an everywhere densenetwork of transportation facilities; and• The assumption that employment does not cluster when itdecentralizes.The putty-putty assumption is one of the least satisfactory aspectsof the monocentric model. The model ignores the spatial fixity anddurability of housing capital assuming instead that the city isrestructured each period in order to achieve long-run equilibrium[49], [57]. Harrison and Kain [28] were able to overcome theputty-putty assumption by constructing a different logical edificeon the empirical regularity first observed by Colin Clark.Harrison and Kain argued that current spatial structure islogically viewed as an aggregation of historical patterns ofdevelopment under the assumption of durable housing capital.Present density is a function of the weighted density of all pastdevelopment in a particular city. The last decade has seen someimportant theoretical work with dynamic models but thespecifications are often unwieldy and there has been only limitedempirical testing of the models.The lack of specificity regarding the reasons for the existence of- 25 -a CBD has also been the subject of criticism. Wheaton argued thatsimple models of centrality are inadequate and what is required aremodels that explain the attraction of economic activity to cities[47], [57]. A limited number of researchers have responded to thiscriticism by developing non-monocentric models where the locationof employment is endogenously determined (e.g. [47]).Agglomeration economies are fundamental to non-monocentric modelsbut mathematical intractability is often severe with greatdifficulty in solving for equilibrium.Multiple household types have been successfully incorporated intothe monocentric model by a number of researchers including Wheaton[58] and White [64]. In an interesting paper, Steen examined theimplications of relaxing the assumption of ubiquitoustransportation networks [53]. Steen concluded that discreettransportation networks yield a more complex density pattern withhouseholds valuing access both to the transportation route andaccess to the CBD.The central location of employment is the most widely criticizedaspect of the monocentric model according to Wheaton [57]. Thus,employment decentralization was accorded a separate section, 2.4,above.2.5.2 Predictions of the Monocentric ModelBlackley and Follain [7] noted that the monocentric model has been- 26 -subjected to a great deal of testing and criticism." Much of thetesting has focused on the reduced form of the model by testing themonocentric model's ability to predict important spatial phenomena.While such tests may not refute the model's locational equilibriumcondition they may weaken confidence in the monocentric model ifthe model fails repeatedly to predict important spatial patterns.The monocentric model yields at least five important predictions:1. There exists a negatively sloped housing price gradient;2. There exists a negatively sloped land price gradient;3. There exists a negatively sloped population densitygradient;4. There exists a negatively sloped wage gradient; and5. People minimize commuting by travelling inwards on a raybetween their home and job.Each of these predictions has been empirically investigated, mostwith mixed results. The majority of studies have found evidence tosupport a negatively sloped house price and land price gradientalthough there are serious questions about some of the hedonicmodels used to test for a house price gradient.While section 2.3 of this chapter highlighted concern surroundingthe specific functional form of the population density gradient,most of the studies supported the conclusion that populationdensity declines with distance to the city centre. There is,3 This has certainly been the case in the United States. Therehas been considerably less testing of the model in Canada.- 27 -however no similar support for a negatively sloped employmentdensity gradient. Neither Kemper and Schmenner [33] or Schmenner[51] were able to find direct evidence of a rent gradient (andtherefore an employment density gradient) for manufacturing usingregression analyses similar to Muth's [45] model of residentiallocation.It has also proven extremely difficult to obtain data that allowsfor an empirical test of the existence of a negatively sloped wagegradient. Studies by Leigh [36] and Madden [39] provided weakevidence in support of the wage gradient.Perhaps the greatest empirical failure of the monocentric model hasbeen its inability to predict observed commuting behaviour.Hamilton [26] provided the essential test of the monocentric modelin this regard. Hamilton's results indicated that observedcommuting in 14 U.S. cities was eight times that predicted by themonocentric model. This is not surprising considering that thevast majority of urban travel is non-commuting trips. In a sensethe monocentric model is too restrictive to adequately explain thelocation decisions of home owners who base their decisions uponmuch more than commuting cost.2.6 SummaryThe purpose of this chapter was to provide a brief overview of the- 28 -monocentric model. The monocentric model is synonymous with urbaneconomics. The importance of the monocentric model as itsevolution was discussed. Monocentric models have been widelyemployed in urban economics due largely to the simple structure ofthe model and the associated mathematical tractability [13].Sections 2.1 and 2.2 examined the logic behind the model andmathematically derived a simple version of the monocentric modelbased on Mills and Hamilton [42]. Sections 2.3 to 2.5 reviewedsome of the work of those who challenged the assumptions andpredictions of the monocentric model.Section 2.3 discussed the negative exponential functional form.Most of the empirical work concluded that the negative exponentialpopulation density gradient is inappropriate, thereby implying thatthe price elasticity of the demand for housing is not -1. Section2.3 is not an indictment of the model itself but an indictment ofthe use of equation 2.24 rather than 2.23. Section 2.4 describedan important modification to the model by allowing employmentdecentralization. However, the inability to include clusteredsuburban employment remains an important weakness in the model.Finally Section 2.5 provided a brief overview of more generalcriticisms of the monocentric model. Hamilton's critique [26],comparing observed commuting behaviour with commuting estimated bythe monocentric model forms the basis for the remainder of thisthesis.An appropriate epilogue to this chapter is provided by Wheaton[57]. Wheaton argues that while the utility of monocentric modelsin forecasting urban growth is limited they still serve animportant educational function. Even if the assumptions andoutcome of the models are seen as unrealistic the clarity of theirsimple structure has created a new awareness of spatial equilibriumand the role of transportation.CHAPTER 3HAMILTON'S MODEL FOR ESTIMATING WASTEFUL COMMUTINGThis chapter describes and explains the methodology used to comparecommuting behaviour in 23 Canadian cities in 1981. The basic modelwas derived by Hamilton [26]. Employing the same methodology asHamilton's U.S. study allows direct comparison of commutingbehaviour in Canada and the United States.There are four sections in this chapter. Section 3.1 describesHamilton's method for measuring wasteful commuting in cities. Theexposition parallels that given by Hamilton [26] but attempts havebeen made to clarify areas where Hamilton's description wassomewhat opaque. Section 3.2 reviews three methods for calculatingtwo-point estimates of population and employment density gradients.Section 3.3 describes the data used to operationalize the model.The final section concludes the chapter.3.1 Hamilton's Model For Estimating Wasteful CommutingHamilton realized that it is not sufficient to compare the averagecommuting distance (or time) for a sample of cities in order tojudge the relative efficiency of commuting behaviour in each city.Simple comparison ignores extant structural differences amongcities. The distribution of jobs and homes influences commutingbehaviour. Hamilton's model is able to control for each city'sinternal distribution of people and jobs when assessing the abilityof the monocentric model to predict observed commuting.- 31 -3.1.1 The Theoretical Basis of Hamilton's ModelHamilton begins with the standard monocentric model (described inthe previous chapter). Recall from Chapter 2 that individuals areassumed to maximize a utility function, (U), defined over a housinggood, (h), and a composite non-housing good, (g):U - u(h,g)^ (3.1)Utility is maximized subject to a budget constraint in whichcommuting is costly:y- - t± - 400 h(x) + Pgg(x)^(3.2)where: y a household income;• t a the cost of a round trip kilometre;• x a distance from home to the CBD; k a the distance from home to work;• Ph(x) a the price of one unit of the housing good;• Pg = the unit price of the composite good.Notice that the model incorporates a spatial component intostandard consumption theory by specifying Pg as spatially invariantwhile Ph is specified as a function of distance to the CBD, (x).Equilibrium requires that Ph decline as x increases in order tocompensate individuals for the higher transportation costs (i.e.commuting costs) associated with residences more distant from theCBD.- 32 -In the case of complete employment centralization, (i.e. R = x forall individuals), the possibility of wasteful commuting does notexist because aggregate commuting is fixed. There is no possiblereassignment of homes or jobs among existing residents that willreduce aggregate commuting. The average commute is simplyaggregate commuting divided by total employment in the city [pg.1037, 26].When employment decentralizes, but doesn't cluster, householdsmaximize utility by trading off accessibility to x, (rather thanx0), with the price of housing Ph(x).' Land rent, however, variesonly with distance to the CBD (x) as in the case of completeemployment centralization. Hamilton and Mills [42], [43] show thatin the case of non-clustered decentralized employment, there existsa wage gradient, with slope -t, that leaves the rent gradient andthe locational equilibrium unchanged from the case of completeemployment centralization. Lower commuting costs associated witha suburban job result in a lower wage rate at the suburban joblocation [pg. 114, 42].As long as the population located on the ray k F in Figure 3.1exceeds the total demand for workers of a firm at St, wages at icwill be (w* - kt), which is Set less than wages at the CBD (w*). Thediscount amount, (ft), is exactly equal to the savings in commuting1 The importance of the assumption that suburban employmentdoes not cluster is discussed in Chapter 2 of this thesis. Seealso White [62].- 33 -costs for an employee who lives at x, and works at x instead ofworking in the CBD. Individuals who work at k will always chooseto live on the portion of the ray between F in order tominimize both Ph(x) which declines with distance from the CBD andtransportation costs. 2The market solution minimizes aggregate commuting within a city ifevery household chooses to live on the suburban side of the ray x,k F for all possible x. Any violation of this criterion makespossible a house or job swap that reduces total commuting resultingin a pareto improvement. The solution that minimizes aggregatecommuting is not unique, however. Any two workers on the ray k -F can swap houses (or jobs) as long as both remain on the suburbanside of their respective jobs [pg. 1038, 26].The introduction of a negatively sloped wage gradient in thepresence of decentralized employment preserves the originallocational equilibrium which is based solely on the trade-off ofcommuting costs and housing costs. However, employmentdecentralization raises the possibility that aggregate commutingcan exceed the minimum necessary. Wasteful commuting, impossiblein the case of complete employment centralization, becomes possiblewhen the model allows for decentralized employment. The matchingof households and jobs is no longer irrelevant, but can effect the2 If a firm at x demands more workers than are living on kF the suburban wage rate will be greater than (w* - kt) but lessthan w*. This must be the case. For the firm at x to attractworkers it must compensate them for either backward orcircumferential commuting. See Chapter 2, Section 2.4 for moredetails as well as White [62].- 34 -FIGURE 3.1CITY WITH DISPERSED EMPLOYMENTCBD - x,"/V\.,....„,..._ __--------"'■ CBD^- Central Business District (x 0 ).■ x 1^- household location of individual i.■ F^■ the urban boundary.■ i^  suburban employment location.-w■   wage rate at the CBD.■ x i ^  commuting cost for person at x^working at xo ' 1■ (x1 - x)1^  commuting cost for person at x 1 working at x.■ it^- commuting savings from working at X.■ (w . - it)   wage rate at x.amount of aggregate commuting in a city.3.1.2 Calculating the Optimal Average CommuteIf all households are assumed to behave as individual utilitymaximizers, it is relatively easy to calculate the minimum requiredaverage commute for any city. The minimum required commute equalsthe average distance between homes and jobs throughout themetropolitan area.When all jobs are located in the CBD the average commute can beestimated as the aggregate distance of all people from the CBDdivided by the total population: 3A - —1 11 xP(x) dxP (3.3)where:A a the average distance of individuals from the CBD;•^P(x) m the population at any distance, (x) from the CBD; P m the total metropolitan population; R m the urban boundary;• x m distance from the CBD.3 Hamilton's model assumes that the participation rate doesnot vary with distance from the CBD. This assumption is necessaryso that the average distance of the population from the CBD can beinterpreted as the average distance of workers from the CBD. Ifthe participation rate increases with distance from the CBD then"A" will likely underestimate the average distance of workers fromthe CBD; if the participation rate declines with distance from theCBD "A" will overestimate the average distance of workers from theCBD.- 36 -The logic behind equation 3.3 is straightforward. P(x)dx can beinterpreted as the number of people in a ring of width dx locatedx kilometres from the CBD. In a circular city this can beexpressed as:P(x) dx D(x)^x cbc^(3.4)where: ^D(x) E population density x kilometres from the CBD; ^27 E the circumference of a circle in radians (27360'). 4The right hand side of equation 3.4 can be substituted intoequation 3.3 to replace the term P(x)dx yielding:A - P 0f xD (x) 2Tcx clx211 f: X 2 D (X) CLIC(3.5)The total population of the city can be obtained by integratingequation 3.4 from 0 -■ R:4 The circumference can vary from 0 to 360 degrees to accountfor variations in the land available for development amongdifferent cities. Typically 27 - a the number of radiansunavailable for urban development. When 0 = 27, 360' are availablefor development; 0 = it implies only 180' are available fordevelopment.- 37 -P f: P(x) dx0- f0 D(x) 2 .7c x dx  27t f D(x) x dx0 (3.6)All that remains unspecified in equation 3.5 is the functional formof the density function, D(x), and the diameter of the city, R.The majority of urban economic research has assumed a negativeexponential population density function (e.g. [17] [18], [28],[38], [42], [43], [45], [58], [59], [61]):5D(x) - Do e -Y"^ (3.7)where: Do a the population density at the city centre; y a the slope of the population density gradient; e a the natural logarithm; and x a distance from the CBD.Substituting equation 3.7 into 3.6 for D(x) yields:5 Because Hamilton wanted to test the efficacy of themonocentric model he correctly maintained the simplest assumptions.Most researchers begin with the simple model although they usuallymodify the basic model somewhat. White [62] and several otherresearchers subsequently accused Hamilton of creating a "straw man"which Hamilton then proceeds to knock down in the paper estimatingwasteful commuting [25], [26].- 38 -YP2^2nDO —2 -y.kX2 e (3.9)A^2it f k x2 Do -yx dxP 02nDo r±.- x2 e _yx dxP JO(3.8)Integrating equation 3.8 by parts yields: 6In order to estimate A, values for D,, y and x are required.Values for R are estimated using equation 3.7 once y and D 0 areknown.' Estimation of D0 and y is discussed in section 3.2 of thischapter.Hamilton interpreted employment decentralization as the potentialsavings in aggregate commuting arising from jobs moving closer toresidences over time. To measure the potential commute savings foreach city Hamilton calculated the average distance of jobs from theCBD. This was completely analogous to the calculation of theaverage distance of people from the CBD described in equations 3.3to 3.9. The average distance of jobs from the CBD can be written:6 Appendix 1 provides the detailed integration.' See the section titled Choosing the Urban Boundary inChapter 5 of this thesis for estimates of R using different densitygradient parameters.- 39 -B = —J1 r xJ(x) dx (3.10)where: B = the average distance of jobs from the CBD; J(x) = employment at any distance from the CBD; J = total metropolitan employment; R a the urban boundary; and x = distance from the CBD.As in the case of population, J(x)dx can be interpreted as thenumber of jobs in a ring of width dx located at distance x from theCBD. In a circular city this can be expressed as:J(x) dx a E(x) 2n x cbc^ (3.11)where: E(x) = employment density x kilometres from the CBD; and 2r = the circumference of a circle in radians (2r =360').The right hand side of equation 3.11 can be substituted intoequation 3.10 to replace the term J(x)dx yielding:B- o xE(x) 2nx clx- J f 2 2 X E(x) dx 0(3.12)- 40 -Total employment in the city can be obtained by integratingequation 3.11 from 0 -+ R:J fg J(x) dx- f E(x) 21rxdxf- 27c f2 E(x) x dx(3.13)Like the population gradient, the employment density gradient,E(x), is assumed to be a negative exponential function:8E(x)^E0 e-ax^ (3.14)where: E0 = the employment density at the city centre; S = the slope of the employment density gradient; e a the natural logarithm; and x = distance from the CBD.Substituting equation 3.14 into 3.12 for E(x) yields:B^2g f ir- x2 Eoe -ifx dyJ 02gEo ril x2 e _a, clx (3. 15)8 Hamilton's assumption of a negative exponential employmentgradient is problematic. While there is both theoretical and someempirical support for a negative exponential population densitygradient, there is neither for a negative exponential employmentgradient. This issue is discussed in more detail in Chapter 2.Important references include: [10], [11], [19], [21], [27], [33],[41], [50], [51], [55], [56].- 41 -Integrating equation 3.15 by parts yields:B - 2 - 2nE8J .7z0 2 e -a2 (3.16)Hamilton defined the difference between A and B is as the minimumpossible average commute, given the underlying urban structure assummarized by equations 3.7 and 3.14. Thus optimal commute, (C),is defined:C - A - B^ (3.17)The parameters D„ E0, y and d control for differences in theexisting distribution of people and jobs within individual cities.Thus the model can be used to answer the question:"Given the current distribution of homes and jobs, isobserved commuting behaviour consistent with thepredictions of the monocentric model?".In order to answer this question C must be compared with observedcommuting behaviour in a sample of cities.3.1.3 Calculating the Average Random CommuteOnce the failure of the monocentric model to predict observedcommuting behaviour was confirmed, Hamilton introduced the concept- 42 -of random commuting in order to illustrate the degree to which themonocentric model failed. Calculating the average random commute,(E), for a city with a finite radius, R, turns out to bemathematically tedious. The estimate is much less tedious if themean random commute is calculated for a city without limits. It canbe shown that allowing the city boundary to approach 03 imparts onlya small upward bias in the estimate of random commuting, (E). Bothprocedures are described in an appendix to Hamilton's paper [26].Only the infinite boundary estimate is described here and laterutilized in Chapter 5.To begin, suppose that households and jobs are distributedthroughout the city according to equations 3.7 and 3.14. The cityis assumed to be an everywhere dense network or radial roads andbeltways such that every household is located at an intersection.Commuters take the shortest route to work. The city is circularwith 27r - 0 radians unavailable for development. Homes and jobsare distributed according to the population and employment densityfunctions given in equations 3.7 and 3.14. By integrating twiceover homes and jobs the average one way commute is defined as:E 2 (y + 8) + 2M  y2 + 3Y8 1Y8^(y + 8) 2 (3.18)where:E = the average one way random commute for a city without limits;and- 43 -M -.7=- a function of 0 such that:(1) s 2^M = ()8- 22 s. < 2n^M= 3 4(TO (3.19)02= 2n^M = -2Basically, the procedure^specifies^a^distance function^andintegrates that function over house and job locations then dividesby the total number of commuters. The assumption of an everywheredense network of roads imparts a small downward bias into estimatesof E.3.2 Two-Point Estimates of Employmentand Population Density GradientsIn section 3.1 Hamilton's method for measuring the minimum possibleaverage commute in any city was described. Recall that the minimumpossible average commute, (C), was defined as the differencebetween the average distance of homes from the CBD, (A), and theaverage distance of jobs from the CBD, (B). In order to solveequations 3.9 and 3.16 (for A and B) Hamilton required estimates ofD0, y, E, and 6.There are two different methods available for estimating thedensity gradient parameters in equations 3.7 and 3.14. 9 Bothmethods assume density, D(x), declines exponentially with distance,9^Each broad method has a number of variants but,essentially, all are subsets of the two methods described below.- 44 -x, from a predetermined city centre.'°Muth [45] suggested taking the natural log of both sides ofequation 3.7 or 3.14 and estimating y or 6 using OLS:In D(x)-ln Do-yx^ (3.20)For each city estimated using the Muth method, data for D(x) and xmust be collected at the census tract level.Rather than estimate equation 3.20, Hamilton used values for y, D„E, and 6 calculated by Macauley [38]. Macauley in turn used amodified version of Mills' two-point estimation technique.In his pioneering work, Studies in the Structure of the Urban Economy, [43], Mills developed the two-point estimation technique.Two-point estimation is actually a misnomer since the techniqueinvolves the use of two large integrals.'° See Chapter 2 for the implicit assumptions embedded in anegative exponential gradient. It is important to disentangle twoseparate but related issues. Questions regarding theappropriateness of the two-point technique as an estimator of y andD, are distinct from questions about the use of the negativeexponential function as a descriptor of population or employmentdensity. The latter is a fundamental debate in urban economicswhile the former issue is a methodological wrangle. Both Mills'two-point method and Muth's regression-based alternative proceedunder the assumption D(x)=D0e"x is an accurate descriptor of urbandensity. Papers contributing to the debate surrounding theappropriateness of the negative exponential density functioninclude Brueckner [8], Griffith [25], Harrison and Kain [28], Kau,Lee and Chen [31], Kau and Lee [32], Kim and McDonald [34] andHamilton and Mills [42]. Kau and Lee, for example, found that 6 ofthe same 14 SMSAs used by Hamilton [26] were not well characterizedby a negative exponential density function. Kau and Lee obtainedtheir result using a Box-Cox test for functional form.- 45 -The logic behind Mills method is straightforward. 11 Assume thatequation 3.7 is an accurate depiction of population density in aparticular city and that the metropolitan area is circular with 2r- 0 radians unavailable for development." 13 In such a CMA, thenumber of people, n(x), in a ring of width dx, located x miles fromthe city would be:n (x) dx D(x) 4)x c:bc^ (3.21)The total population, N(k), within k miles of the CBD is simply theintegral of equation 3.21 from 0 k:N(k) - fk n(x)cbc.0 (3.22)Substituting the right hand side of equation 3.21 into 3.22 gives:N(k) -fk D(X),XdX0_^Do e -Yx 4)x dx-^fk e -Yx X d_X.(3.23)Integration of equation (3.23) yields:11 For greater detail see Mills [42] (Chapter 3) or Edmonston[16], [17].12 In equation 3.7, Do represents density at the city centreand y is a measure of the rate at which density declines withdistance from the CBD. If y is large density falls off rapidly; ify is small density falls off slowly.13 The implicit circularity of this assumption was modifiedby Edmonston [17]. See section 3.2.3 of this thesis for details.- 46 -N(k) - ^° [1- (1 + y e -Ykly 2 (3.24)where:• N(k) E total population from 0 k;• 0 E radians of land available for development;• k E the radius of the metropolitan area;• e E the natural logarithm; and• D0, y a parameters to be estimated.The basic insight provided by Mills was recognizing that equation3.7 could be estimated with only two observations provided by thecentral city - suburb data. The estimation technique is much lessonerous than alternative techniques which use large samples ofcensus tract data. The following quotation from Mills summarizesthe elegance of the Mills method:"The basic insight exploited in the present chapter isthat if the negative exponential density function is anaccurate representation, its estimation does not dependon where the central city boundary is drawn or on whetherits location changes over time. Furthermore, since it isa two-parameter family of curves, it can be estimatedwith the two observations provided by the cental city -suburb data.It must be emphasized that the cental city - suburb dataare not merely a sample of two observations. Theyprovide two exhaustive and exclusive integrals of thedensity function and thus make use of the entirepopulation of data. There is no reason to believe thatthey provide less accurate estimates than would a largesample of census tract observations." [pg. 35, 43].In contrast to Hamilton's work, there were no recent estimates ofthe population and employment density gradient parameters for asubstantial number Canadian CMAs that could be used for this- 47 -thesis. Thus, it was necessary to solve equation 3.24 for both thepopulation and employment density gradients in each Canadian city.As presented above equation 3.24 is not solvable because it is anon-linear equation with two unknown parameters (D0 and y). Threeseparate methods for solving 3.24 are presented below.3.2.1 The Mills Estimation TechniqueTo solve equation 3.24 Mills noted that letting k co implies:^ - N (3.25)y2where N m total metropolitan population. Substituting N intoequation 3.24 yields:N(1e)- N[1- (1+yk^)e-Tx] .^(3.26)where:• N(k) m central city population;• N m SMSA (CMA) population;• k m radius of the central city;• y m the parameter to be estimated.Equation 3.26 is a non-linear equation in one unknown, y.Non-linear equations are solved using numerical methods. Once y isestimated it can be substituted into 3.25 to calculate Do.Data for N, N(1) and k for each of 23 CMAs was obtained from the1981 Census of Canada and equation 3.26 was solved iteratively for- 48 -Estimates of y are then substituted into equation 3.25 tosolve for Do.Rearranging 3.25 to isolate Do yields:„..,^= (3.27)In order to solve for Do an estimate of 0 is required. Millssolved for 0 by assuming that each city was (semi)circular and thenequated the known area of the city to the appropriate value of 0. 15If we define 2r - 0 as the radians excluded from development ineach city, when 2r = 0 the entire circle is available fordevelopment. The area of a circle, (At), is defined as:A, - nr 2^(3.28)Rearranging the expression 2r = 0 to isolate it implies that it =0/2. Substituting into equation 3.28 for it and isolating 0 yields:4) - 2 [-2A Ir (3.29)Data for the area of the city, (A„), and the radius were obtainedfrom the 1981 Census of Canada. 0 was estimated from equation 3.2914 The algorithm employed is described below in Section 3.3.3.15 Recall that the circumference of a circle is equal to 2r(360 degrees).- 49 -and then substituted, with y, into 3.27 in order to solve for D o.Finally, it is necessary to calculate k, the distance at whichpopulation density declines to 100 people per square mile (or 36.61per square kilometre). 16 y and Do were substituted into equation3.7 and D(x) was set equal to 36.61 persons per square kilometre.The non-linear equation was then solved for k using numericalmethods.The procedure used to estimate the employment density gradientparameters, Et, and 6, was similar to the procedure described forpopulation density gradient parameter estimates. Employment wassubstituted for population in equations 3.24 to 3.29.3.2.2 The Macauley Estimation TechniqueMacauley modified Mills' technique in order to remove a slightupward bias in Mills' estimate of y and 6. Mills' assumption thatthe urban boundary was infinite was a simplification which allowedhim to solve equation 3.24. Mills recognized that allowing k 00would impart a slight upward bias to estimates of y and D o but heconjectured that the bias would be small."16 The population density figure of 100 people per square milewas chosen arbitrarily by Hamilton, as the density dividing ruralfrom urban.17 The upward bias in the Mills method implies that hisestimates of y and 6 underestimate the suburbanization of homes andjobs, respectively. This was shown to be the case for the 1981Canadian estimates included in Chapter 4 of this thesis. Estimatesusing Mills' technique yielded values for y and 6 that wereslightly greater than values obtained with Macauley's method. The- 50 -The method used by Macauley is identical to Mills' method up toequation 3.24. Rather than solve equation 3.24 by letting kMacauley, [38], avoided the Mills bias by estimating two separateequations that are similar in form to 3.24: one for the centralcity and one for the metropolitan area.The first step in Macauley's method is to isolate D, in one of thetwo equations:N(k2) y2Do -441 - ( 1 + yk2 ) e -Yk2 ](3.30)D, is then substituted back into the second equation.^Theresulting expression is the ratio of metropolitan area to centralcity population:N(Ici)^[1 - (1 + yki )N(k2 )^[ 1 - (1 + yk2 ) e Yk2 ]where:N(k,) 7.-= metropolitan area population;N(k2) a central city population;k, metropolitan radius; andk2 ..---. central city radius.(3.31) difference between Mills and Macauley estimates was not large norwas it statistically significant.Macauley [38] also confirmed Mills' conjecture. Macauley'sestimates using both Mills' original and corrected techniquesindicated that the original estimates of y and 6 were only slightlylarger than corrected estimates.- 51 -To estimate 3.31 data for N(k,), N(k2), k 1 and k2 were obtained fromthe 1981 Census of Canada. Equation 3.31 is a non-linear equationin one unknown (y) and was solved using numerical methods.Estimates of y were substituted back into 3.30 to solve for Do. 0was estimated as in the Mills technique.3.2.3 The Edmonston Estimation TechniqueBoth Mills' and Macauley's estimates of y, D„ S and E0 implicitlyassume that all cities are (semi) circular. Irregularly shapedcities were excluded by Mills [43] and Macauley [38] in an ad hocfashion. Because of the limited number of cities in the Canadianurban system, removal of irregularly shaped cities was deemedimpractical. Furthermore neither Mills nor Macauley providecriteria for evaluating which cities are irregularly shaped andshould, thus, be excluded from the sample.Edmonston [17] developed a technique that attempts to overcome thisthe need to exclude irregularly shaped cities from the Millsmodel." Edmonston incorporated a method that measures the averagedistance to the urban boundary from the CBD, for a variety ofdifferent city shapes. Like Macauley, Edmonston solved for y inequation 3.31. However, in order to implement Macauley's model k,18 The Mills estimation procedure is used to describe both hisoriginal and corrected technique. Mills developed the technique soit seems appropriate to refer to it as the Mills technique.However, Macauley provided the corrected estimates that Hamiltonused in his estimates of wasteful commuting.- 52 -and k2 were measured from census maps using a large compass. 19Rather than measure k, and k2 directly Edmonston suggested codingeach metropolitan area's shape based on its current political (andtherefore, statistical) boundaries. Possible shapes included:1. Circle2. Square3. Hexagon4. Rectangles of various length to width ratios:1 x 21 x 41 x 61 x 81 x 10.Given the metropolitan (central city) land area, the coded shapeand the assumption that the CBD is centred within the region, k1(k2) can be estimated as the average distance to the boundary. Forexample, if a city is circular the area is given by equation 3.28.Rearranging 3.28 to isolate r yields: 2°r^Ac^ (3.32)- (0.5642) A;19 Edmonston described this method of measuring the centralcity and metropolitan area radius as "time consuming and not alwaysas accurate as desired." [17].20 The symbol, r is being used to denote the radius or averagedistance to the urban boundary. Using the notation of equation3.31, r k, or k2 for the CMA and central city, respectively.- 53 -The average distance to the boundary for other shapes wascalculated in a similar manner, based on the formula for the areaof each shape. Table 3.1 summarizes the derivation of r (i.e. k,or k2) for eight stylized city shapes.Equation 3.31 requires a value for both k, and k 2 so both the CMAand central city shape had to be coded. In the majority of casesthe shape was similar for both the central city and the CMA. Thecomplete data set is discussed in Section 3.3. With Edmonston'smethod it was still necessary to estimate 0 in order to solve 3.30for Do. Edmonston suggested using a protractor and census maps todetermine the radians available for urban development.Table^3.1:^Radius^CalculationBased on Eight City ShapesSHAPE Calculatedk, and k2Circle r = (0.5642)4A,Square r = (0.5611)4A,Hexagon r = (0.5629)4A0Rectangle1 x 2 r = (0.5416)4A,1 x 4 r = (0.4909)4A,1 x 6 r = (0.4523)4A,11 x 8x 10rr==(0.4247)4A,(0.4023)4A,Source: Edmonston [17].It is obvious from Table 3.1 that the Mills' assumption ofcircularity is not likely to substantially impact gradient- 54 -estimates for cities coded as square or hexagonal. Rectangularcities, particularly those with length to width ratio exceeding1:2, are more likely to be affected by the assumption ofcircularity. Using the 1981 census boundaries, cities with widthto length ratios exceeding 1:2 included Kitchener, Saint John, St.John's, Thunder Bay, Victoria and Winnipeg.3.2.4 Two-Point Versus OLS Density Gradient EstimationMuth's regression method (equation 3.20) has been widely applied(e.g. Alperovich [4], [5] and Muth [45]), to estimate populationdensity gradients.^The paucity of employment data atsub-metropolitan geographic levels has precluded OLS estimation ofemployment gradients using Muth's method. Hamilton's [26] remarkregarding Muth's method illustrates the tacit belief present inmuch of the urban economics literature:"There is a widespread view that the Mills' two-pointestimates of density gradients are inherently inferior tothe Muth technique of regressing log density on distance"[26].White [60] examined carefully the legitimacy of the prevailingattitude toward two-point estimation techniques. White argued thatthe bias in favour of OLS arises, in part, because the statisticalproperties of the OLS estimator are well understood, in contrast tothe two-point estimate's unknown distributional properties.There are a number of concerns regarding OLS estimates of y andmounting empirical evidence to suggest that "researchers using- 55 -two-point estimates need not apologize profusely" [60]. While itis often noted that the two-point method ignores potentialinformation, Hamilton argued the converse is also true: OLS ignoresprior information by not constraining the integral of the estimateddensity function to "add up" to the actual metropolitan areapopulation. Indeed the "thrust of Mills' method is to note thatthere is only one parameter vector which satisfies the priorinformation for an exponential density gradient" [pg. 1045, 26]. 21McDonald and Bowman [40] found constrained OLS yielded estimates ofy that were steeper than ordinary OLS estimates. 22 McDonald andBowman's result is consistent with other empirical work suggestingthat Mills two-point estimates are, on average, steeper than Muth'sregression estimates [26], [28], [42], [43].A second difficulty with OLS estimates of y is a potentially severeupward bias generated by a systematic relationship between censustract geographic areas and population density at any given distancefrom the city centre [6], [20]. The relationship arises becausesparsely populated city districts are consolidated to a greaterdegree than densely populated districts, in order to form censustracts of approximately uniform population. Thus, sparselypopulated districts tend to be under-represented in a random sampleof census tracts [20]. 23 Frankena suggested using weighted least21 See Hamilton's [26] footnote #12 for greater detail.22 Constrained estimation forces the integral to "add up" tototal population. See McDonald and Bowman for more detail on theconstrained estimation procedure employed.23 ^relevant point is that the data required to estimatey are available by census tract, not city district.- 56 -squares (WLS) to correct the bias, and Anderson [6] employed aBox-Cox transformation of the dependant variable, D(x), in order todetermine the proper weights for WLS estimation.White [60] noted a third, often overlooked, limitation of OLSestimates of population density gradients. While OLS estimates ofy are BLUE (Best Linear Unbiased Estimates), when the standardassumptions regarding the error structure of equation 3.7 are made,estimates of Do are biased. OLS estimates of In Do are BLUE but,under the natural log transformation, estimates of D, are not.This limitation is usually overlooked because most researchers havebeen more interested in the gradient parameter, Y. than theintercept parameter, Do.White's [60] paper provides perhaps the strongest endorsement oftwo-point estimation. Given the impossibility of determining thedistributional properties of two-point estimators White insteadconducted a Monte Carlo simulation experiment. An artificial citywas created, in which both the density gradient and the randomerror term applied to the population density of each metropolitandistrict was known. 24 White then compared the ability of OLS andtwo-point estimators to predict the "true" y.White had no prior information regarding the correct errorstructure for equation 3.7 Assuming a multiplicative errorstructure ensured that OLS estimates were BLUE and thus superior to24 See White [60], (pg. 298-299), for greater methodologicaldetail.- 57 -two-point estimates.^However, the magnitude of the differencebetween OLS and two-point estimates of y was less than 5 percent.When White assumed an additive error structure, OLS estimates wereinferior to two-point estimates. White concluded that, overall,two-point estimates performed as well as OLS estimates. Two-pointestimates of y appeared to be biased downward no more than 4-5percent, while the bias in D, depended on the assumed errorstructure (multiplicative versus additive). 25For this thesis two-point estimates were believed appropriateestimators for y, D0, 6 and E, for four reasons:1. Hamilton used two-point estimates and one of the primarygoals was to compare my results with Hamilton's results;2. The evidence reviewed above suggested two-point estimatesare at least as good as OLS estimates;3. The data requirements for two-point estimates were farless daunting than for regression estimates; and4. The data necessary for OLS estimates of 6 and E, are notavailable.3.3 Data SourcesData for this study came from a variety of sources. Employment andpopulation density gradient parameters were estimated for twenty-two Census Metropolitan Areas (CMAs) and for one Census25 Another of White's results is particularly encouraging.White estimated y using a variety of central city-suburbanpopulation splits. He found y was not sensitive to the particularsplit except for extreme cases. When the central city had 5_ 5percent or ? 80 percent of the total metropolitan area populationthe variability of two-point estimates increased dramatically (pg.303), [60].- 58 -Agglomeration (CA). Figure 3.2 ranks the CMAs included in thisstudy according to their 1981 population. Figure 3.3 lists theabbreviations for each CMA used throughout the remainder of thisthesis.In order to estimate wasteful commuting the following data wererequired:• Total CMA population;• Central city population;26• Total CMA employment;• Central city employment;• Central city land area;• Land available for urban development (0);"• Radius of the CMA, (k1);• Radius of the central city, (k2);• Shape of the CMA and cental city; and• Observed commuting, (D).Population, employment and land area for each CMA and central citywas obtained from the 1981 Census of Canada (Table 3.2). The 1981Census represents the first time that Statistics Canada tabulatedemployment by place of work, as well as place of residence. Thus,1981 represents the first opportunity to estimate employment26 Taking the case of the Vancouver CMA as an example impliesthe central city would include data for the City of Vancouver only,while the metropolitan area includes Surrey, Richmond, Burnaby etc.27 (27r - 0) radians are unavailable.- 59 -FIGURE 3.2LOCATION OF CMAs RANKED BY 1981 POPULATION1^Toronto2^Montreal3^Vancouver4^Ottawa-Hull5^Edmonton6^Calgary7^Winnipeg8^Quebec City9^Hamilton10^St Catherines11^Kitchener-Waterlo12^London13^Halifax14^Windsor15^Victoria16^Saskatoon17^Regina18^Saint John's19^Sudbury20^Chicoutimi-Jonquiere21^Thunder Bay22^Saint John23^Charlottetown1 0Source: 1981 Census of Canada— 60 —FIGURE 3.3CMA AbbreviationsAbbreviation CMATor^TorontoMtI MontrealVan VancouverOtt^OttawaEdm EdmontonCgy CalgaryWpg^WinnipegQue Quebec CityHam HamiltonSC^Saint Catherines-NiagaraKW Kitchener-WaterlooLon LondonHal^HalifaxWsr WindsorVic VictoriaSas^SaskatoonReg ReginaSJs Saint John'sSud^SudburyChi Chicoutimi-JonquiereTB Thunder BaySJ^Saint JohnCha CharlottetownTable 3.2:^Population and Employment Data Used to Estimate DensityGradientsCMA Population^ EmploymentCMA^CC^CC Share^CMA^CC^CC ShareAreaCC CMACalgary 625,966 592,743 94.7%5 325,205 300,310 92.3%5 504.96 5055.96Charlottetown 44,999 15,282 34.0% 18,380 13,460 73.2% 6.99 550.73Chicoutimi 135,172 55,465 41.0% 44,715 40,845 91.3%5 147.42 1132.54Edmonton 657,057 532,246 81.0% 339,075 289,035 85.2% 321.65 4142.51Halifax 277,727 176,871 63.7% 127,660 107,180 84.0% 120.92 2508.10Hamilton 542,095 306,434 56.5% 228,435 153,545 67.2% 122.82 1358.50Kitchener 287,081 189,162 65.9% 132,825 87,460 65.8% 199.79 823.64London 283,668 254,280 89.6%5 131,955 116,675 88.4%5 162.28 1601.91Montreal 2,828,349 980,354 34.7% 1,265,055 615,180 48.6% 158.31 2814.43Ottawa 717,978 351,388 48.9% 347,975 259,675 74.6% 139.44 3998.03Quebec 576,075 166,474 28.9% 237,260 103,485 43.6% 89.05 2817.96Regina 173,226 162,613 93.9%5 79,565 74,605 93.8%5 109.83 3421.58Saint John 114,048 73,389 64.3% 45,865 40,795 88.9%5 322.71 1476.08Saskatoon 175,058 154,210 88.1%5 71,730 67,395 94.0%5 122.04 4749.30St. Catherines 304,353 124,018 40.7% 126,110 78,975 62.6% 94.43 1068.07St John's^154,820 83,770 54.1% 60,460 51,370 85.0% 35.10 1127.47Sudbury 149,923 89,773 59.9% 58,180 45,630 78.4% 262.73 2379.84Thunder Bay 121,379 109,365 90.1%5 54,000 50,785 94.0% 323.46 2032.38Toronto 2,998,947 599,217 20.0% 1,571,455 545,160 34.7% 97.15 3742.94Vancouver 1,268,183 414,281 32.7% 646,435 276,215 42.7% 113.13 2786.22Victoria 233,481 64,379 27.6% 105,130 60,820 57.9% 18.78 488.52Windsor 246,110 192,083 78.0% 96,080 84,975 88.4%5 119.76 768.87Winnipeg 584,842 562,059 96.1%5 284,785 268,275 94.2%5 571.60 2310.03Mean 586,980 271,733 60.2% 278,188 162,254 75.2% 181.06 2311.11Source: 1981 Census of Canada.5 m a CMA with an extreme population or employment split. This is likely to make estimatesof the density gradient parameters less reliable according to Edmonston, Goldberg and Mercer[18] and White [60].- 62 -density gradients for a large sample of Canadian cities. 28Section 3.2 described differences between the population andemployment gradient estimation techniques of Mills and Edmonston.The differences revolved around measurement of 0 and k, and k 2. TheMills estimates of 0, described in equation 3.29, are given inTable 3.3.29 Of course, in order to estimate 0 using equation3.29, an estimate of the CMA radius was required. The "measuredradius" in Table 3.3 was obtained from 1981 Census maps using alarge compass.In contrast to the Mills method, Edmonston devised a system tocalculate the radius of each CMA based on the shape of the CMA.Table 3.3 presents the coded shape of each city and the implied"calculated radius" for each CMA. 3° On average, both the measuredand calculated radii are substantially greater than the values ofF (where population density is estimated to decline to 100 peopleper square mile) presented in Chapters 4 and 5.28 Unfortunately the published census data does not permitestimation of employment gradients by industry or occupation.Disaggregated gradients would be of interest to determine whichjobs are most suburbanized.29 The figures in Table 3.3 have been multiplied through byv. Thus a figure of 6.28319 (i.e. 2r) represents a complete circleof 360 degrees available for development. Remember 2v - 0 wasdefined as the area of land NOT available for development.3° Table 3.3 also presents the "circle radius" for each CMA.This was calculated based on land area published in the census andassuming that the city was a complete circle of 360 degrees.Cities where this figure is substantially different from the otherradius measures are likely cities that are not well characterizedas a circle.- 63 -Table 3.3: Estimation of CMA Radius and the Parameter 0CMA^Shape^ 0CMA^CC^Mills^Edmonston CalculatedRadiusMeasured CircleCalgary 1 x 2 1 x 2 6.25722 6.28319 40.2 38.5 40.1Charlottetown Square 1 x 2 4.30258 5.06145 16.0 12.7 13.2Chicoutimi 1 x 2 1 x 4 5.03978 6.28319 21.2 16.5 19.0Edmonton 1 x 2 Square 5.79842 6.28319 37.8 36.1 36.3Halifax 1 x 2 1 x 2 4.21441 3.66519 34.5 27.1 28.3Hamilton Hexagon Square 5.04793 4.88692 23.2 20.7 20.8Kitchener 1 x 3 Hexagon 5.83645 6.28319 16.8 15.5 16.2London 1 x 2 Hexagon 5.80139 6.28319 23.5 21.7 22.6Montreal Hexagon 1 x 2 5.49693 6.28319 32.0 29.9 29.9Ottawa Square Hexagon 5.84080 6.28319 37.0 31.0 35.7Quebec Square 1 x 2 5.57328 6.28319 31.8 28.8 29.9Regina 1 x 2 Square 5.91969 6.28319 34.0 31.7 33.0Saint John 1 x 3 1 x 2 3.90368 3.75246 27.5 20.8 21.7Saskatoon Square Square 5.99644 6.28319 39.8 38.7 38.9St Catherines Square Square 3.86807 2.35619 23.5 17.7 18.4St John's 1 x 4 1 x 2 3.46780 2.61799 25.5 16.5 18.9Sudbury Square 1 x 2 5.65955 6.28319 29.0 26.4 27.5Thunder Bay 1 x 3 1 x 2 4.51640 4.01426 30.0 22.1 25.4Toronto 1 x 2 1 x 2 4.14443 3.49066 42.5 34.3 34.5Vancouver 1 x 2 1 x 2 3.66367 2.87979 39.0 29.6 29.8Victoria 1 x 4 Square 3.19033 3.05433 17.5 12.0 12.5Windsor 1 x 2 1 x 2 3.84435 2.96706 20.0 13.6 15.6Winnipeg 1 x 3 Hexagon 5.68798 6.28319 28.5 23.4 27.1Mean 4.91616 4.96281 29.2 24.6 25.9Two sources of data provided information on observed commutingbehaviour for a sample of Canadian cities:• The Urban Concerns Survey (1978);• The Vehicle Survey Data (1975).3.3.1 The Urban Concerns Survey (UCS)In 1978 Canada Mortgage and Housing (CMHC), and the Ministry ofState for Urban Affairs (MSUA) conducted a survey of 11,061households located in urban areas with population greater than100,000. The survey covered all 23 cities listed in Figure 3.2 andincluded a question on the distance to work and the time it took totravel the given distance. The survey used a stratified randomsample to ensure broad geographic coverage within metropolitanareas. The sample was stratified in each CMA on the followinglocation criteria:• Central city;• Mature suburbs;• New Suburbs;• Exurban;The strata were of different size for each CMA based upon thepopulation distribution recorded by the 1976 Census. Nationally,the sample distribution was: central city 28 percent, maturesuburbs 29 percent, new suburbs 30 percent and exurban 10 percent.Not all strata types were present in every city. Some CMAs had noexurban observations, for example, while Charlottetown was not- 65 -Table 3.4: Summary of Observed Commuting DataOne Way CommuteUCS^VS^UCS^VS^kilometres minutesAve SpeedUCS VSkm per hourSample SizeUCS^VS# #Calgary 10.64 12.28 23.9 20.1 26.7 36.7 152 528Charlottetown 4.22 10.5 24.1 230Chicoutimi 8.30 14.0 35.6 150Edmonton 10.49 12.36 25.4 19.0 24.8 39.0 207 503Halifax 7.97 19.0 25.2 203Hamilton 15.63 22.5 41.7 175Kitchener 12.09 17.6 41.2 184London 8.42 18.0 28.1 146Montreal 12.23 13.32 24.6 22.9 29.8 34.9 300 1,021Ottawa 10.40 10.89 21.6 18.5 28.9 35.3 269 464Quebec 11.28 11.49 21.2 16.1 31.9 42.8 227 74Regina 8.53 17.4 29.4 181St. John 11.10 17.1 39.0 160Saskatoon 7.95 16.0 29.8 222St. Catherine 10.59 15.9 40.0 122St. John's 8.19 20.3 24.2 246Sudbury 13.61 18.0 45.4 132Thunder Bay 10.12 17.1 35.5 116Toronto 12.25 15.27 25.2 24.2 29.2 37.9 318 952Vancouver 13.57 14.11 24.0 21.0 33.9 40.3 206 846Victoria 10.64 20.6 31.0 205Windsor 10.17 16.1 37.9 167Winnipeg 11.09 24.1 27.6 217Observations 4,535 4,388Mean 10.41 19.6 32.2 197Sub-sample 11.55 12.82 23.7 20.3 29.3 38.1 240 627Notes: UCS a Urban Concerns Survey; VS a Vehicle Survey; Sub-sample E the mean values for theseven city sub-sample of the UCS that allows for comparison of the UCS and VS.- 66 -stratified at all. Most CMAs had some of each strata type althoughproportions were often different from the national share.A summary of observed commuting data is given in Table 3.4. Thetotal number of responses that included a useable answer for thecommuting distance question was 4,535. The average number ofresponses for each CMA was only 197. From this data the averagecommute in 1978 was estimated to be 10.4 kilometres. The averagecommuting speed was 32 kilometres per hour. This includesinformation for all modes of transit.3.3.2 Vehicle Survey (VS)A second source of observed commuting data was obtained from avehicle survey conducted by a private consulting company forEnvironment Canada in 1975. The survey consisted of 7,838observations. In this case each vehicle represented an observationso the number of separate households was approximately 5,800. Thesurvey only included seven cities: Calgary, Edmonton, Montreal,Ottawa, Quebec City, Toronto and Vancouver.A summary of observed commuting data from the vehicle survey isalso included in Table 3.4. The average observed commute was 12.8kilometres or nearly 1.3 kilometres more than the average obtainedfrom the urban concerns survey for the same seven CMAs (sub-samplemean in Table 3.4). Velocity was more than 9 kilometres per hourgreater in the vehicle survey. The differences are not surprisingconsidering that the urban concerns survey includes all types of- 67 -transit while the vehicle survey included only the primaryautomobile in each household. The vehicle survey probably oversampled inner city locations because it was not geographicallystratified.Given these considerations the results of the vehicle survey wereconsidered as confirmation of the veracity of observed commutingdata from the urban concerns survey. To the extent that the urbanconcerns survey may be an under-estimate of observed commutingdistance the estimates of wasteful commuting in Chapter 5 will bebiased downward and the Hamilton model will be biased in favour offinding the monocentric model acceptable.3.3.3 Estimating the Zeros of Non-Linear EquationsMills [43] employed a Newton-Raphson algorithm to solve for y inequation 3.26. 31 This thesis employed two alternative algorithmsto solve for the zeros of the non-linear equations:• DRZFUN based on Muller's method; and• ZERO1 based on Bus and Dekker's method.Two methods were used in order to check for consistency. The firstsolution method was based on an algorithm developed by Muller [44].It is a double precision analogue and had the advantage of notrequiring initial estimates of the roots. Using this method singleroots are generally accurate to within five significant figures and31 Some researchers have had problems with convergence usingthe Newton-Raphson algorithm.- 68 -converge within 20 or 30 iterations.The second method proceeds iteratively by a method combining linearinterpolation, rational interpolation and bisection based on thework of Bus and Dekker. The method will find one root of thespecified function but the root must lie within a user specifiedinterval. Both methods produced estimates of y and 6 that wereidentical when reasonable starting intervals were used for the Busand Dekker algorithm.3.4 SummaryThis chapter reviewed a method for estimating wasteful commutingdeveloped by Hamilton [25], [26]. The model estimates the averagedistance of people and jobs from the CBD using the parameters fromnegative exponential employment and population density gradients.The difference between the average distance of homes and jobs fromthe CBD is interpreted as the optimum average commute. This isthen compared with observed commuting for a sample of cities.Hamilton employed density gradient parameters estimated by Macauley[38] to operationalize his model. There were no readily availablegradient estimates for a large sample of Canadian CMAs so gradientshad to be estimated for this thesis. Three variations of the two-point technique developed by Mills', were discussed in Section 3.2.Section 3.3 described the data required to estimate the modelincluding two sources of observed commuting data.- 69 -CHAPTER 4DENSITY GRADIENT ESTIMATESThis chapter presents estimates of population and employmentdensity gradients for 22 Canadian Census Metropolitan Areas (CMA)and one Census Agglomeration (CA). 1 The key inputs into the modeldeveloped by Hamilton [26] are the parameters from each city'semployment and population density gradient. Hamilton [26] usedestimates of Do, E0, y and 6 provided by Macauley [38] who, in turn,employed a modified version of Mills' [43] two7point technique toestimate the four gradient parameters.There are four sections in this chapter. The first section reviewsprevious two-point estimates for several different sets of cities.The second section presents estimates of population densitygradients for 23 Canadian cities in 1981. Section 4.3 summarizes1981 employment gradient estimates for the same 23 cities. Thefinal section summarizes the chapter.4.1 Previous Two-Point Density Gradient EstimatesIn his pioneering work Studies in the Structure of the UrbanEconomy [43] Mills estimated 360 density functions for 18metropolitan areas in the United States. 2 Cities with irregularly1 Charlottetown was included despite its status as a CA fortwo reasons: (i) observed commuting data were available forCharlottetown; and (ii) to ensure complete provincial coverage.2 Estimates were made for 4 years (1948,1954,1958,1963), and5 sectors (population, manufacturing, retailing, services andwholesaling). Thus, 18 x 5 x 4 = 360.- 70 -shaped boundaries (e.g. San Francisco) or cities considered to bepolycentric (e.g. New York-Newark) were excluded from Mills'sample. Mills' average gradients for each sector and year arepresented in Table 4.1.Table 4.1 : Average Density Gradients, Mills and Macauleyy / YEAR 1948 1954 1958 1963 1970/72 1977/80Population 0.58 0.47 0.42 0.38 0.29 0.24Manufacturing 0.68 0.55 0.48 0.42 0.34 0.32Retailing 0.88 0.75 0.59 0.44 0.35 0.30Services 0.97 0.81 0.66 0.53 0.41 0.38Wholesaling 1.00 0.86 0.70 0.56 0.43 0.37Mills' important conclusions included:• Density gradients in all sectors tended to flatten overtime.• Gradient estimates varied much less than central densityestimates. Gradients ranged from 0.20 to 1.00 whilecentral population and employment densities ranged from6,000 to 60,000 persons per square mile.• y was inversely related to metropolitan population and D,was directly related to metropolitan population. Centraldensity estimates appeared to be more sensitive todifferences in total population than gradient estimates.It is important to recognize Mills' criteria for inclusion ofcities because Hamilton [26] employed a sub-sample of Mills' 18cities in his own work. Additionally, the small size of theCanadian urban system made strict application of Mills' arbitrarycriteria inappropriate for this thesis. Irregularly shapedCanadian CMAs would likely include Vancouver, Montreal, Halifax and- 71 -St. John's using Mills' criteria. 3 If density gradients cannot beefficiently estimated for irregularly shaped CMAs using Mills'(Macauley's) methodology, then including such cities reduces thecomparability between the Canadian results presented here andHamilton's results.Macauley [38] updated Mills' estimates in 1985. 4 The figures for1970 and 1980 given in Table 4.1 (above) are Macauley's estimates.While Macauley's mean y, based on the same 18 SMSAs used in 1972 byMills, was 0.24 Hamilton [26] employed a sub-sample of Macauley'sestimates with an average population gradient of only 0.22. Theconclusions of Macauley regarding her gradient estimates were: Density gradients in all sectors continued to flattenover time but at a decreasing rate; and Employment and population gradients appeared to beconverging.^Macauley's result combined with Mills'results seems to indicate that, initially, populationsuburbanized and then employment followed.Research has shown that Mills' two point gradient estimates arebiased upwards, (i.e. are too steep), because of Mills' simplifyingassumption that population density declines to zero at the urbanboundary [17], [38], [42]. Mills believed the bias would be small3 Mills gave no specifics regarding what constitutes anirregular city shape, so it is not clear which Canadian CMAs wouldbe excluded by Mills but the four just listed would be primecandidates. Making any generalizations about Canadian urbanstructure while ignoring Montreal and Vancouver is clearlynonsensical. For this reason, it was important to include densitygradients estimated using Edmonston's method. Edmonston's methodattempts to control for city shape.4 Data for Macauley's employment gradient estimates wereavailable for 1972 and 1977 rather than 1970 and 1980.- 72 -for typical values of y and k (equation 4.8).^Macauley [38]verified empirically Mills' conjecture.^In all cases Mills'estimates of y were greater than or equal to her own but thedifference between the estimates was small and not significantlydifferent from zero. 5There was one important caveat to Macauley's conclusion that theMills method does not seriously bias estimates of y and Do.Macauley's results held only when SMSA level data were employed.Macauley found Mills' estimates 68 percent larger than her ownestimates if Urbanized Area (UA) data were employed. 6 EdmonstonGoldberg and Mercer, (pg. 213) [18], argued that the StatisticsCanada definition of CMA is more analogous to the U.S. definitionof UA than SMSA. In the light of Macauley's results it isimportant to test the Edmonston, Goldberg, Mercer claim that CMAsapproximate UAs, because the statistical unit employed may have thepotential to radically alter density gradient parameter estimates.The Mills (Macauley) technique requires that the urban boundary be5 Macauley (pg. 259) showed the bias of Mills' estimates is:N(00) -N(k) - (1 +yk) e-Yk.N(co)6 Urbanized Area (UA) is a more rigid concept of "urban".SMSA uses counties as the basic unit of aggregation, while UA onlyincludes areas meeting specific density criteria. The average UAin Macauley's sample was 462 square miles with radius, (k), 13.2miles. For the same cities the average SMSA area was 3539 squaremiles with radius (k) 33.9 miles (pg. 254) [37], [38].- 73 -regularly shaped.' Cities not meeting this requirement were simplyexcluded by Mills and Macauley. Edmonston [17] attempted toovercome this limitation and adapted the two-point method toinclude irregularly shaped cities. 8 Edmonston found a simplecorrelation of 0.92 between estimates of y using his modifiedmethod and those obtained using Mills' method. The correlation was0.98 for estimates of Do. Such close agreement between the twotechniques for cities with regular boundaries indicates thatEdmonston's method should not incorporate any new bias inattempting to include irregularly shaped cities.Edmonston, Goldberg and Mercer [18] applied Edmonston's modifiedgradient estimation method to a sample of 20 Canadian CMAs usingdata for the period 1950-1976. Prior to 1981 data required toestimate employment gradients was not available. Edmonston et.al .compared Canadian estimates with U.S. estimates for the sameperiod. Mean values for y and Do are presented in Table 4.2.Several patterns are evident from Table 4.2:• Canadian cities had higher central densities than U.S.cities in all periods but Canadian and U.S. gradientsconverged since 1970;• Like their U.S. counterparts, Canadian urban populationshave suburbanized (i.e. density gradients flattened) ata decreasing rate since the 1950's; and7 White (pg. 299) [60] showed that an irregularly shapedboundary imparts a small upward bias to y. However the upward biasmay be partially or totally offset by a downward bias that resultswhen the central city is not centred within the CMA. The magnitudeand direction of the net bias depends upon the degree ofirregularity and eccentricity respectively.8 See Chapter 3 of this thesis for specific methodologicaldetails.- 74 -•^Estimates of y and Do for Canadian cities exhibited thesame relationship with total metropolitan population asprevious U.S. estimates: i.e. gradients tended to beflatter in larger cites while central density tended behigher in larger centres. 9Table 4.2: Average Values For y and D,Edmonston Goldberg and Mercer [18]YEARYCDN USDoCDN US1950/51 0.93 0.76 50,000 24,0001960/61 0.67 0.60 33,000 17,0001970/71 0.45 0.50 22,000 13,0001975/76 0.42 0.45 20,000 11,0004.2 Population Gradients: New Canadian EstimatesPopulation gradients for 23 Canadian cities are presented in table4.3. Estimates using Mills' technique as well as estimates usingmethods incorporating two slight modifications to Mills' technique(Macauley and Edmonston) are included. 1° Macauley estimates areconsidered the base case estimates throughout this thesis. Mills9 Consider the following results for 1976:CMA Population250,000250-499,000? 500,000D0 ^0.42^15,000^0.47^17,0000.39^26,000Curiously, middle sized cities had y exceeding the smallest groupof CMAs but the sample size in each group was very small.10 Chapter 3 outlines the methodological differences among theMills, Macauley and Edmonston estimation techniques.- 75 -and Edmonston estimates are included for two reasons:1. To check the accuracy of the estimates obtained with theMacauley technique; and2. To gauge the sensitivity of Hamilton's wasteful commutingmodel to the particular density gradient parametersemployed.It turns out that estimates obtained with the Mills and Macauleymethods are virtually indistinguishable. Edmonston gradients aresteeper with higher central densities than Macauley estimates, onaverage .114.2.1 Macauley EstimatesHamilton [26] used Macauley's estimates of y, D„ 6 and E, tocalculate his measure of wasteful commuting for 14 U.S.metropolitan areas. The same method employed by Macauley is usedto estimate y and D0, for the 23 Canadian CMAs listed in Table 4.3.The population gradient estimates presented in Table 4.3 and Figure4.1 seem reasonable. 12 The average y for 1981 is 0.3290 with amedian of 0.2835. Values range from a low of 0.1306 in Toronto toa maximum of 0.7179 in Regina. The figures for y are consistentwith the results obtained by Mills [43] and Macauley [38] presented11 The implications for estimates of wasteful commuting arediscussed in the next chapter.12 The observations in Figure 4.1 and all subsequent figuresare arranged in descending order according to the total CMApopulation (i.e. from Toronto to Charlottetown).- 76 -Table 4.3: 1981 Population Density Gradient Parameter Estimates.CMA1(MillsDoMacauleyD,EdmonstonD,Calgary 0.3460377 11978.88 0.3460159 11977.53 0.3828995 14606.40Charlottetown 0.6701036 4696.32 0.6699682 4695.64 0.8614193 6589.55Chicoutimi 0.2008953 1082.47 0.1844091 1011.68 0.1657379 780.07Edmonton 0.2835978 9113.79 0.2834614 9107.39 0.3031348 9611.40Halifax 0.3137889 6488.68 0.3137013 6486.59 0.3488784 9230.50Hamilton 0.2634134 7451.39 0.2586705 7312.23 0.3014035 10221.54Kitchener 0.2585306 3295.85 0.2271979 2847.02 0.2629334 3465.41London 0.4929754 11883.06 0.4928113 11876.55 0.5338795 12869.68Montreal 0.1726244 15332.64 0.1687077 15081.19 0.1756098 14352.38Ottawa 0.2193046 5911.99 0.2187339 5897.66 0.2482884 7072.34Quebec 0.1940538 3892.36 0.1917721 3862.96 0.2064706 3980.87Regina 0.7179666 15084.21 0.7179664 15121.40 0.7594731 15915.71Saint John 0.1889714 1043.29 0.1796146 984.39 0.2077067 1411.02Saskatoon 0.5558713 9020.63 0.5558713 9020.63 0.5917340 9755.63St. Catherines 0.2541840 5083.70 0.2500605 5016.86 0.2377767 7916.03St. John's 0.5036675 11325.58 0.5036489 11325.15 0.5661072 18969.25Sudbury 0.2038195 1100.47 0.1991318 1073.00 0.2163606 1142.33Thunder Bay 0.3647303 3575.15 0.3645032 3571.46 0.4006837 4861.30Toronto 0.1328807 12776.96 0.1306086 12666.03 0.1519010 20519.44Vancouver 0.1887848 12336.72 0.1879939 12300.64 0.1988397 17750.81Victoria 0.3683036 9927.21 0.3643718 9861.18 0.4171162 13857.68Windsor 0.3877557 9625.49 0.3852291 9537.88 0.4772608 19110.81Winnipeg 0.3736645 14356.36 0.3730519 14313.34 0.3700665 12768.70Sample Mean 0.3328660 8103.62 0.3290220 8041.24 0.3645950 10293.70NOTES: and D, are parameters from the negative exponential density gradient:D(x) = D, e- m. D(x) represents the population density at any distance x from thecentral city; D, represents the population density (persons per square kilometre) at thecity centre; represents the rate at which density declines as we move away from thecity centre.- 77 -Slope0.8FIGURE 4.11981 POPULATION GRADIENT ESTIMATESMACAULEY ESTIMATION TECHNIQUETor Van Edm Wpg Ham KW Hal Vic Reg Sud TB ChrMt! Ott Cgy Que SC Lon Wsr Sas SJs Chi ^SJCMAin Table 4.1. 13 A median value to the left of the mean indicatesa distribution with a heavy righthand tail (i.e. there are moreextremely large gradients than extremely small gradients).The Macauley type estimates of y presented in Table 4.3 areinversely related to CMA population.'' Regression of y on CMApopulation yields:"y = 0.38555 - 9.63 x 10 -8 POP R2 - .2157(4.01 x 10 -8 )Based on the above estimate, a population increase of 100,000 (i.e.17 percent of the mean 1981 CMA population) would flatten y by only.00963 (i.e. 2.0 percent of the mean y).There are, however, some notable exceptions to the underlying13 The mean y is reasonably close to Edmonston, Goldberg andMercer's estimate for 1976 (Table 4.2). Remember, however, that EGMused a slightly different methodology as well as a smaller sampleof CMAs (20). The implied reduction in y from 0.42 to 0.32 in the5 years between Edmonston's estimates and this work is more rapidthan expected, a priori. The difference is due, partly, tomethodological differences. The mean y obtained using Edmonston'smethod with 1981 Canadian data (0.3646) was somewhat higher thanthe Macauley estimate (0.3290), (Table 4.3).14 The estimated equation is obviously simplistic. WhileMills and Macauley found y tended to be smaller in cities withhigher populations, Alperovich [5] found the opposite once hecontrolled for the land area of the metropolitan area:"Our results show that holding land supply constant, citieswhich are more populous tend to be less suburbanized.Suburbanization is primarily associated with high supply ofland and not with increased population per se" (pg. 293) [5].15 Standard error in parentheses.- 79 -inverse relationship between y and CMA population. Saint John,Chicoutimi and Sudbury are ranked 22nd, 20th, and 19th in terms ofCMA population but had the third, fourth, and seventh flattestpopulation gradients (Figure 4.2). Further, Regina's gradientappears excessively steep (y = 0.7179) compared with Saskatoon (y= 0.5559) given that the 1981 population in the two CMAs is almostidentical (Regina = 173,226, Saskatoon = 175,058).Two of the apparently anomalous cities, Chicoutimi and Saint John,violate the Mills' assumption of circularity. Both city's boundaryresemble an elongated rectangle, rather than a circle. The resultmay be a poor estimate for y in each city. 16 The discrepancybetween Regina and Saskatoon may result from the small proportionof the total CMA population outside the central city in Regina(i.e. what White [60] termed an extreme split). As a rule of thumbEdmonston, Goldberg and Mercer [18] suggested that a reliableestimate of y cannot be obtained using the two-point method for anymetropolitan area with a suburban population of less than 10,000.In 1981, Regina had a suburban population of 10,613 whileSaskatoon's suburban population was 20,848.' 7 No obviousexplanation for Sudbury's anomalous gradient is apparent.Using the Macauley estimation method, the average population16 Recall White's [60] reminder that irregular boundaries andeccentricity will bias estimates of y in opposite directions.17 The term suburban refers to population within the CMA butoutside of the central city. For example, a resident of North Yorkwould be a suburban resident of the Toronto CMA or a resident ofBurnaby would be a suburban resident in the Vancouver CMA.- 80 -Rank23Pop Grad-+-- Empi Grad  CMA Pop5 -3-1FIGURE 4.2GRADIENT SLOPE RANK1981, MACAULEY ESTIMATION TECHNIQUETor^Van Edm Wpg Ham KW^Hal^Vic^Reg Sud^TB^ChrMtl^Ott Cgy Que^SC^Lon Wsr Sas SJs^Chi^SJCMAdensity at the centre of Canadian CMAs is 8,041 people per squarekilometre (20,827 per square mile). Individual estimates rangefrom a low of 984 per square kilometre (2,550 per mile) in SaintJohn, to 15,121 per square kilometre (39,164 per mile) in Regina(Figure 4.3). The median value, 9,021 persons per squarekilometre, was greater than the mean value, indicating adistribution with a heavy lefthand tail (i.e. there are moreextremely small values than extremely large values in thedistribution)." These central density estimates are notinconsistent with those provided by Mills [42], Macauley [38], andEdmonston, Goldberg and Mercer [18]. The mean central densityreported in Table 4.3 is almost identical to that reported byEdmonston, Goldberg and Mercer for 1976 (Table 4.2).As expected, central population density is positively related tototal CMA population. Regressing D, on CMA population yields:Da - 6350.9 + 2.88 A: 10-3 POP^R2^.2411(1.11 x 10-3)Based on this estimated equation, an increase in CMA population of100,000 would increase central density by 288 people per squarekilometre (i.e. a 17 percent increase in population would yield a3.6 percent increase in D0 , at the mean). 19 D, appears slightly" This is the opposite result compared to the estimates ofy which were skewed toward larger values.19^The estimate of 2.88 x 10-3 is remarkably close toAlperovich's figure of 2.46 x 10-3 (pg 292) [5]. However, theequation estimated by Alperovich controlled for other variables,- 82 -more sensitive than y to changes in total CMA population. 2° Millsalso found that D, was more sensitive than y to changes inpopulation. (Chapter 3) [43].As with the estimates of y, there are some obvious exceptions tothe overall positive relationship between D, and CMA population.Figure 4.4 illustrates that the population central densityrelationship breaks down, particularly in the middle of the urbanhierarchy. 21 The cluster of CMAs including Quebec City (Que),Hamilton (Ham), St. Catherines (SC), and Kitchener-Waterloo (KW)ranked far too low, while Windsor (Wsr), Victoria (Vic), Saskatoon(Sas), Regina (Reg), and St. John's (SJs) rank too high relative totheir population ranking. The largest and smallest CMAs behavecloser to expectations, with population and central densityrankings much more equal.Most researchers who have employed the two-point method to estimatepopulation density functions have been more interested in estimatesof y than D0. Even the technique's pioneer, Mills [43] did notdiscuss his estimates of central density in detail. While at firstit seems perverse that the Canadian estimates presented above ranksuch as metropolitan land area. In an earlier footnote it waspointed out that Alperovich found controlling for land areareversed the effect of total metropolitan population on y. Thisdoes not seem to be the case for estimates of Do.20 A 17 percent increase in population reduced y by only 2.0percent.21 Observations below the diagonal line in Figure 4.4 have acentral population density that is higher than expected given theCMA population while observations above the diagonal line have acentral population density that is lower than expected.- 83 -FIGURE 4.3CENTRAL POPULATION DENSITY ESTIMATE1981, MACAULEY ESTIMATION TECHNIQUEPersons Per Square KilometreTor Van Edrn Wpg Harn KW Hal Vic Reg Sud TB ChrMU Ott Cgy Que SC Lon Wsr Sas SJs Chi SJCMAFIGURE 4.4CENTRAL DENSITY RANK1981, MACAULEY ESTIMATION TECHNIQUERank2321 - — Pop Den5319 - Empl Den17 - CMA Pop15 - el.1 1^i^1^1^1^i^i^i,^1^1^I^i^1^I^1^1Tor^Van Edm Wpg Ham KW^Hal^Vic^Reg Sud^TB^ChrMtl^Ott^Cgy Que^SC^Lon Wsr Sas SJs^Chi^SJCMAthe central population density in smaller CMAs (e.g. Regina and St.John's) among the highest in Canada, this is not entirelyinconsistent with Mills' results (pg. 40, Table 11) [43]. Mills'results showed smaller cities such as Columbus and Toledo had amongthe highest central population densities, ranking ahead of largercities including Boston and Pittsburg.There are at least three potential explanations for the divergenceof D, from the rank-size relationship:1. In order to estimate D0, the radians of land availablefor development, 0, must be known. 22 Most proceduresused to measure 0 are ad hoc, contributing an element ofuncertainty to estimates of D0. In contrast 0 is notrequired to estimate y. In the results presented abovein many of the CMAs in which D, seemed anomalous,(Figures 4.3 and 4.4), 0 is not equal to 2r (e.g.Victoria, Windsor, Hamilton and St. Catherines). 232. Estimates of D, for CMAs with "extreme population splits"tend to exceed the expected central density based on therank-size rule. Consider Table 4.4 below. 243. Three of five cities with unexpectedly low values for D0are part of a dual (polycentric) CMA: Ottawa-Hull, St.Catherines-Niagara Falls and Kitchener-Waterloo. Theother two CMAs both contain a large secondary city:Hamilton (Burlington) and Quebec (Levi).22 Estimation methods are described in detail in Chapter 3 ofthis thesis. For the present it is sufficient to recall thefollowing equation:N(k2) y2Do(1+yk2)e-Yk2]One of the differences between the Edmonston and Macauley estimateswas the measurement of 0.23 Recall that 2r is the maximum number of radians available:i.e. 2r = 360 ° . Thus, if a CMA is circular, 0 = 2r and 0 does nothave to be estimated in order to calculate D0.24 White suggested that ? 80 percent of the population in thecentral city constitutes an "extreme split" [60].— 86 —Table 4.4 : Cities With Small SuburbanPopulation and Large Values For D,CMA^Suburban^Pop.% of CMALondon 29,338 10.3Regina 10,613 6.1Saskatoon 20,848 11.9Winnipeg 22,783 3.9One test of the reasonableness of the estimates for y and Do is toexamine the predicted distance at which population density declinesto 100 people per square mile. Both Macauley [38] and Hamilton[26] used this criterion and both argued that it was reasonable toexpect the predicted distance (denoted F) to be less than or equalto the political (statistical) boundary used in the census. Whilethe cutoff value of 100 people per square mile is arbitrary, it isalso reasonable: densities below 100 people per square mile are notnormally thought of as urban. Figure 4.5 compares the predicteddistance, (F), with the measured radius, (G), taken from StatisticsCanada maps, for each CMA. 25The average predicted radius (F) for the Canadian sample of citiesis 18.6 kilometres while the average measured radius, (G), was 29.2kilometres. In only three cases was F beyond G: Kitchener (by25 If y and Do are known and D(x) is assumed to be equal to100 persons per square mile (or 38.6 persons per square kilometre)F can be calculated by substituting into:D(x) = Doe-'"`and solving for x=F.- 87 -FIGURE 4.5^.kMEASURED VERSUS CALCULATED CMA RADIUS2.1 km), Montreal (by 3.4 km), and Toronto (by 2.0 km). A priori,it was expected that F would more closely approximate the politicalboundary, G, in larger cities. Figure 4.5 partially confirms thisexpectation. However, for the Canadian sample of cities, aregional pattern seems to dominate the size relationship. In moredensely populated central Canada, F and G converge, while thegreatest absolute differences occur in the prairies. 26 The latterresult is not surprising considering that the 1981 legal(political) civic areas of Calgary, Edmonton and Saskatoon arelarger than Montreal, Toronto or Vancouver.4.2.2 Edmonston EstimatesPopulation gradients obtained using Edmonston's method arepresented in Table 4.3 and Figure 4.6. On average, estimates usingEdmonston's method are steeper than the Macauley estimates, andexhibit higher central population density. The mean y is 0.3646and the median is 0.3031, indicating a distribution skewed towardlarger values of y (i.e. skewed right). Estimates range from0.1519 for Toronto to 0.8614 for Charlottetown. The mean estimatedy is smaller than the 0.42 reported by Edmonston, Goldberg andMercer in 1976 using the same methodology. While continuedsuburbanization is expected to flatten the density gradient overtime the implied rate between 1976 and 1981 appears excessively26 The five smallest absolute differences were: Toronto (1.96km), Kitchener (2.13 km), Victoria (2.29 km), Hamilton (2.93 km),and Montreal (3.37 km). The five largest absolute differenceswere: Saskatoon (29.99 km), Regina (25.68 km), Calgary (23.62 km),Edmonton (18.53 km), and Halifax (18.17 km).- 89 -Slope1 .0FIGURE 4.6POPULATION GRADIENT ESTIMATES1981, EDMONSTON ESTIMATION TECHNIQUETor Van Edm Wpg Ham KW Hal Vic Reg Sud TB ChaWI Ott Cgy Que SC Lon Wsr Sas SJs Chi^SJCMArapid. Two factors most likely account for this: The 1981 estimates presented in Table 4.3 include 3 CMAsnot included in the EGM study; and Several boundary changes occurred between 1976 and 1981reducing the comparability of the samples stillfurther. 27The mean difference between the Macauley and Edmonston estimates ofy is 0.03557, equivalent to 9.8 percent of the mean Macauleygradient. The difference is significantly different from zero, atthe a = .01 level (t = 4.10). 28 In only three instances areMacauley estimates steeper than Edmonston estimates: Chicoutimi,St. Catherines and Winnipeg. The greatest absolute differencesoccur in Windsor (24.9 percent), Charlottetown (22.0 percent),Hamilton (14.2 percent) and Victoria (12.6 percent).Differences in the rank ordering of the CMAs are more likely toinfluence estimates of optimum commute (using Hamilton's method)than absolute differences between Macauley and Edmonston estimates.27 One of the supposed advantages of the two-point method isthat estimates are not affected by changes in political boundariesover time. The claim of insensitivity to political change isfounded upon three restrictive assumptions, however: The city is circular; The negative exponential function provides an exact fit;and All municipal annexations are circular (pg. 60) [17].In practice, violation of any of these assumptions will reduceinter-temporal comparability.28 The mean absolute difference was 0.03852 or 10.6 percentof the mean Macauley gradient. This difference was significant atthe a = .01 level (t = 4.77).- 91 -Systematic differences between the Macauley and Edmonston methodsshould not affect estimates of wasteful commuting if thedifferences apply to both the population and employment densitygradients." Using Hamilton's notation, differences in A and Bshould cancel out in the estimate of waste, (C). 3° Figure 4.7confirms that Edmonston estimates of y are essentially anorder-preserving transformation of the Macauley estimates.Like the Macauley estimates, Edmonston estimates of y are inverselyrelated to total CMA population. A simple regression yields:y - 0.4295 - 1.12 x 10 -7 POP .2055(4.74 x 10 -8 )The estimated regression coefficient indicates that a populationincrease of 100,000 (17.0 percent of the mean 1981 CMA population)would flatten the mean gradient by .0112 (3.1 percent of the meany). Thus, Edmonston gradients are slightly more sensitive tochanges in population than Macauley gradients. Recall that a 17percent increase in CMA population flattened the mean Macauleygradient by only 2.0 percent.The Edmonston estimate of mean central density is 10,293 (Table 4.329 ^course both the gradient and central density rankingsshould be preserved between methods.30 See Hamilton's footnote #2 (pg. 1038) [25], [26] for asimilar argument regarding the measure of 0 used to calculate A, Band C.- 92 -people per square kilometre, (26,827 per mile). Values range from780 per square kilometre, (2,020 per mile) in Chicoutimi, to 20,519per square kilometre, (53,145 per mile) in Toronto (Figure 4.8).The median value is 9,757 per square kilometre. In contrast to theMacauley estimates, Do is skewed toward large values (i.e. skewedright). The mean value is 34 percent larger than the mean valuereported by Edmonston, Goldberg and Mercer [18] for 1976. Like theMacauley estimates presented above, Do exhibited a directrelationship to total metropolitan population. Simple regressionyields:Do - 8229.2 + 3.52 x10-3 POP R2 - .2085(1.50 x 10-3)This implies that a population increase of 100,000 (17 percent)would raise mean central population density by 352 people persquare kilometre (3.4 percent). 31The mean difference between the Macauley and Edmonston estimates ofDo is 2,252 people per square kilometre. This is 22 percent ofthe mean Macauley estimate of Do." The difference isstatistically significant at the a = .01 level (t=3.73). As withThis is almost identical to the 3.6 percent for Macauley-type estimates.32 The percent differential for central density is almosttwice that for y. This result is not surprising. One of the maindifferences between the Macauley and Edmonston methods was themeasurement of 0 (see chapter 3 of this thesis for completemethodological details). Recall that 0 is needed to estimate D obut not to estimate y (compare equations 4.9 and 4.10) above).- 93 -FIGURE 4.7COMPARISON OF EDMONSTON AND MACAULEYPOPULATION GRADIENT ESTIMATE RANKINGSy, in only three instances are Macauley estimates greater thanEdmonston estimates: Chicoutimi, St. Catherines, Winnipeg. Thegreatest absolute differences are in Windsor 50.1 percent, St.John's 40.3 percent, Toronto 38.3 percent, Vancouver 30.7 percentand Victoria 28.8 percent. In each of these cities 0 * 2r. Figure4.9 illustrates that while the CMA rankings of estimates of Doobtained using the Macauley and Edmonston are similar, they areless so than the rankings for estimates of y. Significant Do rankchanges occurred for Windsor (9th Macauley, 2nd Edmonston),Winnipeg (3rd, 10th), Montreal (2nd, 7th), Toronto (4th, 1st) andRegina (1st, 4th).Overall it appears that the Macauley and Edmonston methods rank theCMA gradients similarly. There are greater discrepancies in theranking of central population density. However, Edmonstongradients are steeper and have higher central density in all butthree CMAs. If these results extend to the respective employmentgradients, differences are expected to cancel out when Hamilton'smethod is applied in to estimate wasteful commuting, (C).4.2.3 Mills EstimatesThe results of this thesis employing Canadian data confirm Mills'conjecture that the bias in estimating density gradients resultingfrom the simplifying assumption that population density is zero atthe city periphery is small. Macauley [38] found the bias wassmall when SMSA data were used, but the bias was much larger if- 95 -FIGURE 4.8CENTRAL POPULATION DENSITY ESTIMATE1981, EDMONSTON ESTIMATION TECHNIQUEmore compact Urbanized Area (UA) data were used. This discrepancyis of interest because Edmonston, Goldberg and Mercer [18] arguedthat the Statistics Canada definition of CMA more closelyapproximates the U.S. Census Bureau definition of UA than SMSA.Results presented above are not consistent with the Edmonston,Goldberg and Mercer claim.In each CMA Mills estimates of y are greater than or equal to theMacauley estimates presented in Table 4.3. The mean difference is.00384 or 1.2 percent of the mean Macauley estimate of y. In onlytwo CMAs is the difference large: Kitchener, 12.1 percent andChicoutimi, 8.2 percent. Results for estimates of Do are similar.The mean difference is 62.4 people per square kilometre, or 0.77percent of the mean Macauley estimate of Do. The only largedifferential is Kitchener, at 13.6 percent.Mills and Macauley estimates of y and Do are essentially the same.The ranking of CMAs is preserved in all cases and absolutedifferences are small. It can also be concluded that Canadian CMAsmore closely approximate the U.S. definition of SMSA than UA.4.3 Employment Gradients: Canadian EstimatesEmployment gradient estimates for a sample of Canadian cities in1981 are presented in Table 4.5. Prior to the 1981 census even themodest data requirements of two-point estimation could not besatisfied for employment gradients. Thus, these are the onlyCanadian employment gradient estimates of which I am aware.- 97 -Previous U.S. employment gradient estimates ( e.g. Edmonston [17],Macauley [38], Mills [42], [43] ), were disaggregated by employmentsector and are not directly comparable to these results.4.3.1 Macauley EstimatesEmployment gradient estimates obtained using Macauley's method arepresented in Table 4.5 and Figure 4.10. It was expected that theabsolute value of 6 would exceed y in every CMA. If the oppositewere true, it would mean that population was more centralized thanemployment, violating the central assumption of the monocentricmodel (i.e. the assumption that there exists a trade-off betweenaccessibility, and housing consumption). For three CMAs (Calgary,London and Winnipeg) the estimate of 6 is less than the estimate ofy (i.e. the employment gradient is flatter than the populationgradient).Despite the individual anomalies, on average, the results seemplausible. The mean 6 in 1981 is estimated to be 0.4666 with amedian of 0.3563. This is 42 percent steeper than the meanpopulation gradient estimate. Values for 6 range from 0.1974 inToronto to 1.4436 in Charlottetown. A median value to the left ofthe mean indicates a distribution with a heavy righthand tail (i.e.there are more extremely large gradients than extremely small).This was also the case for y.Like population gradients, employment gradients are inverselyrelated to CMA population (Figure 4.10). Regression of 6 on CMA- 98 -Table 4.5 1981 Employment Density Gradient Parameter Estimates.CMA Mills6^E,Macauley6^Eo 6EdmonstonE,Calgary 0.3128020 15085.28 0.3127501 5083.83 0.3461048 6200.16Charlottetown 1.4435890^8902.33 1.4435890 8902.34 1.8560400 12509.68Chicoutimi 0.5814989^3000.12 0.5813880 2999.16 0.5975073 2542.18Edmonton 0.3142159^5773.54 0.3141539 5771.78 0.3359407 6090.77Halifax 0.4760949^6866.02 0.4760936 6865.99 0.5298379 9777.95Hamilton 0.3212670^4670.69 0.3191913 4634.22 0.3711974 6466.61Kitchener 0.2591098^1527.91 0.2279318 1320.93 0.2637403 1607.66London 0.4749254^5130.32 0.4747097 5126.55 0.5142696 5555.24Montreal 0.2302758 12203.53 0.2291056 12146.07 0.2389976 11574.80Ottawa 0.3562899^7562.80 0.3562756 7562.39 0.4048457 9077.54Quebec 0.2694734^3091.33 0.2690403 3087.08 0.2900491 3183.80Regina 0.7218454^7003.44 0.7218454 7003.44 0.7585494 7286.35Saint John 0.3244227^1236.60 0.3232313 1229.21 0.3848479 1815.73Saskatoon 0.6838427^5593.97 0.6838425 5593.96 0.7279616 6049.76St. Catherines 0.3849621^4831.61 0.3844290 4835.03 0.3809130 7855.38St. John's 0.9359563 15273.01 0.9359562 15273.01 1.0529500 25604.43Sudbury 0.2922224^877.85 0.2912226 873.62 0.3167404 931.00Thunder Bay 0.4235179^2144.59 0.4234428 2143.92 0.4664317 2927.71Toronto 0.1978160 14837.47 0.1974853 14819.42 0.2308162 24062.52Vancouver 0.2348140^9728.74 0.2345953) 9721.10 0.2498919 14089.96Victoria 0.6941540 15878.23 0.6941023 15876.97 0.8092844 22557.74Windsor 0.5009179^6271.09 0.5002666 6257.90 0.6251126 12678.34Winnipeg 0.3380818^5722.72 0.3370394 5691.55 0.3328864 5040.95Sample Mean 0.4683520^6661.44 0.4665950 6644.33 0.5254310 8933.41NOTES: 6 and E, are parameters from the negative exponential density gradient:E(x)=E0 e-6x. E(x) represents the employment density at any distance, x, from the city centre;E, represents the employment density (jobs per square kilometre) at the city centre; 6represents the rate at which density declines as we move away from the city centre.- 99 -FIGURE 4.9COMPARISON OF EDMONSTON AND MACAULEYCENTRAL POPULATION DENSITY RANKINGS3 -1 i^t^f^l^II^1 1^1Tor Van Edm Wpg Ham KW Hai^Vic Reg Sud^TB^ChaMtl^Ott^Cgy Que^SC^Lon Wsr Sas SJs^Chi^SJCMA- 100 -population yields:"8 = 0.5607 - 1.60 x 10 -7 POP^R2 - .1952(7.10 x 10 -8 )The estimated equation indicates that an increase in population of100,000 (i.e. 17 percent of mean CMA population) would flatten themean employment gradient by .0160 (i.e. 3.4 percent of the meanestimated 6). Thus, as population grows, 6 appears to flattenslightly more quickly than y. This is consistent with the notionthat jobs follow people to suburban locations and is alsoconsistent with Macauley's contention that population andemployment gradients converge over time [38].While Figure 4.10 confirms that employment gradient estimatesflatten with CMA size, as population gradients estimates did, therelationship appears weaker for the employment gradient estimates.Once again several small cities, Sudbury, Chicoutimi, Thunder Bayand Saint John, have inexplicably flat gradients.Using the Macauley estimation method, average employment density atthe centre of Canadian CMAs in 1981 is 6,644 employees per squarekilometre (17,209 per square mile). E0 ranges from a low of 874per square kilometre (2,263 per mile) in Sudbury, to 15,877 persquare kilometre (41,121 per mile) in Victoria (Figure 4.11). Themedian value, 5,692, indicates a distribution with a heavy" Standard error is in parentheses.- 101 -FIGURE 4.10EMPLOYMENT GRADIENT ESTIMATES1981, MACAULEY ESTIMATION TECHNIQUErighthand tail (i.e. there are more extremely large centraldensities than extremely small).Like central population density, central employment density ispositively related to total CMA population. Regressing E, on CMApopulation yields:E0 - 5011.3 + 2.78 x 10 -3 POP R2 - .2477(1.06 x10 -3 )The estimated equation indicates that an increase in CMA populationof 100,000 would increase the mean central employment density by278 people per square kilometre (i.e. a 17 percent increase inpopulation yields a 4.2 percent increase in E0, at the mean).Thus, E, appears to be more sensitive than 6 to changes in totalpopulation.34The relationship between E0 and total CMA population appearsstrongest at the upper end of the urban hierarchy. In medium sizedcities, (e.g. Winnipeg, Quebec City, Hamilton, St. Catherines andKitchener), E0 is lower than expected based on the simple rank-sizerelationship. Some of the smallest cities have much higher thanexpected central employment density (e.g. Victoria, Regina, St.John's and Charlottetown).34 A 17 percent increase in population reduced 6 by only 3.4percent.- 103 -FIGURE 4.11CENTRAL EMPLOYMENT DENSITY ESTIMATE1981, MACAULEY ESTIMATION TECHNIQUE4.3.2 Edmonston EstimatesEdmonston employment gradient estimates are presented in Table 4.5and Figure 4.12. Four CMAs have estimated employment gradientsthat are flatter than their population gradients (i.e. 6 < y). Inaddition to Calgary, London and Winnipeg (the same three as in theMacauley estimates) Regina has an estimated 6 < estimated y. 35 Themean estimated 6 for 1981 is 0.52543. This is 44 percent steeperthan the mean population gradient (y) and 13 percent larger thanthe mean Macauley estimate of 6. Individual estimates of 6 rangefrom 0.2308 in Toronto to 1.8560 in Charlottetown. The medianvalue, 0.3848, is to the left of the mean by a large amount,indicative of a strongly right-skewed distribution. Regression of6 on total CMA population yields:8 - 0.6359 + 1.88 x 10 -7 POP R2 = .1693(9.09 x 10 -8 )The estimated equation indicates that an increase in population of100,000 (i.e. 17 percent of the mean CMA population) would flattenthe mean gradient by .0188 (i.e. 3.6 percent of the mean 6). Thisis almost identical to the result for the Macauley estimates.Thus, using Edmonston or Macauley estimates, 6 appears to flattenmore quickly than y as CMA population increases. Smaller cities35 Essentially the population and employment gradients wereequal in Regina. The difference occurs in the third decimal place.For the other three cities the population gradient estimate was5-10 percent larger than the employment gradient estimate,regardless of which estimation technique was used.- 105 -FIGURE 4.12EMPLOYMENT GRADIENT ESTIMATES1981, EDMONSTON ESTIMATION TECHNIQUEsuch as Sudbury, Chicoutimi, Thunder Bay and Saint John haveinexplicably flat gradients based on the rank-size relationshipbetween 6 and CMA population, regardless of which gradientestimation technique is employed.As was the case for estimates of y, the Edmonston estimates of 6appear to be an order-preserving transformation of the Macauleyestimates. Figure 4.13 demonstrates the close agreement betweenthe rankings of 6 for the two estimation methods. The similarityin rankings implies that Hamilton's wasteful commuting methodologyshould not be unduly sensitive to the technique used to estimatethe population and employment gradients.The average estimated employment density at the centre of CanadianCMAs is 8,933 employees per square kilometre (23,137 per mile)using Edmonston's method. This is 34 percent greater than theMacauley estimate. In percentage terms, the difference in theestimates average central density is three times greater than thedifference in the estimate of average 6. In addition to thesignificant absolute difference in E,, Edmonston estimates do notappear to be a strictly order preserving transformation of Macauleyestimates of E. This is illustrated by Figure 4.14. Therelationship between the ranking of estimates of D, (Figure 4.4)appears to be much closer than the relationship for the estimatesof E0. The greatest discrepancies occur near the centre of theurban hierarchy (e.g. Hamilton, St. Catherines, Windsor andHalifax).- 107 -l'00FIGURE 4.13COMPARISON OF EDMONSTON AND MACAULEYEMPLOYMENT GRADIENT ESTIMATE RANKINGSFIGURE 4.14COMPARISON OF EDMONSTON AND MACAULEYCENTRAL EMPLOYMENT DENSITY RANKINGSOnce again, central employment density is positively related tototal CMA population. Regressing E, on CMA population yields: 36E0 - 6348.9 + 3.75 x 10 -3 POP^R2 = .1622(1.83 x 10 -3 )The estimated equation indicates that an increase in CMA populationof 100,000 would increase mean central employment density by 375employees per square kilometre (i.e. a 17 percent increase inpopulation yields a 4.2 percent increase in E0 , at the mean).Thus, E, is slightly more sensitive than 6 to changes in total CMApopulation."4.3.3 Mills EstimatesMills and Macauley estimates of employment gradients are almostidentical. In all cases Mills estimates of 6 are greater than orequal to Macauley estimates (Table 4.5). The mean difference isonly .00175 or 0.4 percent of the mean estimated Macauley 6.Again, the only large differential is in Kitchener (13.6 percent).The result for estimates of E0 is similar. The mean difference isonly 17.1 employees per square kilometre, or 0.25 percent of themean Macauley estimate of E€.36 Standard error in parentheses.37 The Edmonston and Macauley estimates of E0 appear equallysensitive to changes in CMA population.- 110 -4.4 SummaryThe purpose of this chapter was to present estimates of populationand employment density gradients for 23 Canadian cities using datafrom the 1981 Census of Canada. The estimated parameters are keyinputs for the model used to determine optimal and wastefulcommuting behaviour. It was argued that two-point estimates ofpopulation density are at least as good as estimates obtained usingalternative techniques. Furthermore, two-point estimation is theonly technique available to estimate employment gradients, due toa lack of geographically disaggregated employment data.Density gradients were estimated using three variations of thetwo-point method. The results are summarized in Table 4.6. Theaverage estimated y ranged from 0.32-0.36, while the mean estimatefor 6 ranged from 0.46-0.52. The average central populationdensity ranged from 8,000 to 10,000 people per square kilometre andthe average central employment density ranged from 6,600 to 9,000employees per square kilometre.Mills and Macauley estimates were virtually indistinguishable.Edmonston and Macauley estimates exhibited some significantdifferences. In all cases y and 6 were inversely related to CMApopulation, while Do and Eo were directly related to CMA population.All parameter estimates were inelastic with respect to CMApopulation: sensitivities ranged from 0.15-0.20.Table 4.6: Summary of Macauley and EdmonstonGradient EstimatesElasticityWith Respect DistributionMean Median^to Population SkewMacauley^Summary0.3290 0.2835^-0.12 RightDo 8,041 9,021^+0.21 Left6 0.4666 0.3563^-0.20 RightE0 6,644 5,692^+0.25 RightEdmonston Summary0.3646 0.3031^-0.18 RightDo 10,294 9,756^+0.20 Right6 0.5254 0.3848^-0.21 RightEo 8,933 6,466^+0.25 RightResults presented in this chapter were consistent with previousU.S. and Canadian estimates. However, several CMAs had estimated1981 population gradients that were steeper than the estimatedemployment gradients. This is perverse and violates one of the keyassumptions of the monocentric model.CHAPTER 5ESTIMATES OF WASTEFUL COMMUTING IN 23 CANADIAN CMAsThe purpose of this chapter is to compare optimal and actualcommuting behaviour for 23 Canadian cities using a model developedby Hamilton [26].1 When Hamilton applied his model to a sample ofU.S. cities, he found aggregate commuting to be almost eight timesthe amount predicted by the simple monocentric model. On thestrength of this result Hamilton questioned the validity of themonocentric model as a description of location behaviour in cities.This chapter uses Hamilton's model to answer two basic questions:• Is commuting behaviour in Canadian cities consistent withthe predictions of the monocentric model?; and• Do Canadian commuters behave in a manner significantlydifferent from their U.S. counterparts after controllingfor differences in the structure of urban areas?. 2Chapter 2 highlighted the importance of commuting behaviour in themonocentric model, or in any model that purports to explain urbanresidential and job choice location (pg. 1097) [61]. Monocentricmodels have been widely used in urban economics due to their simple1 Hamilton's methodology was reviewed in detail in Chapter 3of this thesis.2^Goldberg and Mercer [23] meticulously documenteddifferences between cities in Canada and the United States. Theynoted U.S. cities tend to be more dispersed with lower centralpopulation density. A host of reasons are given for thedifferences. Even accepting that structural differences exist,does not necessarily imply that Hamilton's measure of waste, (D-C),should be greater for U.S. cities. Underlying structuraldifferences would, perhaps, be reflected in larger estimates for Ain the U.S. cities, but not necessarily in more "waste" if bothjobs and homes are more suburbanized in the United States.- 113 -structure and perceived explanatory power. If the monocentricmodel fails to predict fundamental economic behaviour, such ascommuting, the model must be deemed to have crossed the linebetween simplification and over-simplification. Use of a modelcannot be justified by simplicity alone.'Comparing the performance of the monocentric model in severaldifferent countries is also an important task. Internationalcomparison provides an excellent crucible in which to test therobustness of basic economic theories.' Comparing the monocentricmodel's performance in Canada and the United States can beinterpreted as a weak test of theoretical robustness, becauseCanada and the United States are socially, culturally andpolitically similar.The remainder of this chapter is comprised of five sections.Section 5.1 is presented as the base case. The Macauley populationand employment gradients from Chapter 4 are used to calculate the3 In describing his minimum requirements for a scientificmodel Stephen Hawking takes what he calls a simple minded view ofwhat a model must do:"It must accurately describe a large class of observations onthe basis of a model that contains only a few arbitraryelements, and it must make definite predictions about theresults of future observations." Stephen Hawking, A BriefHistory of Time. 4 Marc Bloch, an eminent European historian argued that:"Correctly understood the primary interest of the comparativemethod is...the observation of differences, whether they areoriginal or the result of divergent developments from a commonorigin." Marc Bloch, Toward a Comparative History of EuropeanSociety, in Enterprise and Secular Change. - 114 -optimum commute (C) for 23 Canadian cities. Estimates of optimumcommute are compared with observed commuting behaviour. Section5.2 adjusts the estimates of optimum commute to recognize the factthat most cities have a central area devoted almost exclusively tonon-residential land uses. The third section of this chapterexamines the sensitivity of estimates of optimum commute to thechoice of urban boundary (F). 5 In Section 5.4, estimates of A, Band C, using Edmonston population and employment gradients, arepresented. This is important in order to gauge the Hamiltonmodel's sensitivity to the particular method used to estimatedensity gradient parameters. Section 5.5 concludes the chapter.5.1 Measuring Waste Using Macauley GradientEstimates: Base CaseTable 5.1 is strictly analogous to Hamilton's Table 1 (pg. 1041[26]). The notation is identical to Hamilton's and was describedin Chapter 3 of this thesis. All distances are in kilometres.The average distance of each household from the CBD is given by A.This can also be interpreted as the average commute, if allemployment is located in the CBD. The mean value for averagedistance of households from the CBD in the 23 Canadian cities is5 In order to calculate A and B equations 3.9 and 3.15 wereintegrated to a finite distance, Cc. Hamilton chose R to equal thedistance at which population density, D(x), declined to 100 peopleper square mile. Given the arbitrary nature of Hamilton's choice,it was prudent to evaluate the impact different values of k had onestimates of A, B and C.- 115 -Table 5.1: Optimal and Actual Commute Characteristics Using 1981Macauley Gradient EstimatesCMA A B C D E F GCalgary 5.466 5.896 -0.430 10.64 11.01 16.58 40.2Charlottetown 2.689 1.381 1.306 4.21 3.19 7.17 16.0Chicoutimi 8.209 3.434 4.775 8.30 13.45 17.71 21.2Edmonton 6.596 6.088 0.509 10.49 12.23 19.27 37.8Halifax 5.858 4.147 1.711 7.97 9.18 16.33 34.5Hamilton 7.151 6.060 1.091 15.63 12.70 20.27 23.2Kitchener 7.614 7.599 0.015 12.09 16.36 18.93 16.8London 3.849 3.962 -0.113 8.42 7.04 11.79 23.5Montreal 11.304 8.643 2.662 12.24 19.42 35.37 32.0Ottawa 8.356 5.561 2.794 10.39 13.72 22.99 37.0Quebec 9.258 7.189 2.070 11.29 16.72 24.02 31.8Regina 2.657 2.645 0.011 8.54 4.36 8.32 34.0Saint John 8.388 5.872 2.516 11.11 15.94 18.03 27.5Saskatoon 3.362 2.844 0.519 7.96 5.40 9.81 39.8St. Catherines 7.234 5.120 2.114 10.59 11.71 19.46 23.5St. John's 3.747 2.134 1.614 8.19 4.61 11.28 25.5Sudbury 7.679 6.211 1.469 13.62 15.80 16.70 29.0Thunder Bay 4.840 4.372 0.468 10.13 8.76 12.42 30.0Toronto 14.520 10.067 4.453 12.25 24.02 44.46 42.5Vancouver 10.072 8.358 1.714 13.56 17.49 30.66 39.0Victoria 5.150 2.877 2.273 10.65 6.79 15.21 17.5Windsor 4.864 3.917 0.947 10.18 7.60 14.30 20.0Winnipeg 5.104 5.517 -0.414 11.09 10.02 15.86 28.5Sample Mean 6.69 5.21 1.48 10.41 11.63 18.56 29.2Adj. Mean 6.98 5.23 1.75 10.47 11.97 19.14 28.9Hamilton Mean 13.55 11.76 1.79 14.06 19.46 36.18 n.a.NOTES: All distances in kilometres. A = necessary commute if complete centralization ofemployment is assumed; B = potential commute savings resulting from employmentdecentralization; C = optimum commute (i.e. A - B); D = observed mean commute from the 1977urban concerns survey; E = average commute randomly assigning jobs to houses; F = radius atwhich population density declines to 38.61 people per square kilometre (i.e. 100 per squaremile); G = actual radius of the CMA; n.a. = not available; Adj mean excludes Calgary, London,and Winnipeg because these CMA's had values for C less than zero.6.69 kilometres. Excluding CMAs for which B A yields a slightlygreater mean distance - 6.98 kilometres. 6 The mean for Hamilton'ssample of U.S. cities was 13.55 kilometres. By itself, it is notparticularly significant that the estimate of average distance ofhouseholds from the CBD in Hamilton's sample was more than twicethat for the Canadian cities. The cities included in Hamilton'ssample were larger, on average, than the Canadian cities and it isreasonable to expect the average distance of households to increasewith total CMA (SMSA) population. Regressing the estimate ofaverage household distance from the CBD on CMA population confirmsthe expected relationship with total CMA population: 7A - 5.01 + 2.87 x 10 -6 POP .6033(5.07 x 10-7 )The estimated equation implies that an increase in CMA populationof 100,000 (17 percent of the mean CMA population) would result inan increase in A of 0.287 kilometres (only 4 percent of theadjusted mean A).In Table 5.1, B represents the average distance of jobs from theCBD. Hamilton referred to this distance as the potential averagecommute savings attributable to employment decentralization withinthe metropolitan area. The adjusted mean estimate for B for the 20Canadian CMAs is 5.23 kilometres. Hamilton's mean estimate for the6 When Calgary, London and Winnipeg are excluded it isreferred to as the adjusted mean in Table 5.1.7 Standard error in parentheses.- 117 -average distance of jobs from the CBD was 11.76 kilometres. Likethe average distance of households from the CBD (A), the averagedistance of jobs from the CBD (B) is positively related to totalCMA population. Regressing the estimate of the average jobdistance from the CBD on total CMA population yields:B- 3.97 + 2.12 x 10 -6 POP R2 - .5624(4.08 x 10 -7 )The estimated equation implies that an increase in CMA populationof 100,000 (17 percent of the mean CMA population) would result inan increase in the average distance of jobs from the CBD of 0.212kilometres (4 percent of the adjusted mean B).Figure 5.1 illustrates that cities most likely to diverge from therank-size relationship (for either A or B) are second order centres(e.g. Edmonton, Calgary, Winnipeg), or very small cities (e.g.Sudbury, Chicoutimi, Thunder Bay). 8 The POP coefficients in thetwo regression equations estimated above are consistent with thenotion that in Canadian cities population suburbanizes more rapidlythan employment, as city size increases. 98 The cities included in Figure 5.1, and all subsequentFigures throughout this chapter, are arranged from largest tosmallest in terms of CMA population. The population rank is thusan upward sloping straight line, dividing the figure into two equalhalves (triangles). A and B were also ranked from largest tosmallest. A rank of 1 is assigned to the largest value of A or B,23 to the smallest value.9 The POP coefficient in the first equation is 35 percentlarger than in the second equation. This result arises because ywas more sensitive than S to changes in CMA population. SeeChapter 4 for details regarding this result.- 118 -FIGURE 5.1RANKING OF AVERAGE HOUSEHOLD ANDJOB DISTANCE FROM THE CBD BASED ON1981 MACAULEY DENSITY GRADIENT ESTIMATESIn Table 5.1 C represents the difference between A and B, or theminimum possible average commute in each CMA. This distance (C)can be thought of as the average distance between homes and jobs ineach city. Hamilton referred to it as the optimum commute and,despite the normative overtones of his nomenclature, theterminology is maintained in this thesis. The estimated (adjusted)mean optimum commute is 1.75 kilometres for the Canadian cities and1.79 for Hamilton's sample of 14 U.S. cities. In contrast to theaverage distance of households and jobs from the CBD, C is notstrongly related to total CMA population.Consider the following regression of optimum commute on CMA POP: 1°C- 1.04 + 7.46 x 10 -7 POP^R2 - .1765(3.52 x 10 -7 )The longest estimated optimum commute occurred in Chicoutimi (4.77kilometres) and the shortest in Regina (0.011 kilometre). Figure5.2 illustrates the weak relationship between CMA population and C.It would seem reasonable to expect people in larger cities tocommute greater distances than people in smaller cities, onaverage. The mean observed commute for each Canadian city in 1977,1°^Excluding Calgary, London and Winnipeg^yielded amarginally stronger relationship:C- 1.32 + 7.15 2:10 -7 POP^R2 - .2185(3.191(10 -7 )Log-linear and log-log estimation yielded poorer results than thesimple linear estimation.- 120 -FIGURE 5.2RANKING OF OPTIMAL AND OBSERVED COMMUTE1981 MACAULEY DENSITY GRADIENT ESTIMATES(D), is given in Table 5.1. Contrary to expectation no obviousrelationship between observed commuting distance and citypopulation is discernable in Figure 5.2. Regressing observedcommuting distance on population yields:D - 9.69 + 1.22 x 10 -6 POP^R2 - .1626(6.09 x 10 -7 )Unlike optimal commuting, log-linear and log-log regressions ofobserved and ln(D) on the natural logarithm of total CMA populationresults in a much better fit. Consider the following equations:D - -7.42 + 1.40 1nPOP^R2 - .3555(0.412)1nD- 0.27 + 0.16 1nPOP^R2 - .3825(0.045)Figure 5.3 illustrates the variability in both observed and optimalcommuting distances independent of CMA population. The HamiltonCMA has the longest observed commute (15.63 kilometres) andCharlottetown the shortest observed commute (4.21 kilometres).Kitchener and Regina are two small CMAs with optimal commutesestimated near 0, and long observed average commutes (more than 8kilometres).5.1.1 Assessing the Monocentric ModelComparing estimates of optimal and observed commuting distancesprovides an indirect test of the predictive power of themonocentric model. Blackley and Follain [7] viewed Hamilton's workas part of a growing body of indirect tests of the monocentric- 122 -model.^Such indirect tests focus on the monocentric model'sability (or inability) to predict important spatial patterns. Thetests are considered indirect because they focus upon themonocentric model's reduced form.In this study, the monocentric model did a poor job of predictingcommuting behaviour for the 23 Canadian CMAs, as illustrated byFigure 5.3. Table 5.2 summarizes five different measures of themonocentric model's performance. The five measures are defined as:• (D - C) a gross one way difference between the averageobserved and average optimal commute. Larger valuesindicate more waste.• (D/C) a the ratio of average observed to average optimalcommute. For example, Vancouver's observed commute was7.9 times the optimal commute.• (C/D) a the ratio of average optimal to average observedcommute expressed in percentage terms.^While thismeasure is simply the inverse of D/C it has twoadvantages: first, it avoids extreme values associatedwith small C's (e.g. Kitchener, Regina) and second, ithas an easily understood interpretation: it is thepercentage of actual commuting that can be accounted forby the separation of home and work. Smaller valuesindicate more waste.• (A/D) a the ratio of optimal to actual commute under theassumption of completely centralization of employment.• (D/E) a the ratio of average observed commute to averagerandom commute.^Random commute was determined byassigning households to jobs randomly."By each measure the monocentric model did an almost unbelievablypoor job of predicting observed commuting behaviour in 23 CanadianCMAs in 1981. The average wasteful commute (one way), (D - C), is11 See Hamilton (pg. 1042) [25], [26] for details on thecomputation of random commute.- 123 -FIGURE 5.3OPTIMUM VERSUS OBSERVED COMMUTE1981 MACAULEY DENSITY GRADIENT ESTIMATESTable 5.2: Alternative Measures of Wasteful Commuting Derived FromEstimates in Table 5.1CMA^D-C^D/C^C/Dkms^# %A/D%D/E%F/G#TotalPopulationCalgary^11.07 24.8 4.0 51.7 96.7 0.41 625,966Charlottetown^2.91 3.2 31.0 63.8 132.0 0.44 44,999Chicoutimi^3.53 1.7 57.5 98.9 61.7 0.84 135,172Edmonton 9.98 20.6 4.9 62.9 85.7 0.51 657,057Halifax^6.26 4.7 21.5 73.5 86.8 0.47 277,727Hamilton 14.54 14.3 7.0 45.7 123.1 0.87 542,095Kitchener^12.07 822.0 0.1 63.0 73.9 1.13 287,801London 8.53 74.3 1.3 45.7 119.6 0.50 283,668Montreal^9.57 4.6 21.8 92.4 63.0 1.10 2,828,349Ottawa 7.60 3.7 26.9 80.4 75.7 0.62 717,978Quebec 9.22 5.5 18.3 82.0 67.5 0.76 576,075Regina^8.52 742.8 0.1 31.1 195.8 0.24 173,226Saint John^8.60 4.4 22.6 75.5 69.7 0.66 114,048Saskatoon 7.44 15.3 6.5 42.2 147.3 0.25 175,058St. Catherines^8.48 5.0 20.0 68.3 90.5 0.83 304,353St. John's^6.58 5.1 19.7 45.8 177.7 0.44 154,820Sudbury^12.15 9.3 10.8 56.4 86.2 0.58 149,923Thunder Bay^9.66 21.6 4.6 47.8 115.6 0.41 121,379Toronto^7.79 2.8 36.4 118.6 51.0 1.05 2,998,947Vancouver 11.85 7.9 12.6 74.3 77.6 0.79 1,268,183Victoria^8.37 4.7 21.3 48.4 156.7 0.87 233,481Windsor 9.23 10.8 9.3 47.8 134.0 0.72 246,110Winnipeg^11.50 26.8 3.7 46.0 110.7 0.56 584,842Sample Mean^8.93 68.9 14.9 63.6 104.3 0.64 n.aMean w/o -C's^8.72 85.5 17.6 65.9 103.6 0.68 n.a.Adjusted Mean^n.a. 8.07 19.6 n.a. n.a. n.a. n.a.Hamilton Mean^12.28 12.14 12.6 95.9 78.3 n.a. n.a.NOTES: The mean w/o CMAs with negative estimated C values excludes Calgary,^London andWinnipeg; the adjusted mean excludes Kitchener and Regina in addition to the three CMA'sexcluded from mean w/o -C's; n.a. a not applicable.8.72 kilometres, (Table 5.2). The largest values for (D - C) arein Hamilton (14.54 kilometres), Kitchener (12.07 kilometres.),Sudbury (12.15 kilometres) and Vancouver (11.85 kilometres.),(Figure 5.4).Expressed another way, observed commuting in the Canadian cities isan estimated 68.9 times greater than that predicted by themonocentric model. This result is heavily influenced by Kitchenerand Regina where the optimal commuting distance Pe, O. Even afterexcluding Kitchener, Regina and the three CMAs with optimalcommuting distances less than zero, (D/C) is 8.07. Putdifferently, the monocentric model is able to account for only 19.6percent of observed commuting, on average, (C/D, Table 5.2). Thegreatest share of observed commuting that the monocentric modelexplains in any Canadian city is 58 percent in Chicoutimi (Figure5.5). In Toronto, approximately 36 percent of observed commutingis accounted for by the separation of homes and jobs.The abject failure of the monocentric model is further illustratedby comparing the average distance of households from the CBD andobserved commuting distance (Table 5.2). (A/D) represents theratio of the optimal to observed average commute under theassumption that all 1981 metropolitan employment is located in thein the CBD. Appendix 2 indicates that this is clearly anoversimplification. On average, 25 percent total 1981 CMAemployment was outside the central city, (CC) and central city isa much broader geographic concept than the CBD. Even under theclearly extreme assumption of total employment centralization, the- 126 -FIGURE 5.4AVERAGE ONE WAY WASTEFUL COMMUTE USING1981 MACAULEY DENSITY GRADIENT ESTIMATES *FIGURE 5.5OPTIMAL AS A SHARE OF OBSERVED COMMUTE1981 MACAULEY DENSITY GRADIENT ESTIMATES *monocentric model only accounted for two-thirds of observedcommuting (65.9 percent, Table 5.2).The final indication of the monocentric model's poor performance isprovided by comparing the average observed and the average randomcommute. The average observed commuting distance in the CanadianCMAs was far more accurately predicted by the assumption thatcommuting is a random behaviour, than by the monocentric mode1.12Assuming that households choose their home and job locationsrandomly resulted in an average random commute, (E), of 12kilometres (Table 5.1). (D/E) in Table 5.2 indicates the averageobserved commute was 103.6 percent of the average random commute.Despite the close mean values for observed and random commutingdistance the (D/E) variable exhibits wide variation among cities.Observed average commute exceeds random commute in seven of 23Canadian CMAs (Figure 5.6). In some CMAs observed commuting is 11-2 times greater than the random commute (e.g. D/E: Regina 195.8,St. John's 177.7, Victoria 156.7, Saskatoon 147.3 and Charlottetown132.0). Other cities had values of D only that of E (e.g. D/E:Toronto 51.0, Chicoutimi 61.7 and Montreal 63.0).5.1.2 Comparing Canadian and American CitiesOne implicit thesis of Goldberg and Mercer's The Myth of the NorthAmerican City was Canadian cities are more efficient than U.S.cities, because cities in Canada have a more compact internal12 See Chapter 3 for details regarding the random commutingmodel and the derivation of E.- 129 -FIGURE 5.6OBSERVED VERSUS RANDOM COMMUTE1981 MACAULEY DENSITY GRADIENT ESTIMATESstructure [23]. Comparison of the estimates of wasteful commutingpresented in Table 5.2 with Hamilton's results [26] provides anexcellent opportunity to test Goldberg and Mercer's hypothesis thatCanadian cities are more efficient. 13The estimated mean distance of households from the CBD, (A), forthe U.S. sample of cities is almost twice that for the CanadianCMAs (Table 5.3). The difference is statistically significant atthe a = .01 level, (t=5.49). By itself, a difference in theaverage distance of households from the CBD is not veryilluminating. Recall that previously it was demonstrated that theaverage distance of households from the CBD exhibits a strongpositive relationship with total metropolitan population. Resultsfor the average distance of jobs from the CBD are similar. Giventhat the U.S. cities were larger, on average, the average distanceof households and jobs from the CBD was expected to be larger, aswell. In contrast, the mean estimated optimal commuting distancefor the Canadian CMAs is 97 percent of the U.S. estimate of optimalcommuting distance, despite the fact that Canadian estimates of thedistance of households and jobs from the CBD are only 50 percentand 43 percent of the respective U.S. values. The averageestimated optimum commute is 1.79 kilometres for Hamilton's U.S.cities and 1.75 kilometres for the 20 Canadian CMAs. Thedifference in C is not statistically significant (t=0.112). Thisis an interesting result. According to the monocentric model, even13 All subsequent comparisons are based upon Hamilton's sampleof 14 cities and a Canadian sample of 20 cities (i.e. Calgary,London and Winnipeg are excluded) for the purposes of comparison.- 131 -Table 5.3: Summary Statistics Comparing Canadian and American Estimatesof Urban Structure and Commuting Behaviour^Hamilton Sample^Canadian Sample^Diff^t-statisticMean^a2^a^Mean^a2^a^of Means^H,: Diff=0Significance.95^.99A 13.55 12.70^3.56 6.98^8.56^2.93^6.57 5.49 X XB 11.76 9.98^3.16 5.23 5.32^2.31^6.53 6.37 X XC 1.79 0.56^0.75 1.75^1.58^1.26^0.04 0.11D 14.07 1.43^1.20 10.47 6.09^2.47^3.60 5.48 X XE 19.46 22.26^4.72 11.97^30.29^5.50^7.49 4.12 X XD-C 12.28 1.53^1.24 8.72 7.29^2.70^3.56 5.02 X XC/D 12.65 43.66^6.61 17.65^180.58^13.44^5.00 1.39A/D 95.85 809.27^28.44 65.93^445.70^21.11^29.92 3.23 X XD/E 78.25 478.77^21.88 103.59^1651.78^40.64^25.34 2.27 XNotes: a2^7= variance;^a standard deviation;^t-statistics calculated under the nullhypothesis that the difference between the two samples was zero using a two tailed differenceof means test; degrees of freedom = 32.though the U.S. cities are more dispersed than Canadian cities,(higher values for A), U.S. cities do not necessitate increasedcommuting, because jobs appear to have followed people to suburbanlocations, (higher values for B)."Both Hamilton's results, and the results for Canadian CMAs,indicate that, in 1981, the average observed commute exceeded theaverage optimum commute by a large amount. The mean observedcommute for the American sample of cities is 3.6 kilometres greaterthan for the Canadian CMAs. This difference is statisticallysignificant at the a = .01 level (t=5.48). In order to assesswasteful commuting in the two countries, values for (C/D) arecompared. Approximately 12.7 percent of observed commuting inHamilton's sample is necessitated by the separation of home andwork. The corresponding figure for the Canadian sample is 17.7percent. The difference between the estimates of (C/D) is notstatistically significant even at the a = .10 level (t=1.39).Employing a one-sided t-test, the estimate of (C/D) for theCanadian CMAs is significantly greater (at the a = .10 confidencelevel), than Hamilton's estimate of (C/D). 15 Thus, commuters inHamilton's U.S. cities may be marginally more profligate thancommuters in the Canadian CMAs, but the results presented in Table5.3 are not conclusive.14 The result implies that the U.S. cities in Hamilton'ssample are not innately less efficient due to their internalstructure. This is counter to some of the arguments contained inGoldberg and Mercer's [23] analysis.15 For a one-sided test the critical t-value, at the 90percent confidence level with 32 degrees of freedom, is 1.31.- 133 -One reason for the inconclusive result is probably the largevariation in city size in the Canadian sample. The estimates ofwaste (D - C) and (C/D) for the Canadian CMAs have very largestandard deviations (Table 5.3). This is likely due to greatervariation in the size of cities included in the Canadian sample.CMA population ranges from Toronto, near 3,000,000, toCharlottetown at less than 100,000. The large variance in C and(C/D) may account for the lack of a statistically significantdifference between Canadian and U.S. estimates of commuting waste.5.2 Adjusting for the Areal Extent of the CBDThe CBD in the monocentric model is like a black hole; it is asingle point in space without internal dimension, into which alleconomic activity is drawn. Not only is this theoreticallyunsatisfying, it also introduces a potential source of bias intoestimates of the population density gradient. Gradient estimatesobtained using the negative exponential function yield maximumpopulation density at the dimensionless CBD. In reality, thecentral area of most cities is devoted, almost exclusively, tonon-residential land uses. Hamilton [pg. 1044, 26] referred tothis phenomena as a crater in the density function.It is possible to choose an alternative functional form for thepopulation gradient that allows for a density profile with acentral crater. Rather than complicate his model, Hamilton simplyassumed the CBD in each CMA was one mile in diameter and devoid of- 134 -residences. This has the effect of increasing the averagehousehold's distance to the CBD (i.e. A), by one half mile.Optimum commute also increases by one half mile, because theemployment gradient and the average distance of jobs from the CBDare unaffected by the crater in the population gradient.Although Hamilton's adjustment was simple it was clearly ad hoc.Since economic theory does not strongly support any particularfunctional form for the population gradient, Hamilton could havechosen an alternative function. McDonald and Bowman [40] suggestedseveral alternative equations, such as standardized normal orquadratic, that allow for the possibility of a central densitycrater." However, given that many applications of themonocentric model employ a negative exponential function, Hamiltonwas correct to employ the same function for his critique of theusefulness of the monocentric model.Perhaps the greatest weakness in choosing an arbitrary value forthe diameter of the CBD is that it assumes the same size centralarea for each city. While a CBD diameter of 1.0 kilometre may bereasonable for larger cities in the Canadian sample, it is probablytoo large for some of the smaller cities. However, the samecriticism, (i.e. assuming one size fits all), must be extended toa great body of urban literature. It should be stressed that the16 Mills and Hamilton [42) and Chapter 2 of this thesis showedthat the negative exponential density gradient requires the ratherunrealistic assumptions of Cobb-Douglas housing productionfunctions and a price elasticity of demand for housing equal tonegative 1.- 135 -Table 5.4: Alternative Measures of Wasteful CommutingAssuming a CBD Radius of 1.0 Kilometres^VALUE^ RANKD/C^C/D^A/D^D/C^C/D^A/D# % % #^#^#POP#Calgary 18.66 5.36 60.77 2 22 14 6Charlottetown 1.83 54.74 87.57 22 2 6 23Chicoutimi 1.44 69.54 110.89 23 1 2 20Edmonton 6.95 14.38 72.42 7 17 11 5Halifax 2.94 34.02 86.08 19 5 7 13Hamilton 7.48 13.37 52.14 6 18 22 9Kitchener 11.91 8.39 71.26 3 21 12 11London 9.50 10.53 57.58 4 20 19 12Montreal 3.34 29.92 100.56 15 9 3 2Ottawa 2.74 36.51 90.03 20 4 5 4Quebec 3.68 27.20 90.89 13 11 4 8Regina 8.44 11.85 42.84 5 19 23 17Saint John 3.16 31.64 84.47 17 7 8 22Saskatoon 5.24 19.08 54.81 10 14 21 16St. Catherines 3.40 29.40 77.73 14 10 10 10St. John's 3.13 31.91 57.96 18 6 15 18Sudbury 5.52 18.13 63.73 9 15 13 19Thunder Bay 6.90 14.50 57.67 8 16 17 21Toronto 2.25 44.53 126.73 21 3 1 1Vancouver 5.00 20.01 81.64 12 12 9 3Victoria 3.25 30.74 57.77 16 8 16 15Windsor 5.23 19.13 57.62 11 13 18 14Winnipeg 18.92 5.29 55.01 1 23 20 7Sample Mean 6.13 25.22 73.83 na na na naHamilton Mean 5.69 19.83 103.04 na na na na- 136 -base case assumes the same size CBD in all cities: zero. Oneadvantage of assuming a one kilometre radius is that it eliminatesthe negative values for optimal commuting distance in threeCanadian CMAs.In order to simulate a non-residential central area, 1.0 kilometreis added to each CMA's estimated average distance of householdsfrom the CBD. For example, in Vancouver the average increases from10.07 kilometres to 11.07 kilometres; the average distance of jobsfrom the CBD is unchanged and the optimum commute increases from1.71 to 2.71 kilometres. The increase in the optimum commute makescities appear less wasteful. New values for D/C, C/D and A/D arepresented in Table 5.4. The average observed commute for allcities declines from 8.7 times the average optimal commute to only6.1 times optimal, (D/C). The monocentric model now explains 25percent of observed commuting, (C/D). Under the strong assumptionof completely centralized employment, the monocentric model is ableto explain 73.8 percent of 1981 observed commuting in the 23Canadian CMAs.Table 5.4 ranks commuting waste in the 23 Canadian cities.Winnipeg, Calgary and Kitchener rank as the most wasteful citieswith observed commuting at least 11 times the optimal commute."Chicoutimi, Charlottetown and Toronto rank as the most efficientCanadian cities with 69, 54 and 44 percent of observed commuting17 D/C and C/D are opposite sides of the same coin. Highervalues for D/C indicate greater wasteful commuting, while lowervalues for C/D indicate greater waste. Thus, Winnipeg ranked 1stfor D/C and 23rd for C/D.- 137 -explained by the separation of houses and jobs.Two patterns in the rankings warranted further investigation.Figure 5.7 illustrates that there is no strong linear relationshipbetween CMA population and the CMA's wasteful commuting rank. Infact, comparison Figures 5.7 and 5.3 vindicates Hamilton'smethodology demonstrateing the importance of controlling for urbanstructure when assessing commuting behaviour. While the largestCMAs have some of the longest observed commutes, the largest CMAsare not among the most wasteful. The four largest CMAs, (Toronto,Montreal Vancouver and Ottawa), and three of the smallest CMAs(Charlottetown, Saint John and Chicoutimi), are among the mostefficient cities.In order to explore the relationship between metropolitanpopulation and wasteful commuting in more detail the possibility ofa non-linear relationship was investigated. Table 5.5 summarizesthe results of regressing several various measures of commutingwaste on CMA population and the natural log of CMA population.Linear estimations yield weak results. Log-linear and log-logestimates with (C/D) and (D/C) as the dependant variables do notyield better results. Both log-linear and log-log regressions of(D - C) on CMA population yield improved results, compared withother alternative specifications. This latter result is notunexpected because it was shown above that the average observedcommute is related to total CMA population in a non-linear fashion.FIGURE 5.7RANKING OF WASTEFULCOMMUTING (CID) VERSUS CMA POPULATIONTable 5.5: Regression Estimates of the Relationship Between CMAPopulation and Wasteful CommutingDependantVariableInterceptTermCoefficientEstimateIndependantVariableStandardErrorR-squaredD-C 8.65 4.84 x 10-7 POP 7.37 x 10-7 R2 = .020C/D 13.43 3.93 x 10-6 POP 3.73 x 10-6 R2 = .050D/C 106.34 -4.39 x 10-5 POP 6.17 x 10-5 R2 = .024D-C - 5.22 1.11 In POP 0.52 R2 = .179C/D 19.13 -0.27 In POP 2.95 R2 = .000D/C 435.27 -27.87 In POP 47.5 R2 = .016ln(D-C) -0.07 0.17 In POP 0.07 R2 = .228ln(C/D) 1.62 0.14 in POP 0.34 R2 = .008ln(D/C) 4.32 -0.14 In POP 0.34 R2 = .008The implication in the context of Hamilton's model appears to bethat the monocentric model does not perform well over a certainrange of city sizes. There may be other variables that betterexplain variations in wasteful commuting and thus that account forthe poor performance of the monocentric model.5.3 Sensitivity to the Choice of Urban BoundaryIn order to calculate the average distance of households and jobsfrom the CBD (i.e. A and B) the integrals in equations 3.9 and 3.15had to be evaluated over a definite range. Hamilton [26] chose theurban boundary, Si, to equal the distance at which populationdensity declined to 100 people per square mile, (denoted as F byHamilton). 18 Thus, the boundary for each city is unique anddepends upon the population gradient parameters estimated in (seeChapter 4). Although Hamilton's choice of F was arbitrary, itseems reasonable. A density of 100 people per square mile isroughly equivalent to 10 acre lot sizes. Furthermore, as is shownbelow, the influence of the boundary choice on estimates of optimumcommute tends to wash out if the impact of R upon both A and B issimilar.Column F in Table 5.1 represents the estimated distance at whichpopulation density declines to 100 people per square mile in eachCanadian CMA, based on the density gradients estimated in Chapter4. The boundary (F) is calculated using the density gradient18 This is equivalent to 38.61 people per square kilometre.- 141 -parameters presented in Table 4.3, and then substituted for ic inequations 3.10 and 3.16. For comparison, column G in Table 5.1gives the CMA radius as measured from Statistics Canada maps.Figure 5.8 reveals some considerable differences between theboundary measures (F and G). The calculated boundary is greaterthan the measured (political) boundary (G) in only three CMAs:Toronto, Montreal and Kitchener.Hamilton suggested the impact of a particular boundary choice onestimates of wasteful commuting would be minor.19 However, whenHamilton applied his model to a sample of Japanese cities he foundhis assumption of 100 people per square mile as the dividing linebetween urban and rural was inappropriate and yielded cities withinfinite radius. Further, Macauley [38] found density gradientestimates were sensitive to the particular data set employed.Macauley found that using UA data rather than SMSA data causedmajor differences in density gradient estimates for a given set ofcities. Thus it is worthwhile to check the model's sensitivity tothe choice of urban boundary.Table 5.6 presents revised estimates of waste, (D - C), and theshare of commuting that is necessary, (C/D), allowing x to varyfrom F G 00. 2° The average estimated wasteful commute declinesby 0.56 kilometres when x is increased from F to co. The differencewas is statistically different from zero (t=0.671). Comparing the19 See Hamilton (pg. 1040) [25], [26] footnote #6.20 The figures generated using F are identical to those inTable 5.4. They are repeated here to make comparison easier.- 142 -Kilometres50 ^----- Measured Radius-±-- Calculated Radius302010FIGURE 5.8MEASURED VERSUS CALCULATED CMA RADIUS0 1(11^111111111^1^i^111Tor Van Edm Wpg Ham KW Hal^Vic Reg Sud^TB^ChaMtl^Ott Cgy Que^SC^Lon Wsr Sas SJs^Chi^SJCMA" Calculated is where population density = 100 peopie per snuare mile,Table 5.6:CMAAlternative Measures of Wasteful CommutingAssuming Different CBD BoundariesD - C^ C/DF^G cc^F^G ccCalgary 10.07 10.25 10.26 5.36 3.63 3.62Charlottetown 1.91 1.61 1.61 54.74 61.63 61.72Chicoutimi 2.53 1.74 -0.10 69.54 79.04 101.22Edmonton 8.98 8.80 8.80 14.38 16.05 16.11Halifax 5.26 4.80 4.79 34.02 39.75 39.85Hamilton 13.54 13.41 13.17 13.37 14.20 15.77Kitchener 11.07 11.08 11.06 8.39 8.37 8.51London 7.53 7.58 7.58 10.53 10.05 10.04Montreal 8.57 8.76 8.11 29.92 28.40 33.71Ottawa 6.59 5.95 5.86 36.51 42.72 43.59Quebec 8.21 7.68 7.29 27.20 31.94 35.40Regina 7.52 7.52 7.52 11.85 11.89 11.89Saint John 7.60 6.15 5.16 31.64 44.67 53.51Saskatoon 6.44 6.29 6.29 19.08 21.02 21.02St Catherines 7.48 7.17 6.80 29.39 32.34 35.83St John's 5.58 5.36 5.36 31.91 34.59 34.60Sudbury 11.15 9.92 9.44 18.12 27.15 30.66Thunder Bay 8.65 8.37 8.36 14.50 17.36 17.42Toronto 6.79 6.92 6.06 44.53 43.50 50.51Vancouver 10.85 10.60 10.45 20.01 21.85 22.96Victoria 7.37 7.23 7.04 30.74 32.09 33.89Windsor 8.23 8.04 7.98 19.13 20.96 21.56Winnipeg 10.50 10.66 10.67 5.29 3.95 3.85Sample Mean 7.93 7.65 7.37 25.22 28.14 30.75mean values for (C/D) resultsin the same conclusion. The averageshare of observed commuting deemed necessary increases from 25 to31 percent. Again the difference is not statistically differentfrom zero (t=0.974).Figure 5.9 graphically illustrates the insensitivity of Hamilton'stechnique to the choice of urban boundary for individual CanadianCMAs. The result holds for cities throughout the urban hierarchy.The greatest changes in (C/D) occurr in Chicoutimi and Saint John.Recall that in Chapter 4 it was argued both these cities wereexpected to behave erratically, due to their irregularly shapedboundaries. Interestingly, in Chicoutimi allowing the boundary toapproach 03 increases (C/D) to the point where there is no wastefulcommuting.5.4 Using Edmonston Density GradientsChapter 4 highlighted some significant differences between densitygradients estimated using Edmonston's versus Macauley's estimationtechnique. It was argued that, if the differences between gradientestimates were systematic for both the employment and populationgradients, the Hamilton technique for calculating the optimumcommute would not be adversely effected by the choice of gradientestimation technique. Evidence in Chapter 4 suggested the twotechniques ranked the density gradient parameters a and b similarlyacross the 23 CMAs. However, differences between the D, and E °parameters appeared less systematic. Although Edmonston populationand employment gradient estimates are steeper than Macauley- 145 -FIGURE 5.9OPTIMAL AS A SHARE OF OBSERVED COMMUTEALLOWING THE URBAN BOUNDARY TO VARYestimates, the effect on estimates of wasteful commuting should benegligible, if estimates of both the average distance of householdsand jobs (A and B) are equally effected because wasteful commutingis defined as (A - B).Table 5.7 summarizes the difference in estimates of several keymeasures of waste using Edmonston rather than Macauley gradientestimates. Values presented in Table 5.7 are for the estimateswhich added 1.0 kilometre to the average distance of householdsfrom the CBD to represent a non-residential core in each CMA. Thiswas described in Section 5.2.The mean difference between the estimated average distance ofhouseholds from the CBD is 0.47 kilometres and the differencebetween the estimated average distance of jobs from the CBD is 0.45kilometres (Table 5.7). Both differences are significantlydifferent from zero at the a = .01 level (t=4.74 and t=6.34).Notice that the difference between estimates of A and B is almostequal. Thus, it is not surprising that the difference between theaverage estimated wasteful commute is not significantly differentfrom zero (t=0.04). The mean estimated waste, (D - C), is notstatistically different between Edmonston and Macauley densitygradient estimates.To understand how this is possible, despite large differences inthe estimates of Do and E0, recall equations 3.10 and 3.16 fromChapter 3:- 147 -Table 5.7: Summary Statistics Comparing Urban Structure and Commuting BehaviourUsing Edmonston Versus Macauley Density Gradient EstimatesMacauley^Edmonston^Difference t-statistic SignificanceMean^a2 Mean^a2^of Means^H0: Diff=0^.95 .99A^7.69^8.40^7.22^7.39^0.473 4.74 X^XB 5.21^4.93 4.77^4.17 0.448^6.34^X^XC^2.48^1.95^2.46^2.02^0.026 0.61D-C^7.93^7.19 7.96^7.30 -0.030 0.04^C/D 25.22 253.09^24.90 260.65^0.650^0.14A/D 73.83 440.60 69.27 414.03 4.560 0.73Notes: a2 M variance; t-statistics calculated under the null hypothesis that the differencebetween the two samples was zero using a two tailed difference of means test; degrees offreedom = 44.B 28 2n E0 -2 -81x eOJ_ ^ x e2^27W°A 2 -y.77YPHamilton [26] showed that the second term in each equation is smallrelative to the first term. Indeed, when distance (x) approachesm 3.10 reduces to 2/a and 3.16 reduces to 2/6. Since Do and E0appear only in the second term in each equation, Hamilton's methodis robust with respect to the gradient estimation technique,despite large variance in estimates of D, and E 0 , as long as a and6 are ranked consistently. This conclusion holds for individualcity comparisons as well as for the average estimate of wastefulcommuting. Appendix 3 presents tables that are analogous to Tables5.1 and 5.2 using Edmonston gradients rather than Macauleyestimates. There are no noticeably large differences for anyindividual CMA.5.5 SummaryThe purpose of this chapter was to compare estimated optimalcommuting distances with observed commuting distances in 23Canadian metropolitan areas in 1981. The methodology used tocalculate the optimal mean commute was developed by Hamilton [26].The overall goals were to:- 149 -• Test the ability of the monocentric model to predictobserved commuting behaviour in a sample of Canadiancities; and• Compare the amount of wasteful commuting occurring in asample of Canadian versus American cities.Observed commuting in 23 Canadian CMAs was found to be six to eighttimes greater than the amount necessary due to the separation ofhomes and jobs. Put another way, the monocentric model was onlyable to explain an average 20-25 percent of observed commuting in23 Canadian CMAs in 1981. The average wasteful commute (one way)is in excess of 8.7 kilometres because, on average, homes and jobsare only 1-2 kilometres apart but the average observed commuteexceeds 10 kilometres. The total failure of the monocentric modelin predicting commuting behaviour was clearly evident for thesample of Canadian cities, as it was for Hamilton's [26] sample ofU.S. cities. In fact, randomly assigning workers to houses andjobs provided a much better explanation of commuting behaviour thanthe monocentric model which is based on an implicit trade off ofaccessibility and transportation costs.Comparison of Canadian and U.S. estimates of wasteful commuting didnot provide strong support for the hypothesis that Canadiancommuting behaviour is less wasteful, due to more efficient urbanform. Approximately 75-80 percent of observed commuting in theCanadian cities was "wasteful", while Hamilton found that 85-90percent of observed commuting in his sample of U.S. cities was"wasteful". A two-tailed difference of means test indicated thatthe difference in means was not statistically different from zero,- 150 -even at the 90 percent confidence level. A one-tailed differenceof means test indicated that wasteful commuting in Canadian citieswas significantly less than wasteful commuting in U.S. cities butonly at the 90 percent level of confidence.This chapter also demonstrated that Hamilton's method forestimating wasteful commuting is robust with respect to thetechnique used to estimate the density gradient parameters.Estimates of wasteful commuting were also shown to be robust withrespect to the choice of urban boundary, within reasonable limits.Difference in estimates of wasteful commuting (i.e. the failure ofthe monocentric model to explain observed commuting) among citiesdoes not appear to be related to the population of the metroppl.itanarea. Differences are more likely explained by other variaKes.CHAPTER 6SUMMARY AND DIRECTIONS FOR FURTHER RESEARCHThe overall goals of this thesis were twofold:1. To test the ability of the monocentric model to predictobserved commuting behaviour in a sample of Canadiancities; and2. To compare commuting behaviour in a sample of Americanand Canadian cities.The technique used to test the ability of the monocentric model topredict observed commuting behaviour in Canadian cities wasdeveloped by Hamilton [26]. Hamilton's model was used to estimatethe average distance of homes and the average distance of jobs fromthe CBD in each of 23 Canadian cities. The difference between theaverage distance of homes from the CBD and the average distance ofjobs from the CBD is deemed to be the minimum possible averagecommuting distance (optimal commute). The optimal commute wascompared with the average observed commute in each city.In order to estimate the average optimal commute in each CMA themodel developed by Hamilton requires estimates of population andemployment density gradients for each CMA.6.1 Density Gradient EstimatesThe population and employment density gradient parameters for eachcity are the key inputs into the model used to estimate optimal and- 152 -wasteful commuting.' When Hamilton [26] estimated the model fora sample of 14 U.S. cities he was able to use density gradientparameters previously estimated by Macauley [38]. Unfortunately,there was not a similar set of density gradient parameter estimatesavailable for a lareg sample of Canadian cities.^Therefore,Chapter 4 of this thesis presents 1981 estimates of population andemployment density gradients for 23 Canadian urban areas.Although it was not the goal of this thesis to estimate populationand employment density gradients, the gradients presented inChapter 4 represent a significant contribution to urban analysisfor three reasons:1. They are the only employment gradient estimates for alarge sample of Canadian cities;2. Both the population and employment gradient estimatescover a much larger sample of Canadian cities than anyprevious work; and3. Three separate variants of the two-point estimationtechnique were compared for 23 Canadian cities.Results presented in Chapter 4 are broadly consistent with previousCanadian and U.S. density gradient estimates. A detailed summaryof the gradient estimates is provided by Table 4.6 in Chapter 4.Despite the overall reasonableness of the parameter estimates therewere at least three cities in which the estimated employmentgradient was flatter than the estimated population gradient:Calgary, London, Winnipeg (and Regina for one estimation techniquebut not the others). Such a result is perverse and violates one of' The model is described in detail in Chapter 3.- 153 -the key assumptions of the monocentric model.The Mills and Macauley gradient parameter estimates were virtuallyindistinguishable as was expected a priori. Edmonston gradientparameter estimates, which attempt to control for the shape of theCMA, were steeper with higher central densities for both thepopulation and employment gradients. It was not possible to assesswhich estimates were superior, however. In all cases populationand employment density gradient parameters were found to beinelastic with respect to total CMA population.6.2 Wasteful Commuting - A Critique of the Monocentric ModelCommuting behaviour must play a fundamental role in any model thatpurports to explain urban residential and employment location.Monocentric models have been widely employed in urban economicanalysis because of their simple structure and presumed explanatorypower.Hamilton's model [26] as employed in this thesis represents part ofa growing body of literature aimed at providing tests of themonocentric model's ability to predict important spatial patterns.Recent research has also focused on criticism of the assumptionsunderlying the monocentric model. 2 If, as was shown by thisthesis, the monocentric model fails to predict fundamental economicbehaviour, such as commuting, the usefulness of the model is in2 See Wheaton [57] for a review of the literature critical ofthe monocentric model.- 154 -question. A model cannot be used just for its simplicity if themodel fails to predict crucial aspects of economic behaviour.How badly did the monocentric model tested in 23 Canadian citiesfail? Observed commuting in the sample of Canadian cities was anaverage of 6.0 to 8.0 time greater than the commuting predicted bythe monocentric model. Put another way, the version of themonocentric model tested in this thesis explained only 20-25percent of observed commuting. In fact, randomly assigning workersto jobs and residences provided a much better explanation ofobserved commuting behaviour in a sample of Canadian cities.6.3 International ComparisonBy replicating the methodology developed by Hamilton [26] thisthesis was able to directly assess the relative performance of themonocentric model in Canada versus the United States.International comparison is an important test of an economicmodel's robustness. In this case, the performance of themonocentric model was uniformly poor in both the United States andCanada.Comparison of estimates for wasteful commuting were used toevaluate the relative efficiency of Canadian versus U.S. cities.If the optimal average commute, as calculated by the Hamiltonmodel, represents the minimum possible average commute, then thegreater the divergence between observed commuting and optimalcommuting the more "wasteful" commuting that is occurring in a- 155 -city. There was only weak evidence to suggest that, on average,observed commuting behaviour more closely resembles optimalcommuting behaviour in Canadian cities. Based on the work ofGoldberg and Mercer [23] it was expected that there would besignificant differences between cities in the two countries.6.4 Criticisms of Hamilton's MethodologyHamilton's model for estimating wasteful commuting incorporatesseveral important assumptions including:• When employment decentralizes it does so in a dispersemanner, without clustering3;• Both population and employment density are wellcharacterized by a negative exponential densitygradient;`• Labour force participation rates are independent ofintra-urban location; and• All jobs and homes are equally desirables.With the exception of the assumption regarding labour forceparticipation rates, none of the key assumptions are specific toHamilton's model but are instead implicit in the version of themonocentric model which Hamilton [26] chose to test. Theassumption that labour force participation rates do not vary withincities was necessary to allow Hamilton to interpret the average3 This issue was discussed in Chapter 2, Section 2.4.4 The functional form issue was discussed in some detail inChapter 2, Section 2.3.5 See Wheaton [58] for a model that relaxes this assumption.- 156 -distance of population from the central city as the averagedistance of the "labour force" from the central city. It isimprobable that the overall participation rate shows strongintraurban variation but it is possible that there are significantdifferences in female participation rates among intraurbanlocations.The strongest criticism of Hamilton's model was put forward byWhite [62]. White focused on Hamilton's assumption of non-clustered decentralized employment. It is in some sense ironicthat the most avuncular critic of Hamilton's method for testing themonocentric model focused on the employment assumption. Whitefinds that there is minimal wasteful commuting when suburbanemployment is permitted to cluster [62]. On the surface White'scritique of Hamilton seems to lead to the same conclusion asHamilton's original work. As soon as White allows clusteredsuburban employment her model becomes polycentric and the land rentgradient is no longer characterized by a single location variable.Thus White's conclusion amounts to the statement that if themonocentric model is really polycentric then there is no wastefulcommuting. 66.5 Directions for Further ResearchAlthough White's critique of Hamilton's work perhaps missed the6 Mills and Hamilton [42] convincingly argue that it isimpossible to reconcile clustered suburban employment with themonocentric model. See Chapter 2 of this thesis, in particularSection 2.4 for a discussion of this issue.- 157 -point Hamilton was trying to make, White's work did point to one ofthe weaknesses in urban economics. While there is a large body ofliterature examining residential choice, there has been much lessinvestigation into the determinants of employment location.Instead centrality has typically been assumed.On the theoretical side, two types of models need to be developedfurther. First, are non-monocentric models which have noprespecified centre allowing the model to determine the locationand number of employment clusters. While these models areintuitively appealing mathematical complexity is likely to limitadvances made. Ogawa and Fujita developed a non-monocentric modelthat yielded some interesting results [47]. When commuting ismodelled as very expensive relative to the transaction costs ofdoing business, the city has a completely mixed economy with nocentre. If the reverse is modelled (commuting is cheap relative totransaction costs) the monocentric result is achieved. Reality is,no doubt, somewhere between these extreme results.The second area of investigation must be in the area of polycentricmodels. In contrast to non-monocentric models, polycentric modelshave several prespecified employment nodes. Although this is lesssatisfying theoretically, it seems more likely to yield empiricallyverifiable results. Dubin and Sung [16], Griffith [25] and Wieand[65] have each developed simple polycentric models and tested theimplications for population density and rent gradients.There is also a great deal of empirical work that needs to be done- 158 -examining the determinants of commuting behaviour. It is clearthat the attempt by individuals to minimize commuting distance(i.e. the monocentric model) can explain only a small fraction ofobserved commuting behaviour. Other variables influencingcommuting behaviour might include: The sex of the household head; The income of the household head; The variable used to measure commuting (distance versustime); The presence of a second wage earner in a household; The demand for other travel; Heterogeneity of jobs and houses and the need for"matching"; and Tenure.The influence of some of these variable on commuting behaviour hasbeen investigated to a limited degree. Frankena [19] and White[63], [64] examined differences in commuting behaviour among menand women for a limited number of U.S. cities. Adler [1]investigated the impact of tenure (rent versus own) on commutingbehaviour. Coulson [12] examined the differential impacts ofchanges in time versus money costs of commuting on the decisions ofindividuals.The data base employed in this thesis would permit a detailedanalysis of individual commuting decisions in Canadian cities.Variables available for analysis include sex of the household head,tenure, whether there is a second wage earner, the number of non-commuting trips to downtown, the place of work for the household- 159 -head and the second wage earner, the occupation of the householdhead, the transit mode of the household head and the length oftenure for each household. It is suggested that the next step inthe analysis might logically be to empirically test the influenceof each of these variables on commuting behaviour in Toronto,Montreal and Vancouver. Results obtained for the three largestcities could then be tested in smaller cities to see if there areimportant difference in commuting behaviour among individuals indifferent size cities.In concluding, the words of Michelle White provide an admirablegoal for urban economists:"It is hoped that in the future urban economists will nothave to characterize any commuting behaviour as wastefuland instead will be able to explain it" [62].Although this is clearly overly optimistic, detailed investigationinto the determinants of commuting behaviour represents a promisingextension to this work.BIBLIOGRAPHY[1] ADLER, Moshe, The Location of Owners and Renters in the City,Journal of Urban Economics 21, p347-363, 1987.[2] ALCALY, Roger E., Transportation and Urban Land Values: AReview of the Theoretical Literature, Land Economics 52(1), p42-53,1976.[3] ALONSO, William, Location and Land Use, Harvard UniversityPress 1964.[4] ALPEROVICH, Gershon, An Empirical Study of Population DensityGradients and Their Determinants, Journal of Regional Science 23(4), p529-540, 1983.[5] ALPEROVICH, Gershon, Determinants of Urban Population DensityFunctions, Regional Science and Urban Economics 13, p289-295, 1983.[6] ANDERSON, John E., Estimating Generalized Density Functions,Journal of Urban Economics 18, p1-10, 1985.[7] BLACKLEY, Dixie M. and James R. 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The Case of theTwo-Centre City,  Journal of Urban Economics 21, p259-271, 1987.APPENDIX 1INTEGRATION OF EQUATIONS 3.8 AND 3.15In Chapter three the following equation, 3.8 was presented torepresent the average distance of homes from the CBD, (A), in aparticular monocentric city:A 2 Tc Do ft x 2 e —y1 dx= (A1.1)Integration by parts is based on the following identity:f d v duu - UV - f V 71-x (A1.2)Now if we define the following: u m x2; and dv/dx m e-fxthen: du/dx = 2x; and v = -(e""/y).Applying the formula described by equation A1.2 to equation A1.1yields:- 166 -Do2 7c^x2 e -yxf g _  2 x e -yx^ _ (A1.3) Multiplying through and simplifying yields:2+ 2D027[ j-ft xe -Yx dxYP 0D0 27cx2 e -Yx (A1.4)YPNow recall from equation 3.6 in Chapter 3 that the total populationof the city, (P) can be expressed as:P 27c f D(x) xdx^ (A1.5)Substituting in the negative exponential density function, 3.7, forD(x) in equation A1.5 yields:P 27cDo f xe -Yx di(^ (A1.6)Substituting A1.6 into the second term in A1.4 reduces A1.4 to:- 167 -Do2n;? e -Y"^2YP^YP2-^ x2700 -2 e -v?Y^YP(A1.7)Equation A1.7 is equivalent to 3.9 as required.One other point must be made regarding Hamilton's model.^Inestimating A and B Hamilton was unconcerned whether 0 = 27r because,as long as 27 is used to calculate P, the 27 in the numerator anddenominator of A1.7 cancel out. This implies, however, that forCanadian CMAs where 27 does not equal 0 it is not correct to simplyuse the population given in the census to estimate A. Insteadpopulation had to be estimated using equation A1.5 as though thecity were a complete circle. This also makes the use of Edmonstondensity gradient estimates based on shapes other than a circlesomewhat inconsistent with the Hamilton wasteful commuting model.APPENDIX 2COMPARISON OF CMA EMPLOYMENT AND CENTRAL CITY EMPLOYMENTBASED ON DATA FROM THE 1981 CENSUSChapter 5 showed that when it is assumed all employment in each CMAis located in the CBD the monocentric model does a much better jobof explaining commuting behaviour. The Table below clearlyillustrates that a significant portion of CMA employment in 1981was not located in the central city. On average, 25 percent ofemployment was located in suburban municipalities. Larger citiestended to have a smaller share of total employment located in thecentral city.Central city is a much broader geographic concept than CBD. UsingVancouver as an example, the central city refers to the City ofVancouver (this would include downtown and the Broadway corridor)The CBD, as it is used in the monocentric model, refers to thedowntown area only. Thus, it is reasonable to conclude that asignificant proportion of total metropolitan employment was locatedoutside the central business district in 1981.CMA CMAEmploymentCentral CityEmployment^ShareCalgary 325,205 300,310 0.9234Charlottetown 18,380 13,460 0.7323Chicoutimi 44,715 40,845 0.9135Edmonton 339,075 289,035 0.8524Halifax 127,660 107,180 0.8396Hamilton 228,435 153,545 0.6722Kitchener 132,825 87,460 0.6585London 131,955 116,675 0.8842Montreal 1,265,055 615,180 0.4863Ottawa 347,975 259,675 0.7462Quebec 237,260 103,485 0.4362Regina 79,565 74,605 0.9377Saint John 45,865 40,795 0.8895Saskatoon 71,730 67,395 0.9396St. Catherines 126,110 78,975 0.6262St. John's 60,460 51,370 0.8497Sudbury 58,180 45,630 0.7843Thunder Bay 54,000 50,785 0.9405Toronto 1,571,455 545,160 0.3469Vancouver 646,435 276,215 0.4273Victoria 105,130 60,820 0.5785Windsor 96,080 84,975 0.8844Winnipeg 284,785 268,275 0.9420Mean 278,188 162,254 0.7518APPENDIX 3ESTIMATES OF WASTEFUL COMMUTING DERIVED USING EDMONSTONDENSITY GRADIENT ESTIMATESChapter 5 presented estimates of wasteful commuting based ondensity gradient parameters obtained using Macauley estimationtechniques. This appendix presents estimates of wasteful commutingbased on density gradient parameters obtained using Edmonstonestimation techniques. The tables presented in this appendix areanalagous to Tables 5.1 and 5.2 in Chapter 5.Table A3.1: Optimal and Actual Commute Characteristics Using 1981Edmonston Gradient EstimatesCMA^A^B^C D E F GCalgary 4.977 5.381 -0.404 10.64 9.70 15.50 38.5Charlottetown 2.140 1.077 1.064 4.21 2.37 5.97 12.7Chicoutimi 8.640 3.343 5.297 8.30 14.71 18.14 16.5Edmonton 6.187 5.705 0.482 10.49 11.31 18.20 36.1Halifax 5.365 3.743 1.622 7.97 7.94 15.70 27.1Hamilton 6.240 5.255 0.984 15.63 10.69 18.51 20.7Kitchener 6.715 6.702 0.013 12.09 13.92 17.10 15.5London 3.554 3.660 -0.105 8.42 6.35 10.88 21.7Montreal 10.845 8.282 2.563 12.24 18.60 33.70 29.9Ottawa 7.446 4.904 2.542 10.39 11.91 20.99 31.0Quebec 8.635 6.677 1.959 11.29 15.43 22.45 28.8Regina 2.516 2.519 -0.003 8.54 4.00 7.93 31.7Saint John 7.716 5.049 2.667 11.11 13.41 17.33 20.8Saskatoon 3.171 2.677 0.497 7.96 4.93 9.35 38.7St. Catherines 7.792 5.213 2.579 10.59 11.57 23.39 17.7St. John's 3.393 1.898 1.495 8.19 3.60 10.95 16.5Sudbury 7.212 5.753 1.459 13.62 14.43 15.66 26.4Thunder Bay 4.509 4.039 0.470 10.13 7.69 12.07 22.1Toronto 12.674 8.636 4.038 12.25 20.21 41.31 34.3Vancouver 9.640 7.896 1.744 13.56 16.06 30.83 29.6Victoria 4.561 2.470 2.091 10.65 5.65 14.10 12.0Windsor 4.026 3.163 0.858 10.18 5.46 13.00 13.6Winnipeg 5.124 5.551 -0.427 11.09 10.11 15.68 23.4Sample Mean 6.221 4.765 1.456 10.41 10.44 17.73 24.6Adj Mean 6.679 4.868 1.812 10.57 11.05 18.83 23.7Hamilton Mean 14.001 12.199 1.802 14.00 19.46 36.21 n.a.NOTES: All distances in kilometers; A m necessary commute if complete centralization ofemployment is assumed; B a potential commute savings resulting from employmentdecentralization; C m optimum commute (i.e. A - B); D m observed mean commute from the 1977urban concerns survey; E a average commute randomly assigning jobs to houses; F m radius atwhich population density declines to 38.61 people per square kilometer (i.e. 100 people persquare mile); G a actual radius of the CMA; n.a. m not available; Adj mean excludes Calgary,London, Regina and Winnipeg because these CMA's had values for C less than zero.Table A3.2: Alternative Measures of Wasteful Commuting Derived FromEstimates in Table A3.1CMA^D-C^D/CkmsC/D A/D D/E F/G TotalPopulationCalgary 11.04 26.3 3.8 46.8 109.7 0.40 625,966Charlottetown 3.15 4.0 25.2 50.8 177.7 0.47 44,999Chicoutimi 3.01 1.6 63.8 104.0 56.5 1.10 135,172Edmonton 10.01 21.8 4.6 59.0 92.7 0.50 657,057Halifax 6.35 4.9 20.4 67.3 100.3 0.58 277,727Hamilton 14.65 15.9 6.3 39.9 146.2 0.89 542,095Kitchener 12.07 907.7 0.1 55.6 86.9 1.10 287,801London 8.53 80.0 1.2 42.2 132.6 0.50 283,668Montreal 9.97 4.8 20.9 88.6 65.8 1.13 2,828,349Ottawa 7.85 4.1 24.5 71.7 87.2 0.68 717,978Quebec 9.33 5.8 17.4 76.5 73.1 0.78 576,075Regina 8.54 3443.9 29.5 213.6 0.25 173,226Saint John 8.45 4.2 24.0 69.4 82.9 0.83 114,048Saskatoon 7.47 16.1 6.2 39.8 161.6 0.24 175,058St. Catherines 8.01 4.1 24.4 73.6 91.5 1.26 304,353St. John's 6.70 5.5 18.3 41.4 227.5 0.66 154,820Sudbury 12.16 9.3 10.7 53.0 94.4 10.59 149,923Thunder Bay 9.66 21.6 4.6 44.5 131.8 0.55 121,379Toronto 8.21 3.0 33.0 103.5 60.6 1.20 2,998,947Vancouver 11.82 7.8 12.9 71.1 84.4 1.04 1,268,183Victoria 8.55 5.1 19.6 42.8 188.4 1.18 233,481Windsor 9.32 11.9 8.4 39.6 186.4 0.96 246,110Winnipeg 11.52 26.0 3.8 46.2 109.8 0.67 584,842Sample Mean 8.96 109.5 14.6 59.0 120.1 0.76 n.a.Mean w/o -C's 8.76 55.7 18.2 62.7 115.6 0.83 n.a.Adjusted Mean n.a. 8.4 19.2 n.a. n.a. n.a. n.a.Hamilton Mean 12.29 12.2 12.7 95.9 78.3 n.a. n.a.NOTES: The mean w/o CMAs with negative estimated C values excludes Calgary, London, Reginaand Winnipeg; the adjusted mean excludes Kitchener in addition to the four CMA's excludedfrom mean w/o -C's; n.a. E not applicable.- 173 -

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