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Individual choice behaviour and urban commuting Torchinsky, Raymon Lev 1987

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INDIVIDUAL CHOICE BEHAVIOUR AND URBAN COMMUTING by RAYMON L. TORCHINSKY A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF x DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Geography) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1987 © Raymon L. Torchinsky, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) ABSTRACT Urban commuting patterns can be viewed as the spatial manifestation of the outcome of labour market processes. Recent theoretical and empirical work investigating urban labour markets has emphasized the role of spatial wage differentials in mediating the interrelationship between labour supply and demand distributions and the dynamics of land-use change. This thesis represents an extension of such research. A simulation approach to commuting modelling, based on the explicit characterization of the interrelationship between urban location and interaction in terms of labour market processes, is developed. The solution path logic of the simulation model is designed to provide normative commuting outcomes, given the spatial pattern of labour supply and demand, under a wide range of assumptions concerning labour market processes and choice-making behaviour of market participants. An explicit characterization of the labour market, based on the specification of an endogenous behavioural assumption set, defines a model version. Thus, the model ma}' be used to test the ability of various behavioural constructs to explain empirical commuting patterns. The justification and internal logic underlying the development of a specific model version is presented. This version is based on the assumption that the decision by a worker to apply for a job is objectively rational, given that the market environment does not provide certainty as to the outcome of an application. It is shown that such choice behaviour is analogous to the game-theoretic mixed strategy solution to non-cooperative games under uncertainty. The algorithm of the operational model incorporating this approach is detailed. The model was tested on empirical commuting patterns derived from Vancouver Census data, and model results were compared with those obtained from a positive entropy-based model. Commuting predictions exhibited a level of accuracy comparable to that achieved by the calibrated entropy model. ii T A B L E OF CONTENTS Abstract ii List of Figures v Chapter I. INTRODUCTION 1 Chapter II. COMMUTING PATTERNS AND THE URBAN LABOUR MARKET 7 A. Introduction 7 B. Wages and the Intraurban Location of Employment 7 1. Empirical Evidence of Wage Differentials 12 C. Commuting Models and Wage Differentials 15 1. Equilibrium Theory of Urban Wage Differentials ... 16 2. The Linear Programming Commuting Model 19 3. The Entropy Model 20 D. Conclusion 25 Chapter III. INDIVIDUAL CHOICE COMMUTING MODEL 27 A. Introduction 27 1. Simulating the Urban Labour Market - A Normative Approach 30 B. Solution Path Algorithms 36 1. Labour Market Simulation Models 37 2. Analog Models Generating Optimal Solutions 42 C. Description of Model Variables 45 Chapter IV. DECISION-MAKING AND THE ICCM-1 MODEL 49 A. Behavioural Assumptions 50 1. Objective, Subjective and Bounded Rationality 50 B. Uncertainty 58 1. Stability of a Suboptimal Solution 58 2. Determinate and Probablistic Choice Behaviour 62 a. Expected Utility 62 b. Risk Aversion 64 c. Uncertainty and Outcome Probabilities 70 C. The Mixed Strategy Solution 71 1. A Lottery Analogy 71 2. Application to the Labour Market 81 D. Conclusion 85 Chapter V. TWO VERSIONS OF THE ICCM-1 MODEL 86 A. Mixed Strategy and Job Application Modelling 86 1. The Mixed Strategy and Dispersed Worker Residences 88 2. Two Sub-versions of ICCM-1 93 a. ICCM-la 93 b. ICCM-lb 95 c. Allocation Example 98 iii d. Implications of the Solution Path 105 B. Extension co a Multiple-Application Situation 107 Chapter VI. ICCM-1 ALGORITHM 112 Chapter VII. TESTING OF THE ICCM-1 MODEL 119 A. Introduction 119 B. A Summary of the Economic Structure of Vancouver ... 120 C. Data 125 1. 1981 Vancouver Commuting Data 126 2. 1971 Vancouver Commuting Data 132 3. The Travel Cost Matrix 133 a. The Travel Cost Metric 135 b. Measuring Travel Time 138 D. Results of Model Runs on 1981 Data 141 1. Measures of Fit 141 a. The x 2 Statistic 143 b. The R 2 Measure 146 c. The Log-Likelihood Ratio Test Statistic 147 2. Statistical Results: Disaggregated Data Sets 148 a. Comparison of Goodness-of-Fit Statistics 153 b. Differences Between Male and Female Commuting Patterns 155 c. Sensitivity of the Entropy Deterrence Function Parameter 157 d. Analysis of Rent Surfaces 160 e. Analysis of Wage Surfaces 161 f. Stability of the ICCM-lb Solutions 164 3. Predictions Using Aggregated Data Sets 165 E. Comparison of 1971 to 1981 Projections 167 F. Summary 171 Chapter VIII. CONCLUSION 173 A. Discussion 175 Bibliography 177 Appendix: Empirical and Modelled Commuting Patterns 188 iv LIST OF FIGURES TH-1 ICCM Solution Path Algorithms and Model 38 Versions V - l ICCM-1 Example: Net Wage Distribution V-2 ICCM-1 Example: Data V-3 ICCM-1 Example: Strategy Matrix V-4 ICCM-1 Example: Allocation Matrix V-5 ICCM-1 Example: Expected Utility of Choice Strategies VII-1 Occupational Structure of Greater Vancouver 124 VII-2a Municipal Aggregation 128 VII-2b Vancouver CMA: Municipal Zones 129 VII-3 Vancouver Map Pattern Adjustments: 1971 - 134 1981 VII-4 Travel Time Matrix 140 VII-5 Goodness-of-Fit Measures and Direction of 144 Prediction Error VII-6a Model Results: Statistical Analysis, 1981 Data 149 VII-6b Model Results: Statistical Analysis, 1981 Data 150 VII-6c Model Results: Statistical Analysis, 1981 Data 151 VII-7 Sales Occupations: Travel Time Cumulative 158 Frequency Distributions VII-8 Sensitivity Analysis of Entropy Model Beta 159 Parameter VII-9 Analysis of Selected ICCM-lb Generated Rent 162 Distributions VII-10 Analysis of Selected ICCM-lb Generated Wage 163 Distributions VII-11 Model Results: Statistical Analysis, 1981 166 Aggregated Data VII-12 Model Comparison: ICCM-lb / Entropy 169 Projections, 1971-1981 100 101 102 103 104 v CHAPTER I. INTRODUCTION The separation of home and workplace is a fundamental aspect of the organization of urban space in industrialized societies. For the vast majority of urban workers, the trips to and from work, undertaken at the same times and via the same mode and route 5 days a week with little variation, constitute a routine which establishes a significant portion of individual activity space and, in aggregate, creates the bulk of the demand for urban transportation services. Commuting is the physical means of linkage between areas allotted to interdependent functions: zones of production by labour, and zones of reproduction of labour. Commuting flows are thus the spatial expression of the current outcome of the urban labour market. The concept of a commuting trip, as it will be used in this thesis, refers to the travelling a worker performs with regularity between home and workplace, regardless of distance. There is no minimum distance requirement, so that workers who work at their place of residence will be treated as having a commuting trip of zero distance. A distinction is drawn between a commuting trip and the journey-to-work, in that the latter is more inclusive as it does not require that the trip is one of a repetitive series. For example, construction workers and sales representatives may travel to work each day, but their destinations are not fixed. In these cases, it is incorrect to identify their journeys-to-work as commuting trips. Individual commuting trips can be aggregated to produce two types of spatial distribution: commuting pattern and commuting flow. The commuting pattern is 1 INTRODUCTION / 2 the set of stable origin / destination pairs, with each element corresponding to the job and residence location of an employed worker. Subsets of the commuting pattern can be defined by selecting origin / destination pairs according to criteria based on occupation or industry classification, demographic characteristics, etc. Spatial aggregation of the commuting pattern requires an a priori organization of locations into a set of zones. The distribution of trip-end pairs can then be expressed as a matrix whose size is determined by the number of origin and destination zones utilized. The row (origin) and column (destination) sums (or marginals) of the matrix correspond to vectors defining the spatial distribution of trip generation and attraction. The commuting flow is based on the way in which trips are accomplished with respect to the transportation network, and refers to the distribution of mode-specific link volumes. These definitions correspond to the distinction made in transport planning models between the distribution component, defining the demand for travel between origin and destination zones, and the assignment component, concerned with the routing aspect of satisfying travel demand. The focus of this thesis is an investigation of the processes underlying the generation of commuting patterns; the terms pattern and flow will be used interchangeably to refer to the magnitude of inter-zonal interaction. This issue has been a central target of research activity by urban and transport geographers, planners, civil engineers, economists and operations research scientists for decades. The primary impetus behind this activity was the practical necessity to plan the infrastructure required to satisfy the demand for transport services initiated by the post-war boom in automobile ownership levels and the resultant INTRODUCTION / 3 expansion of urban areas. The problem has been viewed traditionally as one of establishing the basis of a static commuting pattern solution to an urban situation in which the land-use pattern (the spatial vectors of trip generation and attraction) is known a priori. The problem of understanding the relationship between land-use and transport supply was not so much ignored as separated from the search for a static solution to the commuting distribution problem. As a result, the methods adopted to model commuting patterns were based on theoretical constructs that had little to do with the processes underlying the generation of empirical commuting patterns. It is the purpose of the research project reported in this thesis to investigate a normative simulation approach to modelling commuting patterns. In order to accomplish this, the research program was divided into three parts. First, a general simulation model is developed, based explicitly on the theory underlying the derivation of urban spatial wage differentials developed by Goldner (1955) and Moses (1962). Specifically, it is assumed that: a) wages paid to workers are homogeneous within plants for a given job category; b) workers apply for job openings in response to the net wage, i.e., the wage offer less commuting costs; and c) employers are cost minimizers, so that plant wages are set at the level required to attract the necessary labour force. Thus, the spatial wage surface and the commuting pattern are inter-related phenomena, determined by labour market processes operating in relation to a given spatial distribution of labour supply and demand. The simulation approach, termed the Individual Choice Commuting Model (ICCM), INTRODUCTION / 4 is in fact a family of model versions, each of which is based on an explicit characterization of the choice-making behaviour of workers in the labour market. The model can accommodate a wide range of behavioural assumptions, and is designed to generate a solution based on a competitive market process as defined by the behavioural assumption set and the (exogenous) labour supply and demand surfaces. The model is normative in that, as market processes are specified prior to simulation, there are no parameters to calibrate. The second part of the research program is to operationalize a specific version of the model, termed the ICCM-1. This model incorporates a characterization of market participants as being objectively rational decision-makers operating in a market environment that is both uncertain and non-equilibrating: i.e,, a worker applying for a job is not assured that the application will be successful, and there is no means available to ensure that the market will attain an equilibrium state. The third aspect of the research is to investigate the ability of this model version to explain empirical commuting patterns. The objective of the testing procedure is to evaluate the potential of the ICCM-1 model as a tool for a) generating testable hypotheses for the deductive investigation of spatial wage and commuting patterns, and b) for predicting commuting patterns in a planning context. Chapter II presents a discussion of previous research that has investigated the relationship between the functional and spatial interaction of labour supply and INTRODUCTION / 5 demand distributions. It is shown that theoretical and empirical work has emphasized the importance of labour market processes in the generation of industrial location and commuting flow patterns, specifically as the spatial manifestations of labour cost differentials. However, the complex nature of this relationship has resisted specification by descriptive means, and theoretical frameworks have, in the main, been based on highly restrictive assumptions. Furthermore, attempts to base interaction modelling on labour market processes have been neither extensive nor very successful; the restrictive assumptions of equilibrium theory have dominated these approaches. Chapter III introduces a flexible approach to modelling static commuting patterns, termed the Individual Choice Commuting Model. The model structure is based on an iteration procedure which explicitly simulates labour market processes in order to generate commuting patterns and spatial wage and rent surfaces. The main feature of the model is that it may be used to establish normative outcomes under a variety of assumptions regarding the behaviour of labour market participants. Thus, it may be used to generate predictions which can form the substantive content of testable hypotheses concerning the impact of choice behaviour on urban spatial phenomena. Chapter IV presents the behavioural logic underlying an operationalized normative version of the model, termed ICCM-1. The approach is grounded in the theory of games to simulate the commuting choices of objectively rational workers in an uncertain environment. Chapter V continues the discussion in terms of the specification of the model as an application' of behavioural theory to the urban INTRODUCTION / 6 labour market. Two sub-versions of the model are presented in order to demonstrate the ability of the simulation approach to generate testable predictions based on specific sets of behavioural assumptions. Chapter VI contains the algorithm of the ICCM-1 model. The results of a series of test runs comparing the abilities of the two ICCM-1 versions and an entropy model to predict commuting patterns are presented in Chapter VII. The test indicates that the behavioural foundation of the normative ICCM-lb model is sufficient to generate pattern predictions that rival the accuracy of a calibrated entropy model. In addition, it is shown that interpretation of the entropy model's deterrence parameters are problematic, while the uncalibrated nature of the ICCM-1 procedure makes interpretation of the results straightforward. CHAPTER II. COMMUTING PATTERNS AND THE URBAN LABOUR MARKET A. INTRODUCTION This chapter explores the relationship between commuting and location patterns, specifically in terms of the determinants of spatial wage differentials and the operation of the urban labour market. It is argued that labour demand and supply surfaces, characterized by clustering of employment locations and dispersal of residential locations, result in a wage surface which is central to the understanding of commuting patterns. Current means of structuring commuting models on the labour market process are discussed. B. WAGES AND THE INTRAURBAN LOCATION OF EMPLOYMENT The land-use patterns that influence commuting patterns are to a large degree determined by the relative impact, in a given situation, of the economies and diseconomies of scale in the location of economic activities. Scale economies create centralized clusters of activity; such economies are bounded, however, in that costs tend to rise after centralization reaches a critical level. The point at which this occurs differs by industrial sector (Czamanski, 1965) and according to the production and transportation technology in use (Fales and Moses, 1967; see Gordon, 1978, for a dissenting view written from a Marxist perspective). A number of factors are involved in producing the spatial clustering of economic activity. Firstly, scale economies can be gained through spatial concentration of production. Internal scale economies are those obtainable as a function of the size 7 COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 8 of individual producing units (plants), as determined by either the specific technology utilized in a single-output production process, or in the level of vertical integration obtained in a multi-product plant. The latter category has been termed economies of scope. (Panzar and Willig, 1975). External scale economies, termed agglomeration economies, are achieved through plant clustering. The importance of location choice as a strategy to control expenditures associated with forward and backward linkages is a significant element in the factor cost minimization theory of industrial location (Weber, 1929; Isard, 1960). Benefits of proximity that are internal to an industry (but external to the individual plant) are economies of localization, while benefits derived from the juxtaposition of different industries are economies of urbanization. Goldstein and Gronberg have argued that, at the urban scale, agglomeration economies are the multi-plant equivalent of economies of scope: Urban areas can be viewed as "vehicles" for spatial integration, in the same way that vertically integrated firms gain efficiency from engaging in multi-output production. It is not simply the scale of activity in the area that is important, but the improvement in productive efficiency from placing related activities at a common location. (G.S. Goldstein and T.J. Gronberg, 1984; page 92) These scale benefits are not limited to the industrial sector of the urban economy, but can be achieved also by the commercial and service sectors (Daniels, 1980). Secondly, scale economies can be gained as a result of the provision of transportation infrastructure, both in terms of the site-specific location of COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 9 trans-shipment facilities and in terms of the relative accessibility consequences of network structure, t This implies that employment concentration may occur not only as a result of the characteristics of individual or inter-related production processes, but also through the common attraction to specific locations of otherwise unrelated activities (Czamanski, 1976). Richter (1970) and Streit (1969) use the neutral term "geographic association" to refer to spatial concentration of activity, regardless of functional cause. Agglomeration provides lower linkage costs, but also can result in increasing land and labour costs. In a survey of manufacturers in Cincinnati, Schmenner (1978) identified land cost differentials as the primary factor in decisions to relocate to decentralized sites. A study of spatial differences in land prices by Richardson, Vipond and Furbey (1974) supports the hypothesis that price gradients are stable over time. Thus, firms obtaining most benefit from agglomeration economies locate at or remain in central sites in spite of high land costs, while those sensitive to scale diseconomies locate at dispersed sites. The importance of spatial differentials in urban labour costs was first noted by Goldner (1955). He suggested that individual workers have "normal preference areas", centred on their place of residence, within which they are willing to work. If employment is sufficiently concentrated so that some workers must commute beyond the boundaries of their preference area, the labour market is in t For example, Coughlin, et al. (1976) and Miller (1974) discuss the impact of truck terminal location; Torchinsky (1981) discusses the effects of rail terminals on industrial concentration; a study by Moses and Williamson (1967) indicates the effects of dock facilities on location patterns; Hoare (1974) investigates the impact of airport location on the spatial pattern of employment; Lachene (1964) analyzes the effect of urban road network design on land-use patterns. COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 10 a situation of "locational incompatibility". Such concentration need not only occur at the urban centre: The deficiencies of labor supply can most clearly be visualized at the central business district and, in some cases, at the periphery of the metropolis. The deficiency in the downtown centre is the result of the extreme concentration of metropolitan services with their large and intensified demand for labor measured against a labor supply curtailed by the workers' locational preferences for jobs farther out from the centre of the city. On the outskirts of the metropolis, individual plants and business centres may mushroom in size, exerting a need for manpower that exceeds the thinly distributed labor supply nearby. (W. Goldner, 1955; page 125) In order to attract sufficient labour, employers offer higher wages to compensate for the disutility of the longer commute: To adapt to this incompatibility, higher plant wage levels act to broaden the workers' preference area and therefore draw workers beyond the limits of their normal locational preferences. (W. Goldner, 1955; page 126). Thus, at any point in the urban area, the supply of labour in relation to demand determines the wage level, which in turn affects the value of the plant location to the firm. In addition, the wage level required to attract workers depends on commuting behaviour - specifically, on the size of preference areas. If plant wage levels are homogeneous, so that all workers in a given plant doing a similar job earn the same wage, then sensitivity of plant location costs to labour supply can be very high. The overall wage level will be determined by COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 11 the wage required to attract the farthest employee, regardless of the distribution of the commuting distances made by other employees. Assuming that the cost of production is a function of the factor costs of capital and labour inputs, and selling prices are determined by a competitive market, Scott (1981) points out that increases in labour costs can have a significant negative effect on the rate of return on capital investment: It seems to be well within the bounds of possibility, for example, for decreases of the order of 20% or 25% in profits to be brought about by increases in the order of only 5% in wages (depending on the relative magnitudes of the capital-labour ratio and the production price). (A.J. Scott, 1981; page 25) Scott continues this line of argument by postulating an intra-urban Heckscher-Olin effect (Scott, 1981; 1983; 1984): concentration of labour-intensive activities in core areas is due to the relative advantage in labour costs those areas enjoy as a result of their greater access to the labour pool. The economies of scale gained by the clustering of functionally inter-related small-scale labour-intensive activities are balanced by the tendency for wage rates to rise in response to the high aggregate labour demand - thus, centralized locations minimize the negative effects on profits resulting from locational incompatibility. Large-scale, capital intensive activities that are relatively footloose due to internalization of scale economies are able to take advantage of lower land costs in peripheral locations as they generate a spatially dispersed demand for labour. This view of the inter-relationship between the spatial distribution of labour supply and demand, the urban wage surface and commuting patterns forms the COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 12 foundation of the research presented in this thesis. Specifically, if it is accepted that, in a static sense, current wage differentials are determined as the outcome of labour market processes operating in response to the spatial distribution of labour supply and demand, then the commuting pattern can be interpreted directly in terms of the labour market. Furthermore, the wage surface and the commuting pattern are simultaneously the cause and the effect of each other; this inter-relationship provides a logical foundation for modelling the spatial labour market. 1. Empirical Evidence of Wage Differentials Empirical evidence of the existence of urban wage differentials is limited, both as a result of the difficulty of obtaining accurate data, and due to the multitude of intervening variables in the determination of wage dispersion (Goodman, 1970). Published government sources rarely disaggregate wage information to a sufficient spatial scale to allow a thorough investigation of the phenomenon. Scott (1981) utilizes Statistics Canada data on manufacturing wages based on municipalities within the Toronto metropolitan area to conclude that wages increase with distance from the urban centre; however, he aggregates employees in all manufacturing industries, so that no control on skill level, education, etc. is possible. In other words, if jobs in labour-intensive industries require less skill than those in capital-intensive industries, wage differentials may result from the distribution of capital intensity rather than from the interaction of labour supply and demand surfaces. Segal (1960) reports similar findings in New York, but again the data is highly aggregated. Wachter (1972) found evidence of a positive wage gradient in the Boston labour market. In contrast, Eberts (1980) finds that COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 13 wages for municipal public employees in the Chicago region, disaggregated by occupation, follow a negative gradient from the centre. Moffitt (1977) reports similar results in an investigation of central city / suburb wage differences. In a detailed study of wage differentials in various industries and occupations in Chicago, Shultz and Rees (1970) found that a dichotomous regional variable was significant in explaining differences in wages in most occupations. They note: The most consistent result is that the South region is a high wage region for blue-collar occupations. We ascribe this result to the concentration of heavy industry, particularly basic steel and petroleum refining, in the South region. ... The whole wage pattern described by the regional variables might be roughly described as a wage gradient that is lowest in the north-east corner of the Chicago area and highest in the southeast. This gradient is related to the pattern of location of employment and residences, since the North and West region has a heavier concentration of residential neighbourhoods and the South and East region has a heavier concentration of nonresidential areas. (A. Rees and G.P. Shultz, 1970; pages 178-179) Seltzer (1951) found a similar spatial pattern of wages in the steel industry in Chicago 20 years earlier; this suggests that such differentials may be highly stable. Kain (1968) suggested that housing segregation may affect the distribution of wage levels in American cities. In a study of the effects of such discrimination in the labour market of San Francisco, Straszheim (1978) found that dummy variables representing 6 urban sub-regions were significant in explaining wage differentials after controlling for education, age and race. In discussing these COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 14 results, he ascribes the regional effect to labour supply / demand factors: Comparisons of the estimated wages for black and white workers reveals that for less educated workers, those with a high school education or less, wage differentials between blacks and whites are highest in the city center and less in the ring, reflecting the supply of black workers residing in ghetto areas. ... Suburbanization of higher income black households is such that the wage gradient for professional jobs for blacks more clearly resembles the negative gradient for white wages than the positive gradient for low income black employment. (M. Straszheim, 1978; page 139) *\ A number of studies have investigated the relationship between wages and commuting distance for individual employees. This research is theoretically problematic in that underlying it is the concept of wage differentials as compensation for individual commuting costs, rather than as compensation for commuting from plant-specific labour shed boundaries. That is, if the distribution of commuting costs incurred by employees for any given plant is normally distributed, the mean commuting cost and the wage level will be correlated over all plants, and a random sample of workers will show a relationship between commuting cost and wage rate. If this is not the case (and it is likely that commuting cost distributions within plants are positively skewed), then a random sample may show no relationship between commuting costs and wages, even though systematic spatial wage differentials exist. Rees and Shultz (1970) find that a distance variable, entered in logarithmic form, is significant in explaining wage differentials in 8 of 10 occupations tested. This result supports the expectation of skewed plant-specific commuting distributions discussed above. Kasper (1983) studied the relationship between commuting cost and wage COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 15 adjustments for a sample of recently moved workers in Glasgow. After controlling for sex, age, experience and training, he found that, in a logged regression, the commuting cost variable was highly significant in explaining wage adjustments, while in a linear form the variable was not significant. Again, this result supports the conclusion that wage levels are related to labour shed boundaries. The effect of commuting shed extent on wage rates may be inferred from studies on inter-urban wage differentials. For example, Izraeli (1977), Hoch and Drake (1974) and Kenny and Denslow (1980) found that wage levels are associated with city size. This relationship may be ephemeral, however, as Miller (1983) found that inter-urban wage differentials were best explained by differences in labour productivity. In a study investigating the dynamics of employment decentralization, Steinnes (1977) found that suburbanization of residences occurred in advance of suburbanization of jobs. He concludes that the view that employment attracts labour is incorrect; rather, increasing supply of labour attracts new employment. This evidence supports Scott's hypothesis of the importance of labour costs to the intraurban production location decision. C. COMMUTING MODELS AND WAGE DIFFERENTIALS There have been two attempts to model urban commuting patterns specifically with reference to the relationship between wage rates and the journey to work. The first of these, based on the linear transportation problem first examined by Hitchcock (1941), generates a globally optimal solution in which the total COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 16 commuting cost in the system modelled is minimized. This approach corresponds to the normative solution of a competitive economy in equilibrium (Stevens, 1961), assuming rational workers with homogenous utility function. The dual of the program provides the shadow-prices, or location rents of residences and workplaces and the wages paid to labour at each employment location, t The second approach is based on the random utility interpretation of the doubly-constrained entropy model (Domenich and McFadden, 1975; Tanner, 1980). This approach is designed to replicate the outcome of a competitive labour market in which the workers are rational decision-makers having non-homogenous utility functions; thus, their rationality is subjective as opposed to the objective variety assumed by the linear programming approach. The entropy model is a special form of a gravity interaction model (Tocalis, 1975). While the gravity model has spawned numerous specific versions that have been applied to commuting problems, the entropy approach is the only one that can be interpreted in terms of labour market processes (Tanner, 1980). 1. Equilibrium Theory of Urban Wage Differentials Both of these models are based on the assumption of a competitive labour market in which rational behaviour under conditions of perfect information generates a stable equilibrium solution. Theoretical research on urban wage differentials has been an offshoot of work done by economists investigating equilibrium conditions of urban structure. For example, the urban land-use models t See, for example, Ford and Fulkerson (1955) and Dantzig (1960) for a discussion of simultaneous solutions to the primal and dual of the transportation problem. COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 17 of Alonso (1964), Muth (1969) and Kain (1962) attempt to derive the equilibrium spatial pattern of urban land-use by establishing the bid-rent curves of cost-minimizing firms and utility-maximizing workers. Unit land values reflect relative accessibility to markets and factor inputs for firms and to jobs for workers; allocation of land rings/sectors to industrial or residential uses is accomplished through the operation of a competitive land market. Workers trade off the disutility of commuting (in terms of both time and money) for the utility associated with other goods, so that the slope of the bid-rent curve for residential land is a reflection of the elasticity of substitution between commuting costs and consumption of space. Thus, in the absence of income effects, land-use patterns are largely determined by the preference of households for low versus high density housing. The process by which the land-use pattern comes into existence requires that workers know the accessibility value of different housing sites so that rational bids can be calculated. This suggests that either all employment opportunities occur at a single location (usually the CBD), or that the allocation of workers to pre-existing employment centres is accomplished prior to the allocation of workers to residences. In both cases workers effectively will minimize their commuting costs in that they will work at the job location as close to their place of residence as possible. Moses (1962) first noted that such an equilibrium entails a wage gradient, as a fixed wage across the urban area means that workers would always be better off to work at any location closer to their place of residence than is the CBD. As a result, dispersed firms could attract labour with reduced wages. He investigated the effect on the wage gradient of various assumptions concerning COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 18 the relationship between commuting distance and utility, presupposing that the wage level offered at the CBD set the peak of the gradient. His analysis went beyond that of Goldner (1955) in that commuting costs were set in a utility framework, replacing the looser concept of a commuting preference area. He established that workers respond to the net wage in deciding on their location of employment; that is, given a wage Wj paid at employment site j, a worker residing at i will gain a net wage N. . by accepting the job at j equal to: (1) N. . = W. - r. . iJ J iJ where r. . is the cost of commuting between i and j. Moses called specific attention to the importance of modelling commuting behaviour on labour market processes: The analysis raises some interesting questions about the logic of traffic assignment techniques used in origin-destination studies, at least so far as work trips are concerned. On the whole, they have tended to be rather mechanical and have ignored the economic content of the problem. The present paper tends to view traffic assignment as a problem in spatial competition involving all competing employment and labor supply zones within a city. Wage rates in all employment zones, and the entire matrix of travel time and money costs are essential to rational allocation of interzonal travel. (L. Moses, 1962; page 62) Ravallion (1979) showed that the wage gradient need not decline with distance from the CBD. By explicitly including labour supply and demand distributions in the analysis, he simultaneously establishes wage differentials and the labour shed boundary between employment locations. He also notes that wages could be expected to rise significantly at the urban boundary, a conclusion supported by COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 19 Scott's empirical study (1981). 2. The Linear Programming Commuting Model The underlying assumption of these studies is that the urban labour market is perfectly competitive, composed of rational utility-maximizing workers with full knowledge. Thus, an equilibrium solution entails an efficient (or Pareto optimal) commuting pattern which minimizes total commuting costs. If the distribution of jobs and workers' residences is established a priori, workers will maximize their net wages and employers will minimize their wage bill if each worker is matched to the job as close as possible to his/her place of residence. Assuming that workers' utility functions are homogeneous and the disutility of work does not vary with the job location, Hochman, Fishelson and Pines (1975) state: Two households which differ from each other in the locations of their places of work and their residences must have the same utility in equilibrium. But it is also known that a competitive equilibrium is a Pareto optimum - that is, no household can increase its welfare level without reducing the welfare level of other households. This necessarity implies that the distribution of work trips must minimize total transportation costs for the given distribution of places of work and residences. For, if transportation costs can be reduced by interchanging either the residences of any two households or their places of work, then resources can be saved and the allocation will be neither a Pareto optimum nor competitive equilibrium. (0. Hochman, G. Fishelson, D. Pines, 1975; page 274) The dual of the linear program maximizes the shadow-prices (rents) associated with employment and residence locations. The residence rent is equal to the net wage (see, for example, the approach of Ford and Fulkerson, 1955), thus wage COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 20 levels can be calculated as the difference between net wages and transport costs. Net wages for any given location are equalized for all residents, so that the variance of rents within a zone is 0. The ability of the linear programming model to predict actual commuting patterns is minimal. The model will always allocate as many workers as possible to jobs within zones; cross-commuting is inefficient, and so cannot exist. Thus, strict boundaries between labour sheds always occur. Hochman, Fishelson and Pines (1975) tested the model with commuting data from two Israeli cities, using travel time to estimate travel cost. Empirical mean times were 25% to 93% longer than predicted times. Similar findings for Toronto, using commuting data from the 1971 Census, are reported by Gera and Kuhn (1977, 1978). The main attraction of this model is that it requires minimal data: only the zonal sums of workers and jobs, and an estimate of inter-zonal travel costs are needed. Thus, its main application in urban research has been as the distribution component of integrated transportation / land-use models which are used to simulate urban growth processes over time (for example, the Herbert-Stevens model and its descendants; Putman (1974) provides a full description of these applications). 3. The Entropy Model The entropy model used for commuting applications is identical to a doubly-constrained gravity model in which the friction factor of distance is in the form of a negative exponential function. Thus, if TV . is the predicted commuting COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 21 flow between i and j, E. is the number of jobs in employment zone j, B . is the J i number of workers in residence zone i, r. . is the cost of commuting between i and j, and B is the friction factor parameter, then: (2) TV . = E . B. D ' . O'. e ' ® T l J i j J 1 J 1 D ' . and O ' . are balancing factors which ensure that the two constraint vectors J * on trip origins and destinations are met: and (3) Z. TV . = E. J V J (4) I. T*. . = B. i i j i In addition, a third constraint is used to establish the value of B. If T. is the empirical commuting flow between i and j, then the model is iterated until: (5) L. . (TV . • T. .) = I. . (T. . • r. .) This constraint ensures that total predicted travel cost is equal to the total empirical travel cost, and can be considered a budget constraint (Nijkamp, 1975). Thus the entropy model is a positive model; the friction factor is calibrated to fit empirical data. The logic of the entropy approach is based on information theory (Snickars and Weibull, 1977; Batty, 1974). Briefly, the model establishes the most likely trip distribution by maximizing the randomness of the commuting pattern, given the constraint set. Thus, the information contained in the predicted pattern is defined entirely by the constraints. It has been argued that the information theoretic interpretation separates the model from the criticism that it is based on an inappropriate analogy to the second law of thermodynamics (see Senior, 1979; Sayer, 1976, dissents from this view). COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 22 The view of the entropy model that is of most interest here is that predicted patterns can be interpreted as the equilibrium outcome of a competitive market in which utilities are subjective and can be estimated by a randomly distributed variable. The random utility interpretation of the entropy approach to commuting modelling is based on the assumption that workers are rational decision-makers in a competitive economy with full information, and their utility functions are distributed according to the log-Weibull distribution. In this way, the behavioural logic of the entropy approach is identical to that assumed by logit binary choice models (Domenich and McFadden, 1975). The discussion below follows the presentation by Tanner (1980), Broughton and Tanner (1983) and Broughton (1981). The main feature of this approach is to replace the interpretation of the j5 parameter as a distance deterrence function (based on the gravity analogy) to the measure of variance in the distribution of population utility functions. Thus, the higher the /? parameter, the smaller is the variance in utility perception. The model assumes that all workers are identical, differing only in where they choose to live and work. Jobs and houses also are identical, differing only in location. A stable equilibrium pattern of commuting travel is established by maximizing the utility of workers: The essential shift of emphasis between the deterrence function approach and the utility-maximizing approach is to replace what is purely a description of the travel pattern by a causal mechanism in which each individual determines where he will live and work by choosing the locations that are best for him. The fact that not everyone chooses the same locations implies that the various costs and COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 23 benefits are not the same for everyone, and the model represents this by means of statistical distributions of certain costs over the population of individuals. A mechanism is required to explain how the totals of homes and Ej jobs are achieved. This is done by introducing rent and salary differentials between the zones. Values of rents and salaries are manipulated until each zone becomes sufficiently attractive to fill the appropriate number of homes or jobs. (J. Broughton and J. C. Tanner, 1983; page 38) The rents and salaries are, in fact, functions of the balancing factors D ' . and O ' . . Thus, given the vectors defining trip origin and destination totals, E. and B . , i j i and the travel cost matrix r • ., any unique value of /3, defining the variance in the perception of workers to commuting costs, rents and salaries, will generate simultaneously a commuting pattern and rent and wage surfaces. The calibrated j3 value, set by the budget constraint, allows model output to be interpreted in terms of revealed preference theory. The theoretical basis of the random utility interpretation rests on the independence of the /3 parameter from the map pattern describing the spatial arrangement of trip origins and destinations. Thus, if the parameter is affected by a specific map pattern, the predicted commuting pattern and wage and rent surfaces cannot be explained in terms of utility theory. This question is as yet unresolved: Curry (1972), Sayer (1977), and Fotheringham (1984) argue that spatial autocorrelation in the map pattern affects the (3 value, while Cliff, Martin and Ord (1974) insist that a doubly-constrained approach is free of map pattern interference. In a study of the behaviour of the model in fitting /3 parameters to 24 urban regions, Griffith and Jones (1979) found that the range of fitted COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 24 values was very wide, and that they were related to spatial structure. Gordon (1985) insists that the problem is one of model specification. The linear programming and entropy models have been linked by Evans (1973), who shows that, when the entropy 3 parameter approaches the limit of infinity, the solution of the entropy model is a global least-cost solution. Thus, if the variance in workers' perception of costs approaches 0, then the efficient solution is also the equilibrium solution (Senior and Wilson, 1974). However, it is clear that this is the case with any form of doubly-constrained gravity model - as the derivative of the distance deterrence factor with respect to distance approaches infinity, model output will be determined by the origin and destination constraint vectors. The major problem with the entropy approach is that the revealed preference of utility variance is only meaningful under the assumptions inherent to the theory of general equilibrium. This theoretical structure has been criticized by Kornai (1971) and Kaldor (1972) on the grounds that it is tautological - the distribution of preferences creates the equilibrium state, but we know this only if we assume the economy to be in equilibrium. It is clear that if the commuting pattern is not the expression of a labour market in equilibrium, then the interpretation of the entropy model only in terms of information theory is correct. C O M M U T I N G P A T T E R N S A N D T H E U R B A N L A B O U R M A R K E T / 25 D. CONCLUSION While it is clear that commuting patterns, by definition, are outcomes of labour market processes, this relationship has not been explored to any great extent in the field of commuting modelling. The fact that models have been used more for prediction in planning applications than as research tools may explain this phenomenon. The two approaches described above suffer from their derivative nature. Linear programming was developed as a means of identifying efficient distribution patterns in the transport of goods; thus, the behavioural content of the model is unrealistic when applied to human decision-making. The re-working of the gravity analogy into entropy theory is more appropriate for trip-making decision situations in which the explicit inclusion of a random element has inherent realism at the scale of the individual, t In testing the applicability of the entropy model to explain commuting patterns, Snickars and Weibull (1977) calibrated a model to 1970 Stockholm commuting data and measured the fit of predicted to empirical flows. They then compared the results to that obtained from a Fratar model, which extrapolates relative zonal interaction from one data set to another. This model was based on the 1965 commuting pattern, and the 1970 prediction was obtained by constraining t For example, Smith (1975) presents an interpretation of the gravity approach in terms of the individual choice theory developed by Luce (1959). The behavioural foundation of the approach is couched in terms of an unconstrained interaction situation: the probability of a decision-maker at location i deciding to travel to a given destination j is based on the relative distances required to travel from i to all substitutable destinations. The probability of travelling to any destination is greater than 0, and probabilities decrease as trip distances increase. This approach may be understandable in terms of a series of decisions of similar type, in which each individual decision refers to a single trip (for example, shopping trip behaviour). The commuting situation is quite different: a single decision (accepting a job) determines the destination of a series of identical trips. COMMUTING PATTERNS AND THE URBAN LABOUR MARKET / 26 flows to match the empirical origin and destination totals of the 1970 data set. The two methods are not equivalent in terms of data input: both require knowledge of one of the empirical matrices (although the entropy model only uses the matrix to calculate total travel costs); the Fratar model does not require a travel cost matrix, while the entropy model does. It should be noted that the Fratar model is devoid of theoretical content, and is simply based on the stability of interaction relationships. They discuss the results as follows: The main conclusion of this application is that the classical gravity model is considerably less powerful as a tool for describing historical changes in trip patterns in the Stockholm region than models based on a priori trip patterns. This holds even though considerable changes in the transport network have occurred in Stockholm during 1965-1975. The results indicate that commuting patterns are not primarily shaped by simple travel time differentials. Their determinant factors may more likely be found among the laws governing the labour market. (F. Snickars and J. Weibull, 1977; page 156) The model presented and tested in the following chapters is an attempt in this direction. CHAPTER III. INDIVIDUAL CHOICE COMMUTING MODEL A. INTRODUCTION The Individual Choice Commuting Model (ICCM) presented in this chapter is designed as a flexible framework for deriving normative commuting flow patterns through a simulation of the spatial urban labour market. Specifically, it has been constructed to replicate market outcomes under defined market conditions reflecting a variety of assumptions concerning individual choice behaviour, information availability and market uncertainty. The model can be extended to provide a means of calibrating endogenous parameters, through the replication of an empirical commuting pattern, in the context of a positive, revealed preference logic. The ICCM structure also allows the construction of versions that generate market outcomes by means of analog, rather than simulation, processes. That is, the algorithm used to arrive at a solution does not purport to simulate the actual operation of the labour market, but the modelled solution can be identified with the one that would arise through defined market processes. Analog versions can be used to provide normative solutions which are globally optimal, and thus can be described externally to their generative (market) process. The distinction drawn between simulation and analog model processes is similar to that Dreyfus (1972), in an investigation of the prospects for the development of computer-based reasoning, makes between cognitive simulation and artificial intelligence. Cognitive simulation is defined as the attempt to build algorithms 27 INDIVIDUAL CHOICE COMMUTING MODEL / 28 that replicate human intelligent behaviour: [Cognitive simulation] is the use of heuristic programs to simulate human behaviour by attempting to reproduce the steps by which human beings actually proceed. (H.L. Dreyfus, 1972; page xxxiii) Artificial intelligence is defined as: the attempt to simulate human intelligent behaviour using programming techniques which bear little or no resemblance to human mental processes. (H.L. Dreyfus, 1972; page xxxiii) The simulation approach taken here thus is subject to the critique Dreyfus levels at cognitive simulation: It is not clear that naive cognitive simulation, as it is now practiced, can have any value at all, except perhaps as a striking demonstration of the fact that in behaving intelligently people do not process information like a heuristically programmed digital computer. (H.L. Dreyfus, 1972; page xxxv) It is not claimed here that the simulation approach replicates human thought processes in detail, but that, in certain situations, simulation through behavioural characterization is the only feasible means of predicting outcomes of theoretical processes. One aspect of this research project is an application and evaluation of behavioural simulation for investigating outcome patterns in decision environments that are too complex to be amenable to analytic methods. The model envisions the existence of an urban environment consisting of defined employment and residence location patterns and a transportation surface. Attention INDIVIDUAL CHOICE COMMUTING MODEL / 29 is directed to the assignment of workers to jobs; that is, the model predicts the spatial allocation of job/worker pairs. The model is static in nature, although, given further information or assumptions regarding the temporal path of urban economic development processes, it can be extended to predict outcomes of dynamic scenarios. The logic of the job/worker matching process used by the model can be decomposed into two sets. The ability of the model structure to accommodate a range of market constructs is derived from the model's primary logic set, which determines the solution path of the simulation. The solution path algorithm utilized by any specific version of the model may be one of four alternatives; a description of these alternatives is the focus of this chapter. The second logic set is based on the propositions required to characterize the behavioural content of labour market processes. This includes considerations of the psychology of workers and employers as economic agents and information flow in the market as well as a market structure typology. The ICCM framework allows for a wide assortment of behavioural assumption sets to be adopted, resulting in the ability to predict commuting outcomes resulting from a variety of definable theoretical market conditions. Each feasible behavioural assumption set, coupled with one of the four solution path algorithms, defines a unique version of the ICCM model. INDIVIDUAL CHOICE COMMUTING MODEL / 30 1. Simulating the Urban Labour Market - A Normative Approach The diurnal pattern of commuter flows in an urban area is the result of the combined effects of employer/employee location decision-making and the operation of the local labour market. The interplay of these factors determines the development dynamics of urban land-use and commuting patterns; here attention is focussed on labour market outcomes when location patterns are established a priori. The problem at hand is to determine the commuting pattern that would result from the operations of an idealized labour market structure. The normative solution is thus purely descriptive; the prescriptive notion of optimality is not inherent to the approach. In other words, the market is idealized only in a reductionist sense. The solution may be used to test (or, more accurately, disprove) the hypothesis that the theorized market adequately represents actual labour market functioning. The success or failure of such a test does not affect the normative status of the model, but does reflect the usefulness of the underlying theory in understanding urban labour markets. The present approach is static in that the derivation of the location patterns is unexplored, and extrapolation to future land-use patterns is not attempted. However, the model may be used for flow prediction. Location patterns accepted as exogenous inputs to the model could be based on anticipated land-use distributions rather than current data, but the model itself is concerned only with one time period. The static nature of the model is satisfactory theoretically if the operation of the postulated labour market is insensitive to the development path of the market's size and range. INDIVIDUAL CHOICE COMMUTING MODEL / 31 The ICCM solution algorithm assumes the existence of a labour market in which workers' commuting costs are compensated by wages; that is, workers are willing to trade longer commuting journeys for higher wages, and employers are required, through competitive bidding for labour, to offer wages sufficiently high to cover commuting costs of employees. It is also assumed that the wage paid to employees in a given job or skill category in a given plant is homogeneous. Workers do not bargain directly with employers to establish individualized labour contracts, but either accept or reject current wage offers tendered by employers. Thus the plant-specific wage level is established as that necessary to induce the last (or most demanding) employee to accept the job offer; the average wage is equal to the marginal wage. This assumption follows from the trade-off argument first presented by Moses (1962), which explicitly identifies residential land rents with residual or net wages. The difference between the wage paid to an employee and the minimum wage the employee would require to accept the job is a rent payment. If workers are homogeneous in tastes, rents are attributable fully to relative location; employers pay all workers the wage at least sufficient to offset the journey-to-work costs of the most distant employee, and employees gain a location rent equal to the difference of the wage level and their personal commuting costs. The ICCM accepts as exogenous data a defined land-use pattern. The development of this pattern is not central to the model; the process is treated as unknown, and may not reflect the activity of a competitive land market. In other words, the type and intensity of land-use may have been allocated to sites through administrative fiat (restrictive zoning regulations, non-market housing INDIVIDUAL CHOICE COMMUTING MODEL / 32 projects, etc.) so that land rents are not determined by consumption preferences of households or alternative-use bids. From the perspective of a given worker, the value of the worker's residential location is distinct from its market valuation (rent). In the absence of consideration of other factors (eg., the value of access to other activities), it is equal to the portion of wages in excess of that required to compensate the worker for the disutility incurred in the actual performance of the job, net of commuting costs. In this way, location rents are viewed as wage residuals. If the postulate that site-specific wage rates are determined by labour-shed range is adopted, it follows that labour supply and demand surfaces defined by the land-use pattern condition commuting flows, wage rates and location values. The ICCM provides a simultaneous solution of these variables. The market operation assumed by the ICCM solution path logic is as follows. Employers set initial remuneration levels for jobs. This level defines a minimum wage in that it is based on the aspatial opportunity cost for the worker of accepting a job. This minimum wage level is referred to as a reservation wage {w). Hasan and Gera (1979) define the reservation wage in the context of search theory: Search theory holds that individuals behave as if they possess a notion of the reservation wage for which they hold out - wage offers below the reservation wage are rejected and the first one equal to or exceeding the reservation wage will be accepted. The reservation wage then represents the worker's asking wage or his wage demand and we use the three terms synonymously. (Hasan and Gera, 1979; page ii) This definition doesn't distinguish between recompense for the disutility of INDIVIDUAL CHOICE COMMUTING MODEL / 33 performing the job itself and recompense for commuting costs - the reservation wage is the minimum wage demanded to cover all costs. In the present context, w is defined as that part of the minimum wage demand which is required to cover all costs except those related to the journey-to-work. Thus, it is the wage demanded to perform a job located at the worker's place of residence. Job attributes are deemed to be advertised in an accurate, effective and non-discriminatory manner, so that the availability or cost of acquisition of information regarding job openings is governed solely by the information flow surface. Workers identify the subset of jobs for which their qualifications meet requirements, and weigh the relative merits of those job offers on the basis of received information. Workers then apply for the job satisfying defined criteria (eg., utility level). The choice is conditioned by the behavioural assumption set, and the particular behavioural model adopted defines an operative version of the ICCM. For example, the model version detailed in the next three chapters, ICCM-1, considers the labour market process in the context of game theory. Under the assumptions adopted by ICCM-1, choice behaviour of workers reflects the (perceived) relative likelihood of the success of the job application. Thus the decision to apply for a job is treated as a gamble in which the payoff values of possible plays are known but the probabilities governing outcomes are not. The uncertainty is due to the inability of individual workers to predict the outcome probability estimates made by potential competitors. This situation requires the explicit consideration of behaviour in terms of relative risk aversion and measures of expected utility. INDIVIDUAL CHOICE COMMUTING MODEL / 34 In general, a market solution is achieved if individual job openings and applications match in such a way that either all vacancies are filled or all workers are employed. Failing this, remuneration levels are adjusted. Employers raise wage offers for jobs that have attracted no applicants, and the process is repeated. Wage offers are not lowered for jobs that are over-subscribed; as the initial wage level is set at a minimum, subsequent adjustments of the wage surface always increase competition for labour, so that an employer can never fill a vacant job at a wage lower than that currently offered. An important consideration is whether evaluation of job offers is deemed to be simultaneous, or whether allocated job-worker pairs are carried over to a subsequent round of matching if the market fails to clear. This issue is discussed in detail below. ICCM simulations assume a competitive economy composed of economic agents acting in accordance with information directly available. From the employers' perspective, knowledge of the effectiveness of a given wage offer is limited to that gained through the result of workers' actions; i.e, either the wage is sufficient to fill the position, or it is too low. There is neither incentive nor mechanism for employers to minimize the global wage bill. Firms are competitive, implying that there is no collusion in setting wage levels, and knowledge of wage levels offered at other employment sites is immaterial. From the workers' perspective, information required to evaluate the relative attractiveness of a particular job at a given site is fully available, but information required to estimate the amount of competition for the job, or the probability of an application being successful, is incomplete. Specifically, workers are cognizant of a surface describing the disutility of travel from their residence ' site to potential INDIVIDUAL CHOICE COMMUTING MODEL / 35 employment sites, but they have at best incomplete information regarding the travel disutility surfaces centred on other residential sites. Similarly, knowledge of the spatial distribution of jobs is available (through advertisements), but knowledge of the distribution of potential competitors for jobs is not. In addition, agents dealing in labour supply or demand arbitrage are formally excluded, so that the operation of a competitive economy with incomplete information cannot be superceded, or transformed into a neo-classical construct, by the insertion of another layer of actors who have full information. It is true that arbitrage in the temporal allocation of labour is not uncommon: eg., many firms supply temporary office help. However, the assumption of a single level of decentralized decision-making regarding the spatial allocation of labour is not unrealistic. If this assumption is removed, the market process is dominated by efficiency considerations, and the final outcome is, by definition, optimal. In this case, an analog version of the ICCM is appropriate. Application for a specific job by a worker in any given round of matching is treated as the most likely realization of a stationary stochastic process. The factors underlying the generation of the transition matrix are defined by the behavioural assumption set; i.e, the manner in which workers interpret and translate information concerning their spatial relationship to the labour market environment, which is defined by the exogenous supply and demand distributions and the (partly) endogenous job utility surface, is a function of the model of individual choice behaviour specified. The labour market environment is non-stationary across rounds of matching as the job utility surface is modified by INDIVIDUAL CHOICE COMMUTING MODEL / 36 wage adjustments determined by the outcome of the previous round. This in turn requires a recalculation of the transition matrix. In order to explicitly model the interdependence between environment and behaviour, which can be described as a type of feedback effect, a simulation approach is required. For any given data set, there is an infinite number of spatial wage distributions that will provide a feasible market solution consistent with the definition above. However, if employers are cost-minimizers in the sense that they will not raise a wage offer above that sufficient to fill the position, and the job-worker matching process is begun with wage offers at the minimum or aspatial opportunity-cost level, the first feasible solution reached via the ICCM solution path will be the normative solution. This consists of a market-clearing relative wage surface and the most-likely commuting pattern given the labour market environment. B. SOLUTION PATH ALGORITHMS The ICCM construct allows four unique logical structures which determine the solution path of a particular model version. This section describes these structures and discusses the appropriateness of three of them for both simulation and analog applications. The 1 fourth structure, while existing as a logical outgrowth of the ICCM approach, appears to be devoid of practical importance. INDIVIDUAL CHOICE COMMUTING MODEL / 37 1. Labour Market Simulation Models The four solution path algorithms can be defined in terms of a matrix having two choices along each dimension (see Figure III-l). The first dimension is based on whether successive allocations utilize memory of preceding allocations. If so, the process is termed sequential; if not, the process is simultaneous. The simultaneous process utilizes the current wage distribution to allocate all workers to all jobs (in accordance with the behavioural assumption set). If the solution is not reached, wage levels are adjusted and the process is repeated; each iteration represents a complete working-through of the market. The sequential process initially attempts to match all workers to all jobs, but, if the solution is not reached, wages are adjusted and the next iteration proceeds with the previous allocation retained. If the model is treated as a simulation, the two processes represent various views of the dynamics of the labour market. The interpretation of the sequential approach is that workers already hired will apply for another job only if the alternative is superior (in terms of the net wage) to present employment. If arbitrage or a secondary labour market is excluded, two scenarios can be constructed to be consistent with this process. First, model iterations represent commuting pattern development in the situation in which employment centres grow through accretion and land use patterns are stable. The model solution then identifies the present normative commuting pattern as conditioned by these development assumptions. Second, if the labour market matching of workers to jobs is assumed to occur simultaneously for all jobs and workers, so that employers continuously raise wage offers until all positions are filled, it is 3 8 F I G U R E I I I - l ICCM SOLUT ION PATH ALGORITHMS AND MODEL V E R S I O N S S I N G L E A L L O C A T I O N R E P E A T E D A L L O C A T I O N S I M U L T A N E O U S : A n a Iog S u b j e c t i v e R a t i o n a l B e h a v i o u r ; w i t h F u l l I n f o r m a t i on (Opt ima I) S i muI a t i on O b j e c t i v e o r S u b j e c t i v e R a t i o n a l B e h a v i o u r ; w i t h U n c e r t a i n t y D y n a m i c A s s u m p t i o n : R a n d o m G r o w t h ICCM-1; S E Q U E N T I A L : N o t U s e d S i muI a t i on O b j e c t i v e o r S u b j e c t i v e R a t i o n a l B e h a v i o u r ; w i t h U n c e r t a i n t y D y n a m i c A s s u m p t i o n : a) S i m u I t a n e o u s A I I o c a t i on b) A c c r e t i on A n a Iog O b j e c t i ve R a t i o n a I B e h a v i o u r ; w i t h F u l l I n f o r m a t i on (Opt ima I ) INDIVIDUAL CHOICE COMMUTING MODEL / 39 reasonable to postulate that workers who have applied successfully for jobs at wages below final levels (i.e., pre-solution iteration allocations) will, in response to subsequent wage offers, apply only for jobs offering a higher level of utility. Thus, the sequential process, in terms of the model's operation, simulates a simultaneous market process. The simultaneous allocation algorithm simulates a more complex market structure, the main feature of which is the persistence of spatial wage differentials relative to turnover in the labour market. The labour market, at any given time, is composed of job openings representing positions that either are newly created or existed previously and are now vacant, and workers that either are unemployed or are employed but looking for a better position. To the extent that a) the spatial distributions of the jobs and the workers composing this labour market are random subsets of all jobs and workers, and b) the mean length of job tenure is short relative to the rate of urban growth, wage levels generated by the simultaneous allocation process will simulate the wage structure and commuting pattern current in the labour market for the majority of workers and employment positions. This idealization of the labour market is most effective in handling a situation characterized by adjustment due to relatively rapid and extensive levels of plant openings and closings, so that the pool of workers actively searching for employment at any given time in a given occupation / skill category is dominated by displaced workers randomly distributed across the population of that category. To the extent that the distribution of employment change exhibits INDIVIDUAL CHOICE COMMUTING MODEL / 40 consistent spatial bias, the model will incorrectly interpret the distribution of available labour. For the purpose of a static simulation of the current outcome of an urban labour market, with no utilization of information as to the development path of the land-use pattern, the simultaneous algorithm is preferred. In terms of the realism of the theoretical structure, the critical issue is the treatment of the effects of urban expansion through employment and residential decentralization. This situation is both a very common dynamic element in the development of urban morphology and a serious problem for any static representation, t Specifically, if employment decentralization is assumed to occur via the opening of new plants in suburban locations and the closing of old plants in central locations, rather than as a result of the movement of plants from central to suburban locations (that is, change in the distribution of employment location is dominated by job termination and creation rather than by transfers) the simultaneous algorithm offers a satisfactory characterization of the labour market process. On the other hand, neither the growth-by-accretion nor the simultaneous market processes envisaged by the sequential simulation algorithm are able to represent satisfactorily commuting patterns resulting from any urban decentralization dynamic. For this reason, the simultaneous algorithm was selected t Note that the discussion here excludes analog modelling of globally optimal normative solutions. These models are, of course, theoretically separated from considerations of market development. Sayer (1976; pages 209-210) discusses this issue in criticizing the use of gravity models for prediction in situations of urban expansion. If the gravity formulation is of the entropy type, he is incorrect in that model calibration is based on the concept of an optimal solution; the fitted fi parameter is as applicable (or inapplicable) in the predictive context as it is in the calibrated context. If the models he refers to are interpreted as representations by means of the gravity analogy, his criticism is well-founded. INDIVIDUAL CHOICE COMMUTING MODEL / 41 for the operational simulation model presented in the next chapter. The second dimension of the matrix determining solution path algorithms concerns the way in which workers are allocated to jobs within iterations. Given the current-iteration distribution of wage offers, the model establishes the probability of a worker applying for a given job (in accordance with the behavioural assumption set), and then determines which applications are successful by randomly matching applications to job openings. At the end of this process, it is possible (and, in fact, is usually the case) that some jobs are over-subscribed, so that in other areas job openings still exist, and that there are unallocated workers for whom the net wage obtainable at at least one open job location is positive. The repeated-allocation algorithm then repeats the process until there are no acceptable job-worker matchings left before adjusting the wage structure and re-iterating. The single-allocation algorithm proceeds directly to the wage adjustment routine without attempting further matchings. The repeated-allocation algorithm is used for all versions of the model which attempt to simulate the labour market. The rationale is that employers would not raise wage offers unless there was an inadequate supply of labour; thus, they would wait for applications from workers for whom their job offer is an acceptable second or third choice before deciding to raise the offer. This algorithm also offers the advantage that interpretation of the solution can be extended, under certain conditions, to include the more realistic situation in which workers apply for more than one job at a time. This issue is discussed at some length in the next section. INDIVIDUAL CHOICE COMMUTING MODEL / 42 2. Analog Models Generating Optimal Solutions Up to this point, ICCM algorithms have been discussed only in terms of simulation versions of the model. There also are two methods of constructing optimizing models, corresponding closely to the subjective choice behaviour interpretation of the entropy approach and to the objective least-cost solution of the linear programming approach respectively. Application of the ICCM construct for generating optimal distributions is not a focus of present research; exploration of the usefulness of the model in this regard is incomplete. The first of these models generates a solution based on the globally optimal distribution generated by the selection of jobs by workers on the basis of maximizing a subjective utility function. Parameters controlling the distribution of the function can be either pre-determined, so that the model provides a normative description of the wage distribution and commuting pattern market outcome, or they can be calibrated by matching the modelled solution to an empirical distribution, so that the model establishes the parameter values reflecting the revealed preference of workers. This model utilizes a solution path algorithm that is simultaneous with single-allocation iterations. The single-allocation procedure ensures that workers apply only for first-choice job openings; the solution then is composed of worker-job matchings that maximize worker utility functions. The advantage of this approach over the entropy model lies in the flexibility the ICCM allows over choice of the subjective utility distribution function while maintaining control over the wage distribution. The entropy model is defined by, INDIVIDUAL CHOICE COMMUTING MODEL / 43 and limited to, the assumption of a log-Weibull utility distribution, which gives rise to the negative exponential distance deterrence function (T. Domenich and D. McFadden, 1975). In a paper examining the relationship of the entropy model to utility theory, Tanner states that: The negative exponential deterrence function has been widely used in transport modelling, in part because it arises naturally in entropy-based models. While it does not necessarily provide the best possible fit to any given body of data, it often provides an acceptable representation. ... When the assumption of a log-Weibull distribution of costs is relaxed, theoretical mathematical analysis appears to become intractable. ... Thus more general models must be derived from a utility structure rather than from a deterrence function formulation. Although mathematical development seems likely to be unrewarding, [the equations defining the mean utility in the system] may lend themselves to a simulation approach. (J.C. Tanner, 1980; pages 6-9) The separation of the solution path from the behavioural assumption set provides the ICCM construct with the ability to simulate the labour market (in the sense indicated by Tanner) by using a variety of utility distribution functions. The second optimizing approach obtainable with the ICCM construct is a model in which the behavioural assumption set postulates the existence of an objective utility function for all workers. Utility is identified with the net wage, and workers apply only for the job offer promising maximum utility. The solution path algorithm is sequential with repeated-allocation iterations. This procedure ensures that each job is allocated to the closest possible worker (in terms of commuting cost), so that the market outcome is a global least-cost solution. INDIVIDUAL CHOICE COMMUTING MODEL / 44 In a sense, this model may be viewed as a relatively inefficient means of solving the both the primal and the dual of the Hitchcock linear transportation problem (Hitchcock, 1941). The advantage of the ICCM approach is that, by explicitly defining application choice rules in the behavioural assumption set, it can derive 'most likely' solutions to degenerate problems. For example, workers may apply to jobs on the basis of: a) random selection among jobs offering equal (maximal) net wages; b) initially maximizing net wages and subsequently minimizing travel costs; etc. This is a significant advantage in that the transportation problem is very prone to degeneracy (R.R. Nelson, 1957; page 406). Miller states that: In problems, therefore, in which the values of the variables in both problems [i.e., the primal and the dual] are of interest in their own right, it is important to recognize that the existence of alternative optima in one problem does not necessarily affect the solution values of the associated problem. One dare not be too optimistic, however. In primal problems which are very likely to be degenerate, the associated dual will probably have non-unique solutions. For example, since the transportation problem is very likely to be degenerate, the solution to the dual of a primal transportation problem will quite probably have alternative optima, i.e., the imputed values will not be unique. (R.E. Miller, 1963; page 163) As the ICCM solves the primal (distribution) and dual (relative wage and rent levels) problems simultaneously, it provides a means of selecting the most likely of a set of non-unique optimal solutions. INDIVIDUAL CHOICE COMMUTING MODEL / 45 C. DESCRIPTION OF MODEL VARIABLES The ICCM translates data defining labour supply and demand surfaces into a normative market outcome. Specifically, the labour market initially can be seen as consisting of: a bounded set of jobs, each fully characterized by a vector of variables relating to objectively-defined attributes such as job description (specific duties, responsibilities), qualification requirements (skills, experience, education), working conditions (physical surroundings, shift hours, security, etc.), monetary benefits (wages, pension plan, bonuses, etc.), non-monetary benefits (conferred status, training potential) and site location (described by a point in Euclidean space); a bounded set of workers, each identified by a vector of characteristics describing qualifications (skills, experience, education), utility (or disutility) associated with or anticipated from objectively-scaled measures of job characteristics (job description, working conditions, monetary and nonmonetary benefits), and site location; and a metric translating the Euclidean surface to a Minkowsky surface (i.e., co-ordinate positions defined in terms of cost/time required to travel from one point to another; Harvey, 1969). In addition, a second Minkowsky surface describing relative ease of inter-site information flows defines the accuracy of workers' utility evaluations of the characteristics of specific jobs. Job/worker pairs for which qualifications required by the employer do not exceed those possessed by the worker constitute elements of a feasible market outcome. A market solution exists when either all jobs or all workers are paired, and the market is cleared. In order to operationalize the model, data must be treated in an aggregated INDIVIDUAL CHOICE COMMUTING MODEL / 46 manner. Specifically, jobs and workers are grouped according to location and occupational / skill criteria. The reduction of the richness of variety exhibited by the range of individual data elements (job descriptions, worker ability and behaviour, relative location and perceived distance, etc.) to a classification scheme based on assumptions of internal homogeneity inherently involves some compromising of correspondence between model variables and reference populations. Correspondence can be diminished either through the constraints imposed by data availability or by the logic assumed to underly modelled processes. A discussion of this issue in relation to the operational ICCM structure follows. The urban labour market, £2, can be subdivided into n sub-markets, m., k = 1 k ...n. Each consists of pools of jobs J^ and workers which match in the sense that any worker in B ^  is qualified to apply for a job in J^. Clearly, the n submarkets overlap considerably, as highly qualified workers would also be able to undertake lower-skilled jobs. In this way (with the possible exception of some professional occupations) the labour market is completely inter-related. The sub-markets become more distinct if the definition of B^ is qualified to include only those workers actively working in or applying for jobs in J^. Data available for urban modelling consists, in most cases, of an aggregated description of the urban employment pattern. Thus, it is possible to know the spatial pattern of actual job-worker matchings in a pre-defined sub-market, but all individual workers and jobs within that market category are treated as being homogeneous in all respects. Specifically, J^ becomes a set of jobs such that each element of J , is homogeneous in all respects except location. Similarly, B, INDIVIDUAL CHOICE COMMUTING MODEL / 47 becomes a set of workers with elements homogeneous in all respects except location. With no further information defining inter-relationships among submarkets, J. defines a job category requiring the type and level of skills, experience and responsibility possessed only by members of the worker set B^, and workers in B, possess only those abilities suitable for employment in J, jobs. The urban labour sub-market represents the process of establishing all J,, B, pairs. ft ft The ICCM approach does not require a balancing routine to come to a feasible allocation, as constraints operate only on the number of jobs and residences in a given location rather than in the submarket as a whole. That is, for example, the model will not allocate more workers to employment zone j than there are jobs in j, but may allocate fewer if there is a shortage of qualified workers. As a result, the ICCM theoretically allows the modelling of a number of inter-related sub-markets simultaneously, given the information required to identify (or infer) the potential submarket membership of individual jobs and workers. The way in which workers then decide on which submarkets to enter can be made explicit through the adoption of an appropriate behavioural model. In practice, the required information is difficult or impossible to obtain; the description and testing of an operational model in the following chapters assumes that submarkets are discrete entities, and thus models a given submarket independently. If workers as a rule apply for jobs at the upper end of their qualifications, it follows that the degree of separation between labour submarkets is a function of labour demand (which can be estimated by the level of unemployment). Generally, the practical approach which follows gains in realism as the urban economy moves towards INDIVIDUAL CHOICE COMMUTING MODEL / 48 full employment. Similarly, locations are aggregated into zones, and the spatial relationship of two zones can only be represented by a measure of mean separation. Information detailing the spatial relationship of jobs and residences located within a zone is nonexistent, so that zones become point locations. As the number of zones increases, the model more accurately reflects the space economy. The final consideration is the definition of employment benefits. As described above, these may take a multitude of forms, both monetary and nonmonetary. The model collapses all employment earnings into the category of wages, without detailing the actual method of payment. CHAPTER IV. DECISION-MAKING AND THE ICCM-1 MODEL This chapter examines the decision-making behaviour underlying the allocation of jobs to workers that characterizes an operational normative version of the ICCM. This version of the model is termed ICCM-1. In subsequent chapters, this approach to choice behaviour will be developed into an operational model, the model's algorithm will be detailed and the results of test runs will be discussed. The theory of individual choice which forms the basis for ICCM-1 was developed with three principles in mind: a) that the theory should reflect realistically the operation of an urban labour market and the behaviour of agents in such a market; b) that it should conform to the minimum requirements of a normative definition; c) that, wherever possible, it be consistent with Occam's razor. The first principle reiterates the motivation for the research project, and stands as the benchmark from which the validity of the theory can be measured. Clearly, there is great potential for conflict between it and the other two principles. The rationale behind the second principle is partly practical, from the point of view that the minimal data requirements of a normative model make such an approach attractive for predictive applications, and partly motivated by the requirements of a generalizable explanation of commuting patterns. The third principle was adopted mainly to aid in the development of an operational model. The ICCM solution path construct provides the foundation for a wide variety of behavioural postulates; simplicity allows for both a clear exposition of the model's basic operation and the development of a firm foundation from which to investigate alternative model versions. 49 DECISION-MAKING AND THE ICCM-1 MODEL / 50 A. BEHAVIOURAL ASSUMPTIONS Three critical assumptions underly the operation of ICCM-1. The first is that the choice situation confronting a worker is defined such that the selection of a job from a set of job openings is an application for, rather than the acceptance of, the job. This implies that the choice is made with the realization that the outcome is uncertain. The second assumption is that workers apply for jobs on the basis of objectively rational choice behaviour. However, uncertainty of outcome implies that such choices may not result in an optimal allocation of workers to jobs. The third assumption is that there is no mechanism available to the labour market to ensure that suboptimal choices are remedied; that is, the effective existence of either a central control agency or competitive labour arbitrageurs is denied. 1. Objective, Subjective and Bounded Rationality To begin with, the realism of assuming objective rationality has been questioned by many researchers in the field of decision-making behaviour. It is the contention here that it is not unreasonable to make such an assumption for present purposes. The importance of the job application decision would lead to careful consideration of the relative benefits of alternatives and, in terms of both the demands of a normative structure and the advantages of simplicity, it is deemed defensible to suggest that such consideration can be objectively characterized. In order to discuss this point, the second assumption (of choice uncertainty) will be suspended. If the worker was viewed as choosing one job from a set of job offers with DECISION-MAKING AND THE ICCM-1 MODEL / 51 certainty as to the success of the application, decision-making behaviour could be defined in terms of: a) the means used to identify and evaluate the relative benefits and disbenefits of the elements (job offers) in the set; and b) the decision procedure applied to select one of the elements. The evaluation stage would require some method of translating factor ratings into a composite measure; this is a relatively straightforward example of the type of problem that is the concern of utility theory. It is sufficient for the sake of the present discussion to assume that job evaluation criteria can be measured, at least subjectively, in terms of a scale based on a utility function, so that each worker is capable of constructing a utility ordering of the set of job offers. It should be noted that the optimal realisation of such an ordering requires full information on the part of the worker with regard to evaluation criteria, in terms of the worker's ability both to identify criteria and to assign utility values that are (at least in the present) accurate. If these conditions are met, a decision can be termed rational if the job offer chosen corresponds to either the element in the set with the maximum utility value, or to any one of a subset of elements that have equal and maximal utility values. Such a decision is objectively rational, in an empirical sense, if there is broad agreement among individuals in the decision-making group (workers) as to evaluation categories and procedures, and these are applied consistently. The definition of rational behaviour implied in the discussion above assumes the existence of decision-makers who have both costless access to complete information and the perceptual and computational ability necessary to analyze fully that information. Such a decision-maker is thus capable psychologically of objective DECISION-MAKING AND THE ICCM-1 MODEL / 52 optimizing rationality. The realism of postulating this form of behaviour has been questioned by Simon (1957), who suggests that human decision-making is characterized more accurately by subjective rationality. He argues that limitations on or peculiarities in individual perceptual or evaluational abilities or premises may result in decisions that are simultaneously subjectively rational and objectively irrational, and, therefore, suboptimal. In defining the principle of bounded rationality, he states that: The capacity of the human mind for formulating and solving complex problems is very small compared with the size of the problems whose solution is required for objectively rational behaviour in the real world - or even for a reasonable approximation to such objective rationality. (Simon, 1957; page 198) It is important here to clarify the role of information in the decision-making process. Simon's approach is based on the decision-maker's handling of information that is available in an objective sense; that is, it exists but either may not be perceived or may not be evaluated in a manner consistent with some objective or external criteria. Thus, rationality is bounded by the innate or psychological constraints unique to the decision-maker, not by the characteristics of the decision-making environment. A somewhat similar concept of subjective rationality has been utilized as the conceptual foundation for the individual choice interpretation of entropy modelling of commuting patterns. In this case, however, rational decisions are viewed as being based on subjective utility evaluations that are informed by individual vagaries in taste rather than bounded by evaluative limitations. Thus, the DECISION-MAKING AND THE ICCM-1 MODEL / 53 entropy construct has been identified with utility-maximizing behaviour, in that choice decisions optimize individual (subjective) utility functions (Senior and Wilson, 1979). In order to avoid confusion, the term subjective rationality will be used to identify this form of decision-making behaviour, in which the distinction between objective rationality lies solely in the lack of homogeneity of evaluative criteria or measurement. Bounded subjective rationality will be used to refer to choice behaviour of the type suggested by Simon: limitations imposed by the inherently restricted abilities of decision-makers are responsible for sub-optimal decisions. Bounded objective rational behaviour will refer to decision-making which is objectively rational given an environment which does not guarantee full information. Simon suggests that bounded subjective rationality leads to decisions based on satisficing rather than maximizing behaviour: The central core of the theory of choice I am advancing here ... is upon simplifying the choice problem to bring it within the powers of human computation. ... The key to the simplification of the choice process is the replacement of the goal of maximizing with the goal of satisficing, of finding a course of action that is good enough. (Simon, 1957; pages 204-205) This view of the behavioural implications of bounded rationality implies that decision-makers are aware of their inability to evaluate alternatives to the degree required to make utility-maximizing choices; satisficing behaviour thus is a DECISION-MAKING AND THE ICCM-1 MODEL / 54 conscious strategy employed to ensure at least acceptable outcomes. Satisficing behaviour involves only a binary classification of possibilities into one group that is acceptable (satisfactory) and one that is not (Isard, et al., 1969). The following discussion will outline two ways in which satisficing behaviour can be operationalized as a choice mechanism. The first approach, in which the actual selection is made randomly from the group of satisfactory alternatives, corresponds directly to Simon's definition of bounded subjective rationality. The second approach treats the selection process as the result of the sequence of evaluation, in that the first job offer encountered that is deemed to be satisfactory is accepted. Should the determinant of the sequence lie outside the direct control of the individual, Simon's model must be expanded to include bounding of rationality motivated by the exigencies of uncertainty in the decision-making environment. For the purpose of modelling satisficing choice behaviour, the two selection processes are equivalent if the evaluation sequence is random. However, these approaches are based on different behavioural scenarios. The first approach (random selection) involves a full evaluation of all elements, but results in only a minimal level of ordering. Simon has argued that the ability of a decision-maker to weigh the relative merits of a large number of choices is limited, so that a full ordering of utility values is impossible. He postulates that in such a case it is more reasonable to base the choice decision on some well-defined level of utility that is known to be satisfactory, and to evaluate possibilities with regard to this level as an external baseline. DECISION-MAKING AND THE ICCM-1 MODEL / 55 The first problem is the issue of how this baseline is established. As the evaluation process is assumed to be inherently circumscribed by subjective limitations, it is unreasonable to postulate a baseline which can be identified by objectively rational criteria or can be other than unique to the individual worker. It is possible to employ a probability distribution to estimate the likelihood of a given (objective) utility value being selected as the baseline, but, without a theoretical justification for the use of a particular distribution, this method admits an indeterminacy that is undesirable from the vantage point of a normative approach. If an objective, homogeneous baseline is used, the problem is analogous to that embodied in the postulate that the normative solution to an entropy model can be identified through the assumption of a preference function whose variance approaches the limit of zero. In both cases, the normative solution to a choice model characterized by a form of subjective rationality requires the use of an objectively defined evaluation procedure. The second problem presented by the random selection scenario is that it is difficult to see why a worker would not be capable of creating a finer classification scheme by repeating the process a number of times with each iteration utilizing the subset of satisfactory job openings obtained from the previous iteration; successive dichotomous evaluation would allow the worker to arrive at an optimal solution via an ordinal ordering.. The long-term and wide-ranging ramifications of the job selection process suggests its importance to the worker; it is unlikely that the additional effort required to assess all job offers in the above manner would be foregone as a matter of course, t For this t Fotheringham (1983) presents a version of a gravity model which is based on the assumption of a two-stage decion-making process, although the iterative DECISION-MAKING AND THE ICCM-1 MODEL / 56 reason, I suggest that satisficing behaviour based on a random selection process is a poor explanation for apparent suboptimal behaviour. The second satisficing scenario, sequential evaluation, also is problematic. If we view the sequencing of job offer evaluations to be a result of the temporal sequence of a worker's receiving (or knowing of) the offers, then the decision to accept the first satisfactory offer in fact exhibits behaviour which not only is rational given a risk-averse psychology, but may also be objectively rational. For example, consider a set of job offers, </, whose elements can be organized into n subsets of J according to the time period in which they are either received or identified, such that J. contains those offers available for selection in the ith time period, and z = ra. If the worker has no knowledge of the elements in the subsets of future time periods, the selection of the first satisfactory offer is an objectively rational decision under conditions proscribed by current knowledge. Indeed, the selection of any but the first satisfactory offer may be irrational, as bypassing a satisfactory offer involves a pure gamble; that is, it is impossible to assign a probability to the event of a better job becoming available in the future. This conclusion is given added weight if we consider that n also would be unknown. Note that if the J. subsets contain more than one i element, the evaluation of each subset constitutes a random selection procedure as discussed above. t(cont'd) nature of this process is viewed in purely spatial terms: destinations are first identified in terms of a general region (or 'macrodestination'), then a specific trip-end is identified within the region. The motivation for this formulation is similar to that discussed here. DECISION-MAKING AND THE ICCM-1 MODEL / 57 It is possible also to view all elements of set J as existing concurrently, but to acknowledge that the worker must incur some expenditure in time or money in order to be informed of their existence. The decision to continue the search after a satisfactory job offer has been identified (in other words, to pay the cost required to be aware of an additional J) is based then on a positive evaluation of the net utility that is anticipated from further information against the cost of acquiring that information. It is impossible to term a decision either to continue or to end search as irrational as the net utility of further search must be unknown. That is, the ability to accurately assign a utility value to further information implies that that information is not unknown, and the decision environment is in fact one of full information. The critical point here is that a choice may be made in this situation (sequential evaluation with incomplete information) that will appear to indicate satisficing behaviour on the part of the worker when analyzed a posteriori, but was objectively rational in terms of the a priori vantage point of the decision-maker. This line of argument leads to the conclusion that the a posteriori appearance of satisficing (or, more generally, suboptimal) behaviour does not lead to the rejection of the assumption of maximizing behaviour. Webber states that: Unless they are very lucky, firms cannot be optimising choices with respect to perfect information; but they can still be optimising their choice with respect to such information as they have. In other words, although a choice may be sub-optimal when examined after the event, it may well have been optimal before the event. Ex post sub-optimal behaviour does not imply satisficing behaviour. (M.J. Webber, 1972; DECISION-MAKING AND THE ICCM-1 MODEL / 58 page 109-110.) B. UNCERTAINTY The discussion above considered choice behaviour under assumptions of various behavioural constructs, with the proviso that choices would be made with the understanding that the outcome was determinate. This clearly is unrealistic. There is a high degree of probability that, given any form of choice behaviour, the matching of job openings and job applications would be imperfect. This implies that a worker's decision to apply for a job cannot be made with the guarantee that no other worker will also apply for that job; outcome uncertainty not only is a characteristic of this choice situation, but the worker applying for a job is likely to be aware that this is the case. The argument represented above defended the assumption of objective rationality as a realistic description of a worker's ability and motivation in making job application choices. The discussion of the implications of assuming objectively rational choice behaviour when the decision-maker is bounded by the knowledge that the outcome of the choice is uncertain will be the main focus of this chapter. 1. Stability of a Suboptimal Solution The third critical assumption of ICCM-1 is that the solution commuting pattern, established through the actions of individual rational decision-makers under conditions of uncertainty, is characterized by stability rather than by optimality. The assumption of stability is embedded in the solution path logic of the ICCM model; any behavioural assumption set defining a version of the model must include it. DECISION-MAKING AND THE ICCM-1 MODEL / 59 Thus, the labour market does not have a mechanism to rectify suboptimal allocation of workers to jobs. This implies not only the formal exclusion of competitive economic agents external to the labour market (arbitrageurs) who are able to profit from arranging a re-allocation of job-worker pairs, but also of workers or employers operating internally as arbitrageurs through co-operative behaviour. The justification for the latter restriction is based on the size and complexity of the urban labour market. Successful co-operation would require a degree of organization that is unlikely to occur; fragmented co-operation among employers or workers through the actions of industrial councils, unions, etc. to influence wage levels is more realistic, but is rejected on the basis that the normative manifestation of such partial attempts would be difficult to establish. For example, collective wage bargaining on a plant-specific scale would have a completely different impact on the spatial distribution of wages as compared to industry-wide bargaining. The ICCM-1 solution can be seen as a base from which various co-operative scenarios can be explored. The argument that suboptimal choices are inherently instable has been used to justify the use of optimizing models to represent the outcome of decision-making under uncertainty. As Webber, quoting Tiebout, states (with reference to location models): Tiebout (1957) has shown that the predictions of maximising models may be accordant with reality even though businessmen may not choose rationally, because the firms in poor locations suffer reduced profits when compared to firms in better locations. Thus, the poorly located firms tend to go out of business. Tiebout argues that society chooses the firms in the best locations. (M.J. Webber, 1972; page DECISION-MAKING AND THE ICCM-1 MODEL / 60 106.) • The fact that models generating least-cost commuting patterns do not accord with reality suggests that reliance on "society" to ensure global efficiency is misplaced in this circumstance. Koopmans (1951) has outlined the necessary and sufficient rules governing decision-making behaviour in a decentralized environment that would ensure the existence of a self-correcting competitive economy. However, full knowledge of relevant information is assumed: The reader will have realized that behaviour according to these rules presupposes a knowledge, on the part of each process manager, of the efficient point set that can be constructed on the basis of those activities involved in the process that he controls. (T.C. Koopmans, 1951; page 464) Furthermore, and most significantly for the present discussion, Koopmans' rules are applicable only in terms of maintaining, rather than generating, a globally efficient allocation of resources: It should be emphasized that, if an inefficient state of resource allocation prevails initially, it is not claimed that adherence by all concerned to the rules stated would lead the mechanism to an efficient point, or even close to such a point, in a stated time interval. To establish such a claim would require a dynamic analysis resting on a more precise dynamic specification of the rules in question. It is claimed only that adherence to the rules - will perpetuate an efficient state once it has somehow come about. (T.C. Koopmans, 1951; page 463) Thus, Koopmans' rules concerning market-oriented decision-making are more applicable to the issue of allocation stability than the generation of allocation DECISION-MAKING AND THE ICCM-1 MODEL / 61 efficiency. The use of the term stability to characterize the ICCM-1 solution is based on two separate concepts. The first refers to the decision-making behaviour imputed to workers choosing among job offers, and is not applicable necessarily to other ICCM versions. The set of choices made in any round of matching job-worker pairs is considered to be stable if it is an equilibrium point. This concept is derived from game theory, and a brief definition is given here: An equilibrium point of a game is a collection of strategies, one for each player, from which no single player can gain by deviating. ... One interpretation of equilibrium points is as collections of strategies that tend to persist over a period of time. ... Equilibrium points correspond to stable modes of behaviour, in the sense that no player can gain by unilaterally changing his indicated stategy. (R.J. Weber, 1981; pages 84-89.) The second use of the term refers to the stability of choice outcomes, and is adopted by all ICCM versions. It is based on the concept of residential location values, analogous to shadow prices associated with the dual of a linear program. For an individual worker, the location value of his place of residence, R', is equal to his net wage less his reservation wage and his commuting costs. A residential zone contains a set of contiguous worker residences which are assumed to exist at a point in space; thus it is possible to speak of a distribution of residential location values associated with a particular zone. The variance of this distribution can be interpreted as a measure of the stability of the commuting pattern which generated it; note that the variance of zonal shadow prices derived DECISION-MAKING AND THE ICCM-1 MODEL / 62 from a linear program is, by definition, 0. More generally, the stability of a commuting pattern solution can be inferred from the distribution of the zonal coefficients of variation of R'. 2. Determinate and Probablistic Choice Behaviour The discussion that follows will present the argument that, by substituting the concept of stability for that of optimality as the diagnostic property of a normative labour market solution, decision-making behaviour based on probablistic ("mixed") rather than determinate ("pure") strategies is required. a. Expected Utility The most common method of approaching the problem of decision-making in conditions of outcome uncertainty is to substitute expected utility for utility as the metric by which choice alternatives are assigned a valuation. Luce states that: The main property required for much of modern decision theory is that in some sense the utility of a gamble should equal the expectation of its component utilities, and this implies an interval scale of utility. As von Neumann and Morgenstern (1944) first showed, we can in fact construct an algebraic choice theory for gambles, provided that the probabilities of events are known, that leads to an interval scale of utility having the expected-utility property. (Luce, 1957; page 76) The concept of expected utility relates the utility of a choice alternative, having DECISION-MAKING AND THE ICCM-1 MODEL / 63 a set of possible outcomes, to the expectations of those outcomes occurring. For example, consider a set of choice alternatives C, with elements s. corresponding to possible and mutually exclusive actions on the part of the decision-maker. Each s. involves a set of outcomes, o., with elements o. , which enumerate the i i i,k n possible unique outcomes initiated by a specific choice. If the probability of an outcome occurring is P(o. ,), and the utility to be gained by the decision-maker i,k should that outcome occur is U(o- )^, then the expected utility of the outcome is: (6) E(o.,) = mo.,) • P(o.,) l,tt l,it l,K and the expected utility of the choice is: (7) E(s.) = Lk Eio.J The rational decision-maker would select his course of action by adopting the s. choice which has the highest expected utility. This implies that, for any pair of alternatives s^, in C, the probability of selecting alternative 1 over alternative 2, written P(s^, s^), must be equal to either 1, 0.5, or 0. Thus, except in the case of alternatives having equal expected utilities and random probabilities of selection, there is a determinate rational choice; the expected utility calculation translates inherent uncertainty of events into certainty of decision-making behaviour. As only one action can be taken, the expected utility of the choice situation is: (8) E(C) = max E(s.) There are a number of problems with this approach, two of which will be discussed at this point. These involve, firstly, the assumption that maximization of a composite function (expected utility) can reflect adequately the simultaneous DECISION-MAKING AND THE ICCM-1 MODEL / 64 evaluation of two independent functions (outcome utility and the level of risk associated with the probability of outcomes occurring), and secondly, the requirement that objective probabilities of outcome events occurring both exist and are known by decision-makers. b. Risk Aversion The first problem is centred on the issue of the rationality of risk-aversion. According to the expected utility hypothesis, risk has no independent cost, but is fully accounted for conjointly with the utility of each outcome. Similarly, there are no direct means of assigning a benefit value to, for example, the thrill that may be obtained from the gambling experience. In this regard, Isard, et al. state: Some attempts have been made to view the [expected utility function] as a utility function over commodity bundles in which one of the commodities is the 'gambling experience' itself; the individual who disapproves of gambling may assign a negative utility to the experience, etc. However, such an interpretation leads to extremely difficult measurement problems. (Isard, et al., 1969; page 181.) It is unlikely that a worker would be influenced by a positive evaluation of the "gambling" experience of the job application process, but the potential disutility associated with the risk of an unsuccessful application is very real. The independent nature of risk can best be illustrated by analysing an example taken from Webber: DECISION-MAKING AND THE ICCM-1 MODEL / 65 Suppose that an expected profits [utility] maximizer is offered two bets, each of which costs $1.00: (i) $10.00 with probability 0.9 or $0.00 with P= 0.1, and (ii) $5.00 with P= 0.7 or $2.00 with P= 0.3. The rational, optimizing decision is to take the first bet. Now suppose that the 'wheel is spun' twice and that the first bet yields $0.00 while the second yields $5.00. Then it is said that the person's choice turned out to be wrong or incorrect: after the event, the choice is regarded as sub-optimal. Such a view is wrong: the choice must be evaluated with respect to the information available when that choice was made, and subsequent events do not alter the correctness or otherwise of the decision. Ex ante and ex post optimality are not equivalent. (M.J. Webber, 1972; page 108) Webber's main assertion - that decisions bounded by the state of information available at the time of evaluation are consistent with the requirements of rational behaviour, regardless of whether they turn out to be optimal - has been discussed above. Clearly, uncertainty of events suggests that the only realistic yardstick of rationality is based on the evaluation of ex ante information. In this example, however, there is an additional element that weakens the validity of the identification of a unique rational decision by means of the postulated correspondence between utility of a choice alternative with its expected utility. Information as to the degree of risk contained in each bet has been ignored -the person making the choice between bets knew that there was a probability of 0.1 that the second bet would turn out to be better, and, more importantly, he knew that the second bet involved no risk whatsoever, while the first bet had potential for loss. The conclusion that choosing the first bet is the only rational decision can only be supported if the bet was one of an extended series of such bets, so that, over the long run, the probability of coming out ahead by choosing DECISION-MAKING AND THE ICCM-1 MODEL / 66 the second alternative approaches 0. On a one-time basis, however, a risk-averse player is always protected, while the risk-taker is not - does the outcome discussed in the example represent the effects of luck, or the potential (and unvalued) benefits of risk-aversion? This is not to say that the second bet is then the only rational choice. The difficulty is that the expected utility concept is insufficient to allow a complete ranking of choices if disbenefits of risk are considered to be an independent component of the decision-maker's utility function. Two problems present themselves in such a case: firstly, is there an objective method of relating the magnitude of risk to the value of a choice alternative; secondly, is the approach to risk evaluation invariant with respect to the relative importance (to the decision-maker) of the choice situation. One approach to this problem is presented by Isard, et al. (1969). A decision-making framework is constructed to identify choice behaviour in a variety of situations. The framework is based on three classes of behavioural assumptions, relating to: a) preferences; b) objectives; and c) guiding principles. The structure is algebraic, so that determinate rational decisions can be identified in a range of choice situations. There is explicit recognition that uncertainty of outcome realisation implies that the criteria determining preference ordering of choices do not correspond necessarily to those determining preference ordering of outcomes. That is, the function defining objective rational choice in conditions of certainty (maximize outcome utility) may not have an analagous form in conditions of uncertainty (maximize expected utility of choice). DECISION-MAKING AND THE ICCM-1 MODEL / 67 Preference assumptions are fundamental to the characterization of strategies used by decisionmakers to identify desirable choices from sets of possible actions. Isard, et al. define a preference ordering as a preference relation between choice pairs which exhibits completeness and transitivity over the set of outcome possibilities. A preference function is an ordering which is capable of numerical representation. A utility function thus is a preference function over outcomes, and an expected utility function is a preference function over prospects (Isard, et al., 1969; chapter 5).t The operational form of preference functions determining optimal choice in conditions of uncertainty is based on the behavioural assumptions termed objectives: One might characterize the objectives of the individual as the "driving mechanism" which directs his behaviour. They embody his concept of the "optimal state of affairs" he hopes to achieve, and they define the nature of his "optimizing" behaviour, which usually takes one of the following forms: i) to maximize the level of some "desirable" properties of outcome possibility sets; ii) to minimize the level of some "undesirable" properties of outcome possibility sets. By focusing on single properties of outcome possibility sets, the individual"s objectives effectively "reduce" or "transform" these sets into more simple elements readily comparable. (Isard, et al., 1969; page 186) An objective function is a means of assigning to outcome possibility sets a unique numerical value reflecting preferences based on the decision property t Isard, et al. use the term ulity function to define any preference function over prospects; expected utility is one specific form of such a function (Isard, et al., 1969; page 177). DECISION-MAKING AND THE ICCM-1 MODEL / 68 deemed critical. Behavioural objectives can be formalized as decision-making strategies. For example, a max-min payoff stategy can be used to identify the option in which the utility of the worst possible outcome is at a maximum - the second bet in Webber's example would be selected as the optimal choice by a gambler using this strategy. The solution is determined solely by the objective of risk-aversion; the magnitude and associated probabilities of the non-minimal potential payoffs available through alternative choices are not considered. Similarly, a max-max payoff strategy reflects the objective of pure risk-taking; the option offering the potential of the greatest payoff, regardless of its probability of occurring, would be selected. Isard et al. discuss a number of single-factor objectives for which decision strategies based on rational choice behaviour can be defined. However, they make no attempt to identify which of these objectives can be considered rational in a given situation. Thus, in the Webber example above, either choice may reflect rational decision-making, depending on what objective underlies the strategy followed; any single objective cannot justify both choices simultaneously. The problem, from a behavioural modelling point of view, then becomes one of identifying either the appropriate objective or the probability distribution describing the relative incidence of various objectives in the target population. The third class of behavioural assumptions, guiding principles, govern the modifications made to individual objectives required by the recognition that other DECISION-MAKING AND THE ICCM-1 MODEL / 69 participants in the decision-making process are motivated, and so are distinct from the unmotivated environment. The recognition of other participants as motivated will tend to guide the individual's behaviour in two important ways. First, it will bring into play certain norms or standards of conduct by which the individual has learned to relate to other participants. Second, it will introduce the expectation by the individual that the other motivated participants have similar norms or standards, and thus may reciprocate action choices made on the bases of these norms or standards. (Isard, et al., 1969; page 205) The guiding principles may be informed by moral or religious standards, or more simply may involve the common assumption that decision-makers recognize the rationality of other participants. Clearly, expected utility is the only preference function which allows both for the assumption of some realistic degree of consistency among a group of decision-makers, while at the same time taking into account both the element of risk inherent in choice situations under conditions of uncertainty and the preference rankings of the outcomes themselves. The weaknesses of this approach, as illustrated by the analysis of Webber's example, are made up for by the ability to utilize it, in the context of game theory, as the basis for determining choice behaviour when the underlying probabilities of outcomes occurring are themselves subject to uncertainty. The second question involves the utilization of various choice objectives, depending on the importance of the choice situation. Intuitively, risk aversion becomes more DECISION-MAKING AND THE ICCM-1 MODEL / 70 appealing as the potential negative consequences of a choice situation become more severe. The job application decision undeniably is one of great importance; should pure risk aversion then be seen as the rational approach? The position taken here is that the importance of the decision justifies the use of an approach to choice behaviour, presented below, which translates expected utility objectives into a defensive strategy. c. Uncertainty and Outcome Probabilities The second area of potential difficulty with the expected utility preference function is that it anticipates that decision-makers know or can calculate the probability of events occurring. Luce criticizes this assumption, stating that: [Expected utility functions] assume that subjects know and deal with the objective probabilities of events as such, rather than with some subjective measure of likelihood. ... The axiomatic structure required to get the desired interval scale postulates a degree of rationality and consistency that is a bit too quixotic to take entirely seriously. (Luce, 1959; page 76) In the labour market situation under consideration, the success probability of a job application depends on the decisions of other workers; thus, objective outcome probabilities do not exist a priori. In this sense, any method of estimating these probabilities must be subjective. However, as will be shown in the next section, estimation of uncertain outcome probabilities implies that some means of predicting the behaviour of others must be adopted, and the use of expected utility preference functions, along with the implied assumptions of rational and consistent behaviour, is the only means of developing a normative structure that DECISION-MAKING AND THE ICCM-1 MODEL / 71 does not rely on revealed preference. C. THE MIXED STRATEGY SOLUTION The discussion above outlined the difficulties encountered in defining a rational decision function which handles adequately both outcome utility and prospect risk. In this section an argument will be presented to justify the use of expected utility maximization as a reasonable objective function to characterize the job application decision. It will be shown that this approach implies a mixed strategy solution, and that a probabilistic interpretation of the solution allows for the construction of a decision model based on objectively rational choice behaviour. In order to illustrate the application of a mixed strategy approach to decision-making, an example analagous to the job application situation will be discussed. 1. A Lottery Analogy Consider a lottery with the following characteristics. There are Y tickets for sale, each selling for the identical price of t. A set of prizes, o., i = 1 to j, is available. Each prize has a monetary value of v., such that: (9) t < V l < v2 < v0 < ... v. All prospective ticket purchasers are aware of the prize values. At the time of purchasing a ticket, the lottery player allocates his ticket to one of the prizes; that is, his ticket is placed in a drum set aside for the prize he has chosen. The prizes are awarded individually through a random selection of a ticket from each drum. If a drum is empty, that prize is not awarded. Tickets are eligible DECISION-MAKING AND THE ICCM-1 MODEL / 72 to win only the prize they are allocated to. Two decisions face the prospective player: a) should he enter the lottery, and b) if he enters, which prize should he select? The lottery can be characterized as a form of non-cooperative zero-sum game with Y players having incomplete information (in that the preferences and strategy sets of other players are unknown; see R.J. Weber, 1980; page 105). The solution to such a game, as will be argued in this section, involves the mixed extension to the minimax theorem of von Neumann and Morgenstern (1944). For the moment question a) will be put aside. Assume that Y is known by prospective players, and that the draw does not take place until all tickets are sold. At the date of the drawing, there are n. tickets allocated to the o. prize, and, by the assumption above, (10) Z.n. = Y If choice strategy is based on the objective of maximizing expected utility, the answer to question b) can be found by calculating the expected utilities associated with each possible prize choice. The probability of winning a given prize is: (11) p(o.) = 1 / n. i i Let C(o.) represent the choice strategy of allocating a ticket to be eligible to win o.. There are two possible outcomes associated with this strategy: either the prize is won, with a value of v.] or it isn't, with a subsequent return of 0. The expected value of C(o.) is: DECISION-MAKING AND THE ICCM-1 MODEL / 73 (12) £[C(o.)] = Lp(o.) • uJ + [(1 - p(o)) • 0] i i i i (13) E[C(o.)3 = p(o.) • This lottery differs from the gambling example of Webber discussed above in that the probabilities of winning various prizes do not exist independent of the choices made by players. As a result, objectives other than one based on maximizing expected utility do not appear to be applicable to this situation. It will be shown below that consideration of the objective of maximizing maximum outcome utility leads to the adoption of a strategy involving an unnecessarily high level of risk. Furthermore, as long as n. > 1, the worst possible outcome of C{o.) is always the same (i.e., 0); this implies that the objective of pure risk-aversion (maximize the minimum return) does not lead to any level of utility discrimination among alternatives. Similarly, adoption of objectives based on a risk analysis in addition to or other than that embodied in an expected utility function is unlikely to be helpful. For example, the utilization of outcome variance as a measure of risk, as suggested by Rapoport and Stein (1972), is of little meaning here, as the variance of outcomes associated with a choice alternative is always determined solely by the value of the prize indicated by that choice. Clearly, E[C(o.)] depends on p(oj), which is a function of the aggregate choice behaviour of all players. If the player assumes that nothing can be inferred about the behaviour of other players, then he would anticipate that the Y tickets will be randomly allocated. This situation would result in a probability function for winning o. of: (14) p(o.) = 1 / (Y/j) DECISION-MAKING AND THE ICCM-1 MODEL / 74 which leads to: (15) E[(C(o.)] > E[(C(o. )] > ... > £[(C(oJ] J J-l 1 Therefore, it would be rational to choose E[{C{ojf\. However, if all players make the same analysis and act accordingly, then (16) p(o.) = 1 / Y; p(o. . . J = 1 J M * 7 This implies that either the assumption of random selection by other players is incorrect, or that the other players are not making the same analysis of the choice situation. The assumptions made concerning the actions of other decision-makers reflect the player's guiding principles (as defined by Isard, et al.) in that they are based on beliefs about the motivation of others, von Neumann and Morgenstern (1944) make the argument that an element of rational behaviour is the assumption that other players are also acting rationally; thus, a guiding principle of this nature is the only one that does not require either acceptance of a belief concerning, or particular knowledge of, the motivations of others. This follows from the implication of symmetry in the theory of zero-sum games: rational analysis of choice outcomes is independent of which player is viewed as the decision-maker. Furthermore, this assumption leads to a choice strategy which is optimal from a defensive point of view, in that it represents the best that can be done in that situation. If other players should deviate from a rational strategy (for whatever reason), then the mis-reading of others' motivations will cause the strategy to be sub-optimal: While our good strategies are perfect from the defensive point of DECISION-MAKING AND THE ICCM-1 MODEL / 75 view, they will (in general) not get the maximum out of the opponent's (possible) mistakes - i.e., they are not calculated for the offensive, (von Neumann and Morgenstern, 1944; page 164.) This can be interpreted to mean that the assumption of motivations in other players that are non-rational is in fact an unwarranted gamble, and that the risk-averse guiding principle is to assume global rationality. If the requirement of symmetry in the choice analysis is accepted, it then is clear that the only pure strategy possible is to select Oj - the prize with the maximum value. However, this is not an equilibrium point, as every player could do better by choosing any other prize (all other strategies offer certainty of winning). In the terminology of von Neumann and Morgenstern, there is no strictly determinate solution. There is, however, an equilibrium point which is associated with a generally determinate solution. An equilibrium point defines a set of strategies whereby no player can gain unilaterally by changing his strategy (R.J. Weber, 1980; page 89). In other words, the strategy of any player would remain unchanged even if full information regarding the strategies of other players was made available. In the lottery example, if the distribution of player choices was such to ensure that E(o.) — K, a constant, then all strategies would be stable, and no player would have an incentive to select another strategy. Thus, if (17) n. = [(v. I I . v.) - Y] i i i i Then, for any prize, • (18) E(o.) = [I. v.] I Y DECISION-MAKING AND THE ICCM-1 MODEL / 76 (19) E(o.) = K i There are two ways in which this equilibrium point can arise. The first involves pure strategies based on subjectively rational decision behaviour. Even though the expected values of all choices at the equilibrium point are equal, there are differences in terms of the amount of pure risk associated with each choice - the greater the value of the prize, the smaller the probability of winning. If, among the population of players, there is a distribution of the valuation of risk-aversion not accounted for in an expected utility function, and this distribution is perfectly correlated with the distribution of prize values, then the equilibrium point can be achieved by each player selecting a strategy based on maximizing a composite objective function measuring pure risk as well as expected value. Of course, the assumptions required for this situation are severe: not only must the distribution of risk-aversion be correlated with that of prize values, but the players must know that this is the case. This is an illustration of the difficulties inherent in projecting behaviour based on subjective rationality - there is no way of establishing the existence of an equilibrium point without having a priori complete knowledge of the distribution of utility valuations, and any set of decisions may define an ex post equilibrium, given a suitably chosen distribution of such valuations. The second approach to the establishment of the equilibrium point assumes objectively rational behaviour, and is based on the concept of a mixed strategy. Let $ be a vector with elements p., i = 1 to j; p, represents the probability DECISION-MAKING AND THE ICCM-1 MODEL / 77 of selecting o^ . $ is a mixed strategy, defined as a probability distribution of choice over all alternatives, such that: (20) I. p, = 1 Thus any pure strategy C(o.) is a special case of with p. . , = 0. The expected value of $ is: (21) = I . [p. • p(o.) • w(o.)] Now, if (22) p. = [v. I I . uJ and symmetry ensured that all players utilized the same (mixed) strategy, then (23) n. = p. • Y (24) / I . = [(v. I Z. v.) • Y] This is the choice distribution required for the equilibrium point solution. It is important to note that the players do not require any knowledge or estimate of Y, but can base their strategy entirely on the (known) magnitudes of v.. Thus, the assumption made above, that V was known, is unnecessary. In fact, even if 1 < Y < j, the mixed strategy solution is unchanged. The question that arises is: what does the mixed strategy $ mean in terms of discrete choices? The lottery involves only one decision, not a series of "moves"; is it meaningful to speak of a single choice in terms of a probability distribution over all alternatives? von Neumann and Morgenstern insist that it is: We make no concessions: Our viewpoint is static and we are analyzing only a single play, (von Neumann and Morgenstern, 1944; page 147.) DECISION-MAKING AND THE ICCM-1 MODEL / 78 It must be emphasized, however, that their justification is based entirely on the grounds that they are investigating the theoretical framework of rational choice behaviour; they are not concerned with the means used by decisionmakers to establish such behaviour. Isard et al. present a concrete example of how a mixed strategy (or, in their terminology, a continuous action space defined over a discrete choice set) can be translated into a specific choice decision: In essence, while feasible action choices may only be finite in number, it is always possible to consider a continuous range of possible probability distributions over action choices. That is, if the participants are able to randomize their actual choice of an action, for example by spinning a roulette wheel to determine the action chosen, then they may also be able to choose their roulette wheels from a "continuum" of possible roulette wheels, each differing by an arbitrarily small amount in the probability weights assigned to various action choices. Thus, if we designate each possible probability distribution (roulette wheel) as an action-mix, and assume that a continuum of action-mixes is open to each participant, then associated with every action space (either finite or continuous) is a continuous space of possible action-mixes. (Isard et al., 1969; page 244) While the roulette wheel analogy may seem far-fetched, it conveys the flavour of the connection between a mixed strategy and a choice decision. An assumption made in postulating the practicality of behaviour based on expected utility evaluation is that decision-makers are capable of making the necessary (objective) calculations. Luce has criticized this approach partly on the basis of the requirement that: subjects know and deal with the objective probabilities of events as DECISION-MAKING AND THE ICCM-1 MODEL / 79 such, rather than with some subjective measure of likelihood. (Luce, 1959; page 76.) In the lottery, the calculations do not appear overly onerous, as a) the utility of a prize is identified with its monetary value, and b) the probabilities of winning a prize are determined solely by the strategies employed by the players. Thus, there are no objective probabilities independent of choice strategies, and any form of strategy development must take into account the effects on outcome probabilities of strategies employed by other players. It is possible that the utility of money may be subjectively distorted, but, in the absence of knowledge of the functional form of such distortion, it is at least reasonable for a player to assume a linear function with normally distributed error, t However, "second-guessing" clearly is a potential factor in the translation of strategy to decision. It may be reasonable to impute an element of pure gambling in the psychology of choice: Players are tempted to select a prize with a high value, but know that, if everyone did so, they would end up in a poor position. Thus, lower-valued prizes are chosen by those players who gamble that others will be "greedy"; higher-valued prizes are chosen by players who rely on the rationality of others. tThis statement would be less applicable if the variance of prizes offered is very large. For example, the greater propensity of people to buy tickets for lotteries offering large prizes and very long odds than for lotteries with smaller prizes and less (objectively) prohibitive odds suggests that the second derivative of the utility of money with respect to gambling prizes is > 0. This observation is in marked contrast to espectations based on the diminishing marginal utility assumptions common in the literature. Bernouilli (1954) suggests that the utility of money is related to the logarithm of its value; Webber (1972; page 101) accepts Cramer's view that utility is related to the square root of monetary value. It is unlikely that the explanation of this phenomenon lies in the positive utility gamblers assign to risk, as all lotteries offer this benefit in abundance. DECISION-MAKING AND THE ICCM-1 MODEL / 80 In any event, the mixed strategy approach throws no light on the ex post analysis of an individual player's actual choice decision, but it provides a means of predicting the aggregate choice distribution (the n. values) by defining the normative solution in terms of objective rationality. To accomplish this, it is assumed that the value of a choice alternative can be measured solely in terms of its expected utility. R.J. Weber states that, in order for a mixed strategy to make sense, it must be that when faced with an uncertain outcome, each player's goal is to maximize his expected utility (R.J. Weber, 1980; page 91). If expected utility does not reflect fully the value players attach to choice alternatives, it is then unlikely that an' equilibrium point exists. The mixed strategy approach requires a restricted definition of choice alternative utility, but in return does not suffer from the circular reasoning characteristic of subjective approaches that depend on revealed preference to establish prospect utility functions. As a result, it allows for the testing of the underlying assumptions in a variety of controlled situations; hypotheses constructed with regard to the rational analysis of prospect utilities are verifiable, and thus are "scientific" rather than "metaphysical", in Katouzian's (1974) terminology. The solution to decision problem a) posed above, as to whether the player should enter the lottery at all, can be found through the use of the concept of a game's value, t This is defined as the expected utility of the equilibrium point strategy (see von Neumann and Morgenstern, 1944). Thus, if the value of the t An exception to this approach is the trivial consideration of a player motivated by the objective of pure risk-aversion (maximize the minimum return): such a player will abstain from entering if it is known or anticipated that at least one other person will buy a ticket. DECISION-MAKING AND THE ICCM-1 MODEL / 81 lottery is greater than the price of a ticket, it is worth entering. The lottery's value, based on a mixed strategy approach, depends on the total value of prizes, which is known, and the number of tickets sold, which may be unknown. Knowledge of the number of tickets is not required in establishing the best strategy, but is fundamental in deciding whether or not to enter at all. 2. Application to the Labour Market The major difference between the lottery example and the labour market is that, in the lottery, the prizes are determined a priori and players choose in response to the prizes offered; there is no mechanism that relates prize values to players' responses. In the labour market, wages are raised only if applications are scarce; i.e., employers do not raise wage levels above that required to attract the minimum number of applicants so as to fill vacancies. If workers know this, can they assume then that the pure choice strategy of applying for the job offering the maximum net wage will equalize competition across the set of job openings, so that expected utility derived from the strategy will be maximized and an optimal equilibrium will be attained? The problem with this scenario is that the wage surface guaranteeing equal probability of job application success (given a maximizing pure strategy) must somehow come into existence, and workers must: a) know that the wage surface is optimal; and b) assume that all other workers know it as well. In other words, how does an efficient price structure come into existence before market transactions occur? This is the reason market interpretations of linear programming solutions must assume full knowledge by all participants: uncertainty DECISION-MAKING AND THE ICCM-1 MODEL / 82 cannot exist, or else suboptimal transactions may occur. Postulating a Walrasian auctioneer is identical to assuming arbitrage, and a tatonnement process is inapplicable to the labour market in that labour power is not sold continuously in small increments, creating an optimal wage structure based on a dynamic equilibrium. The difficulty underlines the conclusion of Koopmans (1951) discussed above, that rules governing decentralized decision-making can be constructed to ensure the stability, but not the creation, of an efficient allocation. In an analysis of the linear programming transportation problem, von Boventer (1961) argues that the least-cost (globally optimal) programming solution is identical to the market solution. He outlines the decision-making procedure required as follows: The market solution is found by price adjustments. The consumers try to buy from the cheapest source of supply. The suppliers at all locations where aggregate supply is less than the aggregate demand at the given prices raise their prices by one unit. If an infinitesimally small price rise would eliminate all the excess demand, these suppliers just wait, as do all those with excess supply at the prevailing prices. As long as the equilibrium has not been reached, there must always be at least one source of supply whose producers can raise their factory prices without inducing a demand deficit. Equilibrium is reached if the suppliers have adjusted their factory prices in such a way that supply and demand are equalized when each customer buys from the supply source offering him the lowest delivered price. (E. von Boventer, 1961; page 33) The question is, how does the market convey information through price signals if participants wait for equilibrium pricing before concluding transactions? Unless DECISION-MAKING AND THE ICCM-1 MODEL / 83 demand for labour greatly exceeds supply, it would require a great deal of faith in the market for a worker to refrain from applying for a job offering a net wage greater than 0 in the anticipation of a higher wage being offered in the (near) future. Thus, the worker is confronted with a wage-offer surface which may or may not correspond to the efficient surface; this uncertainty creates the environment for which the game theoretic approach presented in terms of the lottery example is appropriate. This discussion illustrates the defensive nature of the mixed strategy solution; it cannot guarantee the best result, but it reflects the acknowledgement by the decision-maker of the risks associated with an uncertain environment. An illustration of this type of situation is found in the two solutions to the prisoners' dilemma game (Rapaport and Chammah, 1965). Briefly, two prisoners are interrogated separately, and each is told that, if he admits to the crime and implicates his partner, he will receive a reduced sentence. The prisoners then have two choices which correspond to two solutions. The first is the cooperative solution - deny the crime. Of course, this approach is optimal if done in concert, as they both go free; but it entails a high level of risk if each prisoner is uncertain as to the decision of the other. The second is the competitive solution - admit to the crime and receive a reduced sentence. This corresponds to a defensive strategy, and, even though it is sub-optimal, can be justified in terms of risk aversion under uncertainty. The concept of the value of a choice situation, discussed in the lottery analogy, provides an explanation for the existence of "discouraged" unemployed - a term DECISION-MAKING AND THE ICCM-1 MODEL / 84 applied by Statistics Canada with reference to unemployed workers who have not actively searched for employment for six months, and thus are not enumerated in labour force statistics. If there is a real cost of job search (that is, if the opportunity cost of time is greater than 0 for unemployed workers), then, if the expected utility of the job application strategy is less than this cost, it may be rational to not enter the labour market. Of course, the probability of such a decision being correct depends on the number of workers who come to the same conclusion, which effectively determines aggregate labour supply. If the number of available workers in a given occupation / skill category exceeds the number of jobs, then the distribution of unemployment is based on two factors: a) the distribution of workers not entering the active labour force (in Statistics Canada terminology); and b) the application strategy of workers in the labour force. Linear programming models allocate the residential location of unemployed workers to the least-accessible sites; thus, the map pattern of employment and residential locations determines the spatial distribution of unemployment. Empirical studies, in general, do not support this aspect of labour allocation theory; central city residents often suffer high unemployment rates (see, for example, Daniels (1986)). The approach outlined in the lottery example provides a more realistic basis for investigating the distribution of both unemployed labour force participants and discouraged workers. DECISION-MAKING AND THE ICCM-1 MODEL / 85 D. CONCLUSION The argument presented in this chapter has supported the assumption of objectively rational choice behaviour as a realistic and practical means of simulating the urban labour market. Furthermore, if the labour market is assumed to contain no endogenous self-optimizing mechanism, the decision-making environment is characterized by outcome uncertainty. The discussion of the lottery analogy indicated that a modelling approach, based on game theory, may be applicable to the more complex situation of the labour market. The next chapter extends this logic to develop a behavioural assumption set that, in the context of the ICCM structure, provides the basis for an operational model. CHAPTER V. TWO VERSIONS OF THE ICCM-1 MODEL The game theoretic approach to choice behaviour under uncertainty, presented in the previous chapter, is extended in this chapter to characterize decision-making in the context of the space economy. The results are used to develop two versions of an operational model designed to simulate the allocation process of the urban labour market. A. MIXED STRATEGY AND JOB APPLICATION MODELLING In this section the issue of the applicability of the mixed strategy solution to the job application problem will be discussed. To begin with, the characteristics of the urban job market will be analyzed in terms of the lottery example in order to determine the degree of compatability between the two situations. The value of a job to a worker is defined as the net wage; that is, the wage paid at the workplace less the reservation wage and commuting costs. Then, for a worker resident at location i, the value of a job at location j is: (25) N. . = W. - (w + T. .) The reservation wage, w, is the spatially invariant component of the occupation / skill specific opportunity cost of accepting a job. Thus, if (W. - T. .) < w, then J i\J there is no incentive for acceptance of the job {N. . < 0). The net utility of a job offer is equivalent to the net wage, i.e., effectively, U(J. .) = N. .. A worker confronted with a set of job offers chooses one to apply for based on the expected utility of that decision. Consider the (unrealistic) situation in which all workers of a given skill and 86 TWO VERSIONS OF THE ICCM-1 MODEL / 87 educational level reside in the same area i of the city. Assuming that the cost of commuting to any job location j is perceived by all of these workers to be equal to (the objective cost) r. ., they will have identical utility evaluations of the set of job offers for which they are qualified, and the decision as to which job to apply for will be based on a choice evaluation analogous to that of the lottery example, with the proviso that not all job offers necessarily will entail positive utility; that is, only a subset J, defined as those jobs for which U(J. .) > 0, will be considered as choice alternatives. The objectively rational choice, for every member of the set of workers W., will be the equilibrium mixed strategy $ such that (26) p. = [N. . I (2. N. .)]; for all N. . > 0 J iJ J ij (27) p. = 0; for all N. . <z 0 J iJ This result involves the implicit assumption that information concerning the elements of J is available to all workers in i, so that $ will be a vector with a choice probability for every job offer in J. As the number of potential job offers may be large, this assumption involves an unrealistic demand on the ability of workers to collect and analyze information. However, if it is assumed a) that there is a random probability of a worker knowing of a specific job offer k in J, and b) that, if J > 1, then the subset J' composed of job offers in J that are considered by any individual worker k is > 2, then the probability of a worker applying for a specific job is unchanged, and is defined by the mixed strategy vector <i>. This conclusion follows from the principle of the independence of irrelevant alternatives; the relative choice probabilities between two job offers are determined solely by the relative net wages of the two jobs, regardless of TWO VERSIONS OF THE ICCM-1 MODEL / 88 the distribution of net wages in the full job offer set. t The assumption that a worker considers job offers in J' rather than in J also is in agreement with the assumptions underlying labour market processes required for the applicability of the simultaneous solution path logic discussed in chapter III. Simulation of job-worker matching then can be viewed as an ongoing process, rather than as a one-time market clearing operation. This characterization of the labour market process involves a change in the interpretation of $ from that of the vector of probabilities indicating the optimal mixed strategy under the condition of full information concerning the set J to that of the vector of probabilities reflecting the relative likelihood that a worker will choose a particular job in J, given that the worker is cognizant only of a subset J' in J. In the latter case, the mixed strategy over J' will not be identical to but will be a scalar function of As the present concern is with establishing the probability of a worker applying for one of the full set of job offers, the second interpretation of $ will be adopted. 1. The Mixed Strategy and Dispersed Worker Residences Workers do not live in one location in the city; residences are distributed in some pattern throughout the urban area. Thus, a job at location j offering a wage of W. will provide a vector of net wages N. . that is determined by the J iJ commuting costs between any residential location i and the job site j. The TThis principle forms a cornerstone of Luce's theory of individual choice behaviour. The assumption of its applicability in choice situations constitutes his Axiom 1; the acceptance of this axiom allows for the utilization of a conditional probability approach to choice behaviour. See Luce (1959), chapter 1. TWO VERSIONS OF THE ICCM-1 MODEL / 89 rationale behind the derivation of $ is that all workers (prospectively) competing for a job would derive equal utility from the wage offered to the successful applicant. But, as N. . varies with respect to i, so does U{J. .). This indicates that <t> varies over space, so that a unique vector exists for any (residential) location i. The equilibrium strategy for all locations can be established analytically in the following manner. By definition, all strategies must be optimal, in that no alternative strategy exists which is superior to one in the matrix <f>. In terms of a mixed strategy, this condition is satisfied only if the expected utilities derived from the choice alternatives having positive probabilities of selection in J', for any worker k, are equal. Then, if K. is the expected utility of any job applied for from a residence in i, p. . is the choice probability of a worker in i applying for a job in j, and B. represents the number of workers of the occupation / skill category in question resident at i, then the expected utility of an application for a job in location j gained by a worker resident at location i is: (28) K. = {N. . I L. (p. . • B.)} Two difficulties are inherent in this approach. First, as the K values are indeterminate, it is unlikely that an <t> matrix of choice probabilities which solves the equation set defines a unique equilibrium. With no information feedback mechanism in the job application process, so as to allow applicants to know which equilibrium solution is under consideration by other applicants, this approach to labour market modelling appears to be unrealistic. TWO VERSIONS OF THE ICCM-1 MODEL / 90 This situation can be illustrated by considering the somewhat analogous para-mutual betting problem. An equilibrium strategy is possible only because of the existence of a tote-board which supplies updated information as to current odds, which are based on the relative number of bets made on each horse; this is analogous to knowing, at regular intervals, the relative values of Z. (p. . • i ij B.) for each j job location. Of course, if only one bet can be made, or if all bets made by an individual player must be placed at the same time, this information is only helpful for those players who make the last bets. (This explains why experienced track gamblers wait for the last possible moment to place bets.) The second difficulty is more serious, and follows from the possibility that an equilibrium solution, defined in terms of the identity of K. for all j alternatives in <i> .^, does not exist. An equilibrium solution of this type depends on the existence of a specific relationship between the distribution of N. . and that of the residences of applicants, B., so that some objective function utilized by all applicants will produce p. . mixed strategy probabilities that are an inverse linear function of U(J- .). As U(J- .) varies with i, it is likely that the required function exists only in terms of mean values of U{J. .) for any i. An alternative, although somewhat weaker, definition of an equilibrium solution could then be utilized: if the mixed strategy vector is viewed as a single decision, then the set of mixed strategies $ for which the (aggregate) expected utility of any vector • $ is at a maximum, given the other mixed strategies defined by 3>, then the solution is stable. The principle of substitution implies TWO VERSIONS OF THE ICCM-1 MODEL / 91 that, if the method of arriving at is rational, then the method of arriving at $ is consistent. This definition of an equilibrium solution will be adopted in the following discussion. The ICCM-1 model utilizes a heuristic algorithm to derive an equilibrium solution as defined above. There are two reasons why an analytical approach has been considered to be unrealistic as a theoretical basis for labour market modelling. First, it is necessary to assume that each worker has the ability to gather and assimilate the information necessary to derive a full understanding of the implications of any (potential) choice strategy ' concerning the resultant level of competition to be faced in applying for any specific job. This requires complete information regarding the spatial distribution of competing job applicants and the potential commuting costs of these applicants to all job destinations. This assumption clearly is beyond the limits of credulity. The enormous amount of information required can be appreciated if it is realized that the commuting cost structure depends in part on congestion costs engendered by commuting flows; thus, knowledge of the transportation cost surface must be dynamic as well as spatially complete. Second, even if each worker has the ability to gather this information, the computational ability required to transform it into an equilibrium strategy is beyond anyone lacking sophisticated computational facilities, not to mention an operational algorithm. The problem of incomplete information can be handled by assuming a consistent behavioural construct that reflects the method used by job applicants to overcome data (or computational) limitations. In discussing this approach to games with TWO VERSIONS OF THE ICCM-1 MODEL / 92 incomplete information, Weber states: A standard assumption in the study of noncooperative games is that the structure of a game - the number of players, their strategy sets, their preferences - is common knowledge among the players. What if the final payoff depends on an unknown state of nature, about which each player has some private information? What if each player is uncertain about the preferences of his competitors, and hence about their strategic motivations? An approach suggested by Harsanyi (1967) is to imbed such situations in larger games; the uncertainties are represented by probability distributions over these moves. ... An equilibrium n-tuple of strategies in a game can be viewed as a collection of consistent, rational conjectures: each player's choice of an action, conditioned on his type, must be optimal given his conjectures about the behaviour of others, and all players must make the same conjectures about their mutual competitors. This viewpoint suggests defining a player's randomized strategy in terms of the conjectures his competitors can make about his behaviour - that is, as a joint distribution on action and type spaces. (R.J. Weber, 1980; page 105.) In the labour market situation, we are assuming that preference functions of applicants are objectively identical, and so are known globally. However, payoffs (net wages) are known only for applicants in similar locations. To identify equilibrium strategies, workers need to know the aggregate number of competing applications for each job, which is L. (p. . • £.). As an accurate assessment i V i requires more information than is available, then some consistent estimation technique must be postulated. The approach suggested by Harsanyi, which utilizes probability distributions over players' conjectures, may be effective in the present situation in that the TWO VERSIONS OF THE ICCM-1 MODEL / 93 uncertainty of an applicant's beliefs about the net wage surface perceived by another worker increases as the distance separating the two increases. Thus, inclusion of a stochastic element in the estimation procedure whose variance increases with distance would reflect the distribution of uncertainty. The estimation procedures utilized in the two ICCM-1 versions (detailed below) are based on deterministic rather than stochastic approaches to interpreting available information; the motivation for this choice is practical rather than theoretical, and lies in the advantages of simplicity. 2. Two Sub-versions of ICCM-1 The ICCM-1 model contains two alternative estimation methods; the use of either constitutes the adoption of a specific behavioural assumption, and thus defines a sub-version of the model. a. ICCM-la The approach taken in the simplest construct, ICCM-la, is based on applicants utilizing the distribution of relative utilities they themselves (potentially) gain from the job set offering W. wages as an approximation of the distribution of relative utilities accruing to all applicants, regardless of residence location. The rationale is that, even though this approach may involve serious errors in terms of approximating the net wage distribution as seen from any individual residence location distant from the applicant's, it will be an effective means of estimating the level of competition faced in applying for a given job. Furthermore, just as the uncertainty of estimation increases with distance TWO VERSIONS OF THE ICCM-1 MODEL / 94. between residence locations, if worker residences are not overly clustered, the importance of accuracy of estimation decreases. This follows from the localized nature of urban labour markets: wages are based on the minimum level required to attract a sufficient work force, so that wage levels only become sufficiently high to attract labour from distant locations if there is a severe imbalance between local supply and demand. Localized labour markets imply that estimation of competition based on extrapolation from local conditions is a reasonable approach. This argument is consistent with Scott's (1981) view of the process of urban employment decentralization: spatial wage differentials are critical in determining the relative attractiveness of urban locations for plant expansion or new plant siting; the dynamics of residential and industrial location that create the labour supply / demand surfaces (exogenous to the model) will favour localization if the initial assumption of the wage determination process is accurate. Under version la, mixed strategy vectors are composed of choice probabilities (for each specific job at location j) of: (29) p. . = N. . / Z. N. . If the number of available jobs in destination zone j is Ej, the probability of applying for any job located in j is then: (30) p. . = (£.• N. .) / Z. (E. • TV. .) ij J l J J J ij If the number of workers resident in zone i is B., the number of applicants from residence zone i applying for jobs in j, A. ., is: (31) A. . = B. • p. . TWO VERSIONS OF THE ICCM-1 MODEL / 95 b. ICCM-lb The second approach involves the use of an additional term to adjust the estimate of the level of competition initially provided by the relative net wage perceived from the applicant's residence by taking account of the applicant's spatial position in the labour shed of each employment location. This position factor, V. ., handles two aspects of the ability of workers to estimate the level of competition faced in applying for a given job. The first is that the higher the wage offered, the greater the diameter (in terms of a cost-of-travel map transformation) of the labour-shed; this implies, in absence of knowledge of the distribution of potential competitors, that competition increases with the wage level. The second aspect concerns the relative distance from an applicant's residence to a job site vis a vis the diameter of the potential labour shed of that job location. It is defined as the square root of the percentage of potential applicants to j for whom the value of N. . is greater than that derived by the reference applicant. The reason the square root of this percentage is used is to avoid overcompensation - a) the net wage (the basis of strategy determination) is also a function of position in the labour shed; and b) consistency of estimation among applicants means that the adjustment effect is cumulative. Now, if the cost of travel is a linear function of distance and is the same in all directions from all points (i.e., if the travel-cost map transform is identical to a Euclidean map, up to a scalar constant), and if the distribution of workers in a given occupational / skill category is random, so that the expected density of these workers is everywhere constant, then potential competition unaccounted for 2 by N. . and due to: a) the labour-shed diameter, is W. ; and due to b) the TWO VERSIONS OF THE ICCM-1 MODEL / 96 2 2 5 relative distance of residence to job site, is (T. . / W. )' . J In order to establish a choice strategy with the maximum expected utility, a worker would then increase the probability of applying for a job alternative for which the anticipated competition due to factors a) and b) (additional to that estimated from the net wage calculation) is relatively low, and decrease the probability of applying for a job for which anticipated competition is relatively high. Thus, (32) V. . = 1 / {(T. 2 / W. 2)' 5 • W2} iJ lJ J J (33) V. . = 1 / (r. . • W.) V iJ J The difficulty with this estimation procedure is that the assumptions of an iso-cost transport surface and constant residential density are highly unrealistic. Empirical evidence shows that residential densities typically decline with distance from employment centres. For example, Muth (1969) found a strong positive association between the percentage of manufacturing industry located in the central city of an S.M.A. and the population density gradient, implying that an assumption of constant residential density requires the condition of a highly dispersed or even employment distribution. Furthermore, insofar as transport costs are determined by travel time, it is likely that costs are positively related to the density of the built environment through the intervention of congestion effects. A second factor limits the reliability of the assumption of an iso-cost transport surface. This assumption requires that travel between any two points i and j can be accomplished via a straight-line routing, and that all such routings have TWO VERSIONS OF THE ICCM-1 MODEL / 97 equal capacities and volume delay functions. In reality, the typically hierarchical nature of urban transport networks suggests that supply characteristics are highly uneven. It is reasonable to postulate that the average speed achieved between i and j, as measured in relation to a straight line joining the two trip-ends, is controlled by some factors that are decreasing functions of the length of the trip, and some that are increasing functions of trip length. The first type are based on a) the likelihood that the shortest-path route (in terms of travel time) will tend to deviate more from a (Euclidean) straight line as total travel time increases; and b) the ability of commuters on shorter journeys to make use of a wider range of alternative routings (in order to avoid congestion bottlenecks, etc.). The second type is based on the postulate that commuters on longer trips will be able to make more use of high speed and capacity routings than will commuters on shorter trips. Thus, the compromise position, that average trip speed is a linear function of trip distance (with density held constant) has been accepted as a defensible assumption. In order to combine realism with simplicity, ICCM-lb assumes that the average speed of urban travel, as measured in relation to the straight line distance D. . separating any two points i and j, is an inverse linear function of the square root of population density between i and j (A. .) and the cost of travel ( T . .) is a linear function of travel time, so that: (34) D. . = K, • K0 • {r. . / A. Z5} where K^, are slope constants (and will be ignored as they do not affect the final results). The (relative) area enclosed by a circle with radius D. ., centred at trip origin i, is: TWO VERSIONS OF THE ICCM-1 MODEL / 98 (35) D. 2 = T. 2 / A. . and the estimate of the population living within this area, in terms of relative 2 measurement, is provided directly by T .j . Under these assumptions, the best estimate of the value of V. . would remain 1 I (T. . • W.), recognizing that the reliability of this measure depends on the ^•J J variance of population density subsumed in the aggregate mean measure A. . not being too great. The mixed strategy vector of choice probabilities under version ICCM-lb then is composed of. elements defined in the following manner: (36) p. . = (N. . - V. .) I I . (N. . • V. .) ij ij ij J ij ij for each specific job in j and, analogous to the distribution equations of ICCM-la, (37) p. . = ( £ . • N. . • V..) / Z . (E. • N. . • V. .) ij J lJ lJ J J ij (38) A. . = B. • p. . Note that ICCM-la is identical to ICCM-lb, with V. . = 1, for all ij. c. Allocation Example In this section, an example of the two allocation procedures is presented. The example examines only the first allocation procedure within an iteration, and accepts as given the wage structure defining the context of the iteration. The wage structure assumed here has not been endogenously determined, and is established for demonstration purposes. Only four employment zones and three residence zones are used; the example can be viewed as a description of the modelling operation in one section of an urban area. TWO VERSIONS OF THE ICCM-1 MODEL / 99 Figure V - l represents the wage, travel cost and net wage environment for employment locations a,b,c,d and residence locations 1^ 2,3. Solid lines represent net wages, so that intersections with the dashed lines rising vertically from residence locations indicate N. . values. Figure V-2 presents the data utilized; the fact that ^-B. is set to equal ^-Ej is for ease of analysis only. Figures V-3 and V-4 present the modelled strategy matrix and resulting job application matrix. Note that where N. . < 0, the choice strategy indicates a probability of applying = 0. Figure V-5 presents the expected utility vectors of the choice strategies derived from version la and lb behavioural assumptions. These are equal to, for each residence zone, the product of the net wage obtainable From an employment location, the probability of applying for a job at that location and the probability of the application being successful, summed over all employment locations. The objective of the modelling procedure, given the basic assumption of objective rational behaviour underlying ICCM-1, is to allocate workers to jobs on the basis of a choice strategy which maximizes the expected utility of outcomes. In this example, version lb clearly is superior - the more complex competition estimation procedure does a better job of adjusting for labour-shed position than does the simplified approach of version la. A comparision can be made between ICCM results and the expected utility vector that would be derived from a (pure) choice strategy of applying for the job offering the maximum utility; i.e., the highest net wage. This strategy is inferior F I G U R E V - l ICCM-1 E X A M P L E : NET WAGE D I S T R I B U T I O N 1 a b 2 c D i s t a n c e i n T r a v e l C o s t U n i t s F I G U R E V - 2 I C C M - 1 E X A M P L E : DATA 1 0 1 1 . Wages a t E m p l o y m e n t S i t e s : a 1 0 b c 7 9 d 6 2 . T r a v e l C o s t M a t r i x : a b c 1 2 2 4 3 9 7 1. 4 9 3 2 3 . N e t Wage M a t r i x : a b i 1 8 2 6 3 1 4 4 - 1 - 3 3 4 4 . D i s t r i b u t i o n o f E m p l o y m e n t : a b e d 1 0 0 7 5 8 0 5 0 5 . D i s t r i b u t i o n o f R e s i d e n c e s : 1 1 3 0 2 1 0 0 3 75 1 0 2 F I G U R E V - 3 I C C M - 1 E X A M P L E : MIXED STRATEGY MATRIX M o d e l V e r s i o n l a a b- c d 1 . 6 3 . 2 4 . 1 3 0 2 . 3 5 . 1 8 . 3 8 . 0 9 3 . 1 4 0 . 5 7 . 2 9 M o d e l V e r s i o n l b a b e d 1 . 7 0 . 2 5 . 0 5 0 2 . 1 4 . 1 3 . 6 5 . 0 8 3 . 0 4 0 . 3 8 . 5 8 F I G U R E V - 4 I C C M - 1 E X A M P L E : JOB A P P L I C A T I O N MATRIX M o d e l V e r s i o n l a a b e d Sum 1 8 2 3 1 17 0 1 3 0 2 3 5 18 3 8 9 1 0 0 3 1 0 0 4 3 2 2 75 Sum 1 2 7 4 9 9 8 31 3 0 5 M o d e l V e r s i o n l b a b e d Sum 1 9 1 3 3 6 0 1 3 0 2 1 4 3 6 5 8 1 0 0 3 3 0 2 9 4 3 75 Sum 1 0 8 46 1 0 0 5 1 3 0 5 F I G U R E V - 5 I C C M - 1 E X A M P L E : E X P E C T E D U T I L I T Y OF CHOICE S T R A T E G I E S M o d e l V e r s i o n l a 1 5 . 1 4 2 5 . 1 3 ' 3 3 . 6 0 M o d e l V e r s i o n l b 1 6 . 2 7 2 5 . 7 0 3 3 . 8 3 C o m p a r i s o n : E x p e c t e d U t i l i t y o f P u r e S t r a t e g y 1 6 . 1 5 2 3 . 6 6 3 2 . 2 9 TWO VERSIONS OF THE ICCM-1 MODEL / 105 to that of the ICCM-1 model, as potential competition is ignored. Generally, if the wage structure is not optimal, this result will hold. If the wage structure is optimal, then the pure maximizing strategy will be superior. The justification for the present approach, as discussed earlier, is that workers will not be able to recognize an optimal wage distribution even if it does exist; therefore, the mixed strategy is always rational. Finally, note that the wage structure presented in the example will not provide a model solution: wages at locations b and d, under version la, and location b, under version lb, must be raised, and workers re-allocated according to the choice strategy indicated by a new iteration. d. Implications of the Solution Path The ICCM-1 simulation (for both versions) is simultaneous, and proceeds via a two-level nested iteration - that is, a repeated-allocation solution path algorithm. The outer iteration is defined by the current distribution of wages (WJ. The inner iteration allocates workers to jobs by assuming a random matching of the distribution of job openings (supplied as exogenous data) to applicants (endogenously determined by the strategy matrix <i>). Should the initial result of the inner iteration procedure fail either to allocate all workers or to fill all job openings, and, after the allocation, jobs remain unfilled for which at least one worker would derive a potential net wage > 0, then the allocation procedure is re-iterated, with the distribution of applications for still-open jobs based on conditional probabilities derived from the original $ matrix. When the conditions for reiteration no longer are satisfied, then the wage levels are adjusted and the TWO VERSIONS OF THE ICCM-1 MODEL / 106 entire procedure is repeated. The logic behind this approach is based on a number of considerations. First, employers are assumed to be cost-minimizers, so that it is unlikely that an employer would raise the wage offer unless he had waited a sufficiently long time to be certain that no applications were forthcoming at the lower wage. Second, job application behaviour of workers is assumed to be motivated by a consideration of the risks inherent in the competitive nature of the labour market; one of these risks is unemployment. Thus, it is reasonable to conclude that, should workers be unsuccessful in their initial job applications, they will continue to apply for job openings offering satisfactory wages, rather than .gambling that wages for jobs remaining unfilled will be raised. The concept of the reservation wage is critical in defining the wage level at which the decision to wait for an increase in the wage offer, rather than to apply for a job offering an unsatisfactory wage, is made. The third consideration relates to the ability of the model to simulate realistic choice behaviour. Up to this point, attention has been focussed on the situation in which, should individual workers make multiple job applications, they do so sequentially. The solution procedure described above allows for an extension of the characterization of choice behaviour to which the ICCM-1 model is applicable to include, with certain restrictions, the situation of multiple applications being made simultaneously. A discussion of this interpretation follows. TWO VERSIONS OF THE ICCM-1 MODEL / 107 B. EXTENSION TO A MULTIPLE-APPLICATION SITUATION An important issue of the accuracy of the lottery analogy concerns the realism of postulating a single job application by each worker, corresponding to lottery players selecting a single prize. It could be argued that workers commonly apply for a number of jobs, specifically in order to protect themselves from competition in the labour market. This position leads to a number of ways of characterizing the labour market outcome, some of which are amenable to formulation as behavioural assumption sets in the ICCM framework. However, the ICCM-1 simplification of assuming a single application at a time leads to a model of labour market functioning which is not necessarily incompatible with multiple application assumptions. Even if it is accepted that workers commonly apply for more than one job at a time, it is unlikely that they would apply simultaneously for every available job. The job application process entails costs in terms of time and money (separate from costs of acquiring information). Also, knowledge of the sequence of hiring decisions by employers would likely be unavailable to workers, so that workers cannot anticipate that job offers will remain open until all potential applications have been received. If the number of potential job offers is large, the worker must concentrate effort by selecting a subset of offers for which to apply. The next question involves the issue of motivation underlying multiple applications. If it is accepted that the utility of job offers can be established objectively, then workers have a full preference ranking of the elements of the TWO VERSIONS OF THE ICCM-1 MODEL / 108 subset J' of available jobs for which they have information. But the decision to apply for a number of jobs implies that, at least under certain conditions, workers are willing to accept a job which does not offer the maximum utility in J'. Consider the situation in which a worker has applied for two jobs from the subset J'. The jobs identified by these applications define the content of the subset J" of J'. The worker is offered job J" , but has not yet learned whether an application for job J"^ will be successful; and JV^  < N^. The rational decision as to whether or not to accept job J" would be based on a comparison of the utility of J" ^ (which is the same as the expected utility of the strategy of choosing J"y i.e., E(C^), as the outcome probability of the choice is 1) with the expected utility of declining J" and waiting for the employer to select the successful application for J" (i.e., E{C )). But, without knowledge of the number of applicants for each job, calculation of the expected value of pure strategies can only be made in relative terms; this choice situation requires calculation in absolute terms. Thus, the worker must rely on a subjective estimate of E(C^) to guide the decision; any level of risk-aversion will lead to a decision to accept J"r Following this argument, it is not unreasonable to make the assumption that, should workers apply for a subset of available jobs, the decision to include a specific job in this subset (J") implies a readiness to accept that job should the application be successful. The problem then is one of determining the basis by which this subset is chosen. TWO VERSIONS OF THE ICCM-1 MODEL / 109 If the magnitude of monetary and time costs incurred in applying for jobs are randomly distributed across the available job set (J'), then, under certain conditions, in the same manner in which the probability of applying for a particular job is determined by the mixed strategy vector 4>, the relative probability of a particular job offer being a member of the subset J" is closely estimated by The conditions are: a) as a rule, the number of jobs known by a worker to be available (J') is large in relation to the number of jobs applied for by that worker {J"); b) the distribution of the number of applications made per worker is a randomly distributed variable exhibiting no spatial bias; c) the sequence of hiring decisions made by employers is random. To illustrate the first condition, consider the following example. A worker considers four job offers, with job a providing a net wage of 4, b a net wage of 3, c of 2, and cf of 1. Thus, under the assumptions of ICCM-la, the probability of applying for one of the jobs will be .4, .3, .2, and .1, respectively. This means that the probability of applying for a rather than b, for example, is the conditional probability: (39) p(a,b) = p(a) I (p(a) + p(b)) (40) p(a,b) = .5714 If two jobs are applied for, and the probabilities guiding the selection are determined by $, then the probability of job a being selected is .716, and the probability of b being selected is .608. In this case, p(a,b) = .5408; if three applications are made, p(a,b) = .5146. Of course, if all four jobs are elements of J", then p(a,b) = .5. TWO VERSIONS OF THE ICCM-1 MODEL / 110 Now consider a similar situation, but that o consists of a location with 10 job openings, all offering a net wage of 4, b a location with 10 openings each with a net wage of 3, etc., so that the set J' contains 40 jobs. If one job is applied for, then, for any specific jobs at a and 6, and 6^ , p(a^,b^) = .5714. If two jobs are applied for, then p(a^,b^) - .5702; if four jobs are applied for, p(a^,b^> = .5698. In this situation, the number of applications would have to exceed about 8 for the relative probabilities of inclusion in J" to change significantly. The second condition ensures that the relationship between the relative number of applications from any two residence zones to any two specific jobs will be equal to the ratio of mixed strategy probabilities scaled by the mean number of applications per worker. Thus, the probability of a successful application is unaffected by the number of applications made. The third condition similarly ensures that, if workers accept the first successful application, there is no spatial bias in the probability of acceptance due to the sequence of employers' decision-making. Finally, it is important to note that the characterization of the multiple job application process outlined here involves the somewhat heroic assumption that costs of applying for a job at j for a worker resident at location i are uncorrelated with T-j. If this is not the case, then job applications will show a distance deterrence effect independent of the spatial distribution of N. .. This is similar to the issue of randomness in spatial information flows. In defense of this position, it is unlikely that, in an urban area, the spatial distribution of TWO VERSIONS OF THE ICCM-1 MODEL / 111 costs relating to the acquisition of information or to job application would exhibit a simple distance decay pattern. Furthermore, in terms of modelling the labour market, this is a specification rather than a theoretical problem; if the information and job application cost surfaces were known, they could be taken into account in the function determining the probability elements of CHAPTER VI. ICCM-1 ALGORITHM A. Data Input i Origin zone identifier; i = 1 to m. j Destination zone identifier; j = 1 to n. B. Vector representing the number of workers in each origin zone of a given occupation / skill category. Ej Vector representing the number of jobs in each destination zone of a given occupation / skill category. r. . An m by n matrix of mean inter- and intra-zonal transport costs: inter-zonal costs estimated by morning peak-hour automobile travel time between zone centroids; intra-zonal costs estimated by interpolated mean travel time. Note: Allowable for L. B. to be <, =, or > L. E.. J J B. Initialize Endogenous Variables r = 1 Iteration counter set to- 1. s = 1 Loop counter set to 1. W . | r = 0 Vector of destination zone specific wages. 5 . | r = 0 Vector of destination zone specific wage adjusters. N.j | r = 0 Matrix of net wages gained by a worker resident in i employed in j. 112 ICCM-1 ALGORITHM / 113 V.j | ^ = 1 Matrix of competition adjustment factors. Q.j|^ = 0 • Matrix of iteration-specific allocation of workers to jobs. A. ,| = 0 Matrix of loop-specific allocation of workers to jobs. F. j r s = 73V Vector of workers not yet allocated to jobs. G. , = E. Vector of jobs not yet allocated to workers. j\r,s j P. .i = 0 Matrix of probabilities, where each element refers to the ij \r,s loop-specific probability a worker resident in i will apply for a job in j. H. . I = 0 Matrix representing the loop-specific number of applications of workers resident in i to jobs in j. C. Allocation Procedure The procedure to allocate workers to jobs utilizes two (nested) levels of iteration. In order to avoid ambiguity, the outer level will be referred to as an iteration, and the value attached to a variable within a particular iteration is indicated by the subscript r. The inner level will be referred to as a loop, and the value attached to a variable within a given loop is identified by the subscript s. Thus, for example, the element values in the matrix of application probabilities at any point in the computation is identified by the notation P. ., . The loop subscript (s) is omitted for variables whose values remain unchanged across loops, and the iteration subscript (r) is omitted for exogenous (data) variables. Note that the difference between ICCM-la and ICCM-lb lies solely in the calculation of competition adjustment factors (step 2). ICCM-1 ALGORITHM / 114 Begin Iteration Routine 1) Calculate the net wage distribution: a) N. ., = W.. - r. . ij\r J\r ij b) if N. ., < 0 ; then N. ., = 0 ij\r i j \ r 2) Calculate the competition adjustment factor: a) Model Version ICCM-la: V. ., = 1 b) Model Version ICCM-lb: V. ., = 1 / (T. . • W., ) Begin Loop Routine 3) Calculate application probabilities: a) P. ., = (TV. ., • V. ., • G., ) / j ; | v b) if the denominator = 0 : then P. ., = 0 4) Calculate the number of applicants from each origin zone to each destination zone: ICCM-1 ALGORITHM / 115 H. ., = P. .1 • F., 5) Determine number of successful applicants: a) if Z . H. .i < G.i ; then J *JK,s j |r,s VI r,s ij I r,s b) if Z. H. ., > G., ; then J ij\r,s j\r,s A. ., = H. ., • (G., / Z. H. ., ) i j | r , s V|r>s j|r,s y y k,s 6) Up-date iteration-specific allocation matrix: Q. ., = Q. ., + A. ., 7) Establish values of variables for next loop: a) Calculate number of unfilled jobs for each destination zone: G. , = G., - (Z. A. .. ) j\r,s+l j\r,s j ij\r,s b) Calculate number of unallocated workers for each origin zone: F., , = F . | - (I. A. ., ) c) Re-initialize allocation variables: A. ., = 0 H . ., , , = 0 P - -I MI = 0 8) Test for solution: if Z. G., , 7 = 0 ; or if Z. F. , = 0; i i\r,s + l then solution has been reached - i.e., either all jobs have been ICCM-1 ALGORITHM / 116 filled or all workers have been allocated to jobs. Exit iteration routine; go to solution routine: step 14. 9) Test for continuation of iteration: if Z . Z . A. ., = 0; then exit from iteration r, the wage levels defining iteration r are insufficient to clear the market. Exit loop routine: go to step 11. 10) Next loop within iteration; go to step 3. End Loop Routine 11) Set wage adjuster variable. The wage is increased for a destination zone if the number of applicants generated through the current r iteration to that zone were less than the number of jobs available: a) if G.i , , = 0 ; then 5.. , , = 0 J\r,s+1 J\r+1 b) if G.i ^ 1 > 0 ; then: J\r,s + 1 i) if {G.i , , / E\ > 0.1 ; then j\r,s+l f J\r+1 ii) if {G.i , , IE) < 0.1 ; then j\r,s + l f j\r+l J\r,s+1 j 12) Set variable values for next iteration: j\r+l> j\r J\r+1 ICCM-1 ALGORITHM / 117 b) r = r + 1 c) s = 1 d) Q. ., = 0 e) / / . , = B. i\r,s i f) G.i = E. J\r,s J 13) Next iteration; go to step 1. End Iteration Routine D. Solution Routine 14) Establish solution allocation of workers to jobs: Q'. • = Q. ., ij ij\r 15) Establish solution employment zone wage distribution: W. = W., J J\r 16) Establish solution residential zone location value distribution. This is calculated as the mean net wage gained by workers in each residential zone: R'. = {I. [(W. - r. .) • Q'. .]} I B. 17) Establish distribution of the local location value stability index. This ICCM-1 ALGORITHM / 118 index is the coefficient of variation of residential zone location values: U. = aDl. I R'. i R i i Establish the global location value stability index. This index is calculated as the weighted mean of L\ values: r = {Z. (L'. • B.)} I Z. B. i i t i i CHAPTER VII. TESTING OF THE ICCM-1 MODEL A. INTRODUCTION This chapter presents the results of test runs using the algorithm detailed in the predeeding chapter. The experiment has two purposes: a) to investigate the ability of the two ICCM-1 model versions to explain urban commuting patterns; and b) to compare ICCM-1 predictions with those derived from an entropy model. The data for the runs comes from the commuting pattern of the census metropolitan area (CMA) of Greater Vancouver, as established by the 1971 and 1981 Statistics Canada Censuses. The data has been utilized in a highly aggregated form as the urban area has been subdivided into only 15 zones, corresponding to the constituent municipalities of the CMA. The difficulties introduced through this level of aggregation make the test results less conclusive than would be desired, but they appear to be sufficient to indicate the main features of the output of the models. The entropy model used for comparison purposes is doubly constrained, i.e., for any zone, the sum of predicted trips originating and terminating in that zone are constrained to agree with the empirical totals (see Chapter II). This is accomplished with the use of origin zone- and destination zone-specific balancing factors, 0'. and D'.. The friction factor is defined as a negative exponential function of travel cost (F = e ^ T ^), and the B exponent is determined by constraining the predicted mean travel cost such that it equals the empirical mean travel cost. The algorithm used to calculate the O'., D'. and B parameters ^ J is a bi-proportional Furness procedure (Evans and Kirby, 1974). Thus, calibration 119 TESTING OF THE ICCM-1 MODEL / 120 of the entropy model requires knowledge of a parameter of the empirical trip distribution, while the ICCM-1 model versions are based solely on the map pattern of trip end-points. B. A SUMMARY OF THE ECONOMIC STRUCTURE OF VANCOUVER If a normative commuting model is to be of value, the relative performance of the model should not be correlated with urban economic structure. That is, urban labour market processes are deemed to be independent of the occupational or industrial distribution of sub-markets. In practice, however, the actual conditions in given urban labour markets will coincide more or less accurately to the assumptions underlying model logic. Thus, a full evaluation of model performance must be based on an understanding of the empirical data utilized. This section provides a brief description of the structure of the Vancouver economy in order to establish the validity of model assumptions. The Vancouver economy can be divided into four distinct components. The first is based on the city's geographical location as the major Canadian Pacific port. The traditional role of the Port of Vancouver as the terminus of the two Canadian trans-continental rail lines, and thus as the western trans-shipment point for exports of raw materials (primarily lumber, grain, coal, sulphur and potash), has been augmented in recent years by the increasing volume of Asian manufactured imports (notably automobiles, electronic equipment and machine tools) utilizing Vancouver's facilities as the western entry-point to the Canadian market. However, although tonnage through the port has steadily increased, technological change in loading and unloading procedures, specifically as a result of the TESTING OF THE ICCM-1 MODEL / 121 construction of efficient bulk and container facilities, has meant that the importance of the Port as a direct employer has diminished. In contrast, employment in firms involved in the commercial rather than physical aspects of Pacific trade has greatly increased. The prominence of Vancouver as a trans-shipment centre initiated the growth of the city, and thus established a sufficient demand base to allow economies of scale in the provision of services. As a result, the second component of Vancouver's economy, its role as the primate city of British Columbia, has steadily grown in importance as the resource extraction industries of the province (primarily lumber, mining and fishing) have expanded. Employment in the tertiary sector has rapidly increased, especially in the provision of financial, administrative and professional services not only to resource-based industries but to the provincial economy as a whole. Demand for specialized technical services derived from regional resource industries has enabled the development of an export-oriented engineering sector. Also, industries linked to Vancouver's position as the distribution centre of the region have also grown. It is important to note that Vancouver's role in providing services to the province includes only a small level of public services, as the provincial government is located in Victoria. Another aspect of the importance of the regional economy to the industrial base of Vancouver can be found in the structure of the manufacturing sector. Manufacturing industries are closely tied to the requirements of regional resource-based industries, both in terms of raw material processing (sawmills, veneer and plywood mills, fish processing) and supplying input requirements. TESTING OF THE ICCM-1 MODEL / 122 Writing in 1973, Steed notes: Much of the expansion and diversification into secondary manufacturing in Greater Vancouver over recent years has simply involved increasing integration with the provincial hinterland through the taking up of forward and backward linkages from this primary sector. (G.P.F. Steed, 1973; page 238) The thrust of this comment would still be applicable in 1981; however, by that date the expansion of manufacturing activity had vastly slowed. This was a result both of an influx of cheaper imported machinery, notably logging and mining equipment, the collapse of the local ship-building industry in the face of Japanese competition, and an increase in the level of resource exports being shipped in an unfinished or semi-finished state. The third component of Vancouver's economy is based on its attractive natural setting, the environmental benefits of a limited manufacturing base and, at least in relation to most other Canadian cities, its pleasant climate. The tourist and retirement industries have become major employers. The fourth component of the economy is based on residentiary services and small-scale manufacturing production aimed at the local market. The general prosperity and growth experienced by the province during the decade of the 1970's, coupled with the increasing number of two-worker households, led to heightened demand for local services. In addition, Greater Vancouver includes an area of fertile agricultural land (the delta of the Fraser River) used for market gardening. TESTING OF THE ICCM-1 MODEL / 123 The overall emphasis of the Vancouver economy can be summarized with the use of the 1981 Census statistics. Employment in manufacturing industries accounted for only 14.4% of all jobs, and employment in transportation industries made up 10.9%, while 33.4% of jobs were in service industries, 7.5% in financial, insurance and real estate, and and 19.4% were in wholesaling and retailing industries. Government accounted for 5.8% of employment. Thus, tertiary industries employed over two-thirds of the labour force. Figure VII-1 shows the distribution of employment by occupation for 1971 and 1981 (see below for detailed definitions of the occupational categories). Jobs in white-collar occupations increased from 64.6% of all jobs in 1971 to 72.2% in 1981. Employment growth was most rapid in managerial, professional and clerical categories, underlining the increasing importance of the service sector in shaping the structure of the labour market. Also, it is noteworthy that during this decade of extremely rapid economic growth, the number of jobs in traditional transportation occupations declined significantly in absolute terms. The primary assumption made by both the ICCM-1 model and the entropy model is that wages are set competitively at plant sites solely in response to demand and supply conditions; that is, co-operative wage bargaining across plants or industries is not accounted for. In the case of Vancouver, union activity in manufacturing, transportation and construction occupations and in the public sector traditionally have been strong. However, by 1981 jobs in these categories had declined in importance, and made up only one-quarter of the labour force. Furthermore, disaggregation of the data by occupation allows the separation of 1 2 4 F I G U R E V l l - 1 OCCUPATIONAL STRUCTURE OF GREATER VANCOUVER 1 9 7 1 1 9 8 1 CHANGE OCCUPATION EMPLOYMENT % EMPLOYMENT % 1 9 7 1 - 1 9 8 1 M a n a g e r i a I P r o f e s s i o n a I C I e r i c a I S a I e s S e r v i c e M a n u f a c t u r i ng T r a n s p o r t a t i o n C o n s t r u c t i on O t h e r T o t a I 2 0 , 3 8 0 5 . 1 5 2 , 2 7 6 1 3 . 1 8 1 , 7 1 4 2 0 . 4 5 1 , 9 7 8 1 3 . 0 5 1 , 9 8 5 1 3 . 0 5 2 , 9 1 9 1 3 . 2 3 1 , 0 8 1 7 . 8 2 8 , 3 2 5 7 . 1 2 9 , 4 4 0 7 . 4 4 0 0 , 0 9 8 1 0 0 . 0 6 4 , 0 2 9 1 0 . 1 1 0 4 , 1 3 0 1 6 . 5 1 4 1 , 3 4 2 2 2 . 4 6 7 , 9 4 2 1 0 . 8 7 8 , 2 3 3 1 2 . 4 7 4 , 3 2 9 1 1 . 8 2 4 , 6 6 4 3 . 9 3 5 , 9 0 2 5 . 7 4 1 , 6 9 5 6 . 6 6 3 2 , 2 6 6 1 0 0 . 0 4 3 , 6 4 9 2 1 4 . 2 5 1 , 8 5 4 9 9 . 2 5 9 , 6 2 8 7 3 . 0 1 5 , 9 6 4 3 0 . 7 2 6 , 2 4 8 5 0 . 5 2 1 , 4 1 0 4 0 . 5 - 6 , 4 1 7 - 2 0 . 7 7 , 5 7 7 2 6 . 8 1 2 , 2 5 5 4 1 . 6 2 3 2 , 1 6 8 5 8 . 0 TESTING OF THE ICCM-1 MODEL / 125 potentially problematic categories, except in the case of white-collar public sector jobs. The relatively low level of government activity in the Vancouver labour market minimizes this difficulty. The models further assume that information about job openings is available to all prospective employees, and employers allocate job openings to applicants at random; thus, internal labour markets are not accounted for. The degree to which this assumption is problematic in the case of Vancouver is difficult to determine: the fact that it is a regional centre of commercial activity rather than a national centre may mean that internal markets are not as complex as those in Toronto or Montreal. Finally, the ICCM-1 model assumes that the operation of the labour market does not include an equilibrating mechanism, while the entropy model makes such an assumption. The rapid growth of employment in Vancouver during the decade prior to the Census of 1981 suggests that the assumption of a labour market in equilibrium may be unwarranted. C. DATA Three data sets are utilized to perform the test runs on the distribution models. The first details the 1981 Vancouver CMA commuting pattern, aggregated by census sub-divisions. This data set is disaggregated by gender and by occupational categories, so that model runs can be made to predict the commuting pattern of specific labour sub-markets. The second is comparable data taken from the 1971 census; this material is available only in an aggregated form. The third set TESTING OF THE ICCM-1 MODEL / 126 relates to the travel cost matrix required to specify the spatial pattern of zones. 1. 1981 Vancouver Commuting Data The 1981 data set is taken from the 1981 Statistic Canada Census long-form demographic tape CSD81B30. The tape contains an expanded-to-population table, based on the 5% Census sample, of place of work by place of residence totals for geographical areas defined by census sub-divisions (CSD). The matrix is disaggregated into 22 occupational categories. Included in the table are workers with a stated usual place of work (as of June 1, 1981) and those for whom a work location specified only at the CSD level is available. Workers with no usual CSD place of work are excluded. Statistics Canada suppresses cell entries representing travel between CSD's with resident populations of less than 10,000. In the Vancouver CMA, this means that journey-to-work travel between and within Indian reservations, the University Endowment Lands (UEL) and Lion's Bay are excluded. These trips could be estimated from other sources; specifically from the census tract based journey-to-work tape. However, this tape is not comparable as it includes only those respondents providing work locations capable of being coded to the census tract level; for the CMA as a whole, these represent only 63% of the trips coded to the CSD level. As the repressed cells do not seriously affect the overall pattern of CMA commuting flows, it was decided to work directly with CSD information. The test results show that the lack of information regarding trips within the UEL affects the ability of the models to satisfactorily handle trips with end-points in this zone. TESTING OF THE ICCM-1 MODEL / 127 The total workforce employed and living in the CMA and included in this data set comes to 629,105. This represents approximately 97% of the overall CMA workforce, including those having no usual place of work. The 22 CSD's were aggregated into 15 zones based on municipalities. Figure VII-2a illustrates the CSD to municipality aggregation procedure, and figure VII-2b shows the spatial zoning system used. The major problem introduced by the high level of spatial aggregation is that the relative level of intra-zonal trips is high - for the entire workforce, 49% of commuting trips are within zones. Broughton (1981; page 8) suggests that a level of intra-zonal trips over 25% of the total affects the calibration of an entropy model, but it should be noted that he was able to meet this restriction in modelling Manchester commuting patterns by utilizing 371 zones. Snickars and Weibull (1983) investigated the properties of an entropy model based on a Stockholm data set consisting of 12 zones, in which over 54% of commuting trips were intra-zonal. For the purpose of comparing the ability of different types of models to explain commuting patterns, rather than calibrating a specific model for a planning application, the degree of error introduced by an aggregated data set should not be prohibitive. The 22 occupational categories were aggregated into 9 composite groupings, with an attempt to maintain a high degree of homogeneity. Statistics Canada suppresses information on very small flows by randomly rounding matrix values to the nearest 5; thus, entries with values of 0 or 5 could represent true data of anywhere between 0 and 10. This means that small entries are unreliable, and that the problem increases with occupation disaggregation. This difficulty led F I G U R E VI I - 2 a M U N I C I P A L AGGREGATION M U N I C I P A L I T Y • B u r n a b y C o q u i 11 am D e l t a New W e s t m i n s t e r N o r t h V a n c o u v e r ( C i t y ) N o r t h V a n c o u v e r ( D i s t r i c t ) P o r t C o q u i t I am P o r t Moody R i c h m o n d S u r r e y C i t y o f V a n c o u v e r W e s t V a n c o u v e r U n i v e r s i t y E n d o w m e n t L a n d s M a p l e R i d g e a n d P i t t M e a d o w s L a n g Iey CENSUS S U B D I V I S I O N 1 5 - 0 2 5 B u r n a b y 1 5 - 0 3 4 C o q u i t l a m 1 5 - 8 0 5 C o q u i t I am 1 (R) 1 5 - 8 0 4 C o q u i t l a m 2 (R) 1 5 - 0 1 1 D e l t a 1 5 - 8 0 2 T s a w a s s e n (R) 1 5 - 0 2 9 New W e s t m i n s t e r 1 5 - 0 5 1 N o r t h V a n C i t y 1 5 - 8 0 6 B u r r a r d I n l e t 3 (R) 1 5 - 8 0 7 M i s s i o n 1 (R) 1 5 - 0 4 6 N o r t h V a n D i s t r i c t 1 5 - 0 3 9 P o r t C o q u i t l a m 1 5 - 0 4 3 P o r t Moody 1 5 - 0 1 5 R i c h m o n d 1 5 - 0 0 4 S u r r e y 1 5 - 0 0 7 Whi f e R o c k 1 5 - 8 0 1 S e m i a h o o (R) 1 5 - 0 2 2 V a n c o u v e r 1 5 - 8 0 3 M u s q u e a m 3 (R) 1 5 - 0 5 5 W e s t V a n c o u v e r 1 5 - 8 0 8 C a p i l a n o 5 (R) 1 5 - 0 1 8 UEL 1 3 - 0 1 1 M a p l e R i d g e 1 3 - 0 1 8 P i t t M e a d o w s 1 1 - 0 2 5 C i t y o f L a n g Iey 1 1 - 0 1 9 L a n g l e y D i s t r i c t F IGURE VI I - 2 b GREATER VANCOUVER - M U N I C I P A L ZONES t o Z o n e C e n t r o i d s : . TESTING OF THE ICCM-1 MODEL / 130 to aggregated groupings that include somewhat disparate categories - the Professional occupational category (see below) illustrates this result. The categories are further disaggregated by gender, and this distinction was maintained in the model runs. The reason for this is that differences between male and female commuting behaviour has been noted in the literature (e.g., Cubukgil and Miller (1982)), and occupational segregation has been identified as a potential explanatory factor (Hanson and Johnston, 1985; page 216). The latter paper calls for more research into this area: In sum, we know that home-work travel distances and travel times differ for men and women, but the reasons for this consistent finding remain murky. In part, this is because it is not yet clear how work-trip distances differ among income or occupational categories for working women. In particular, it is difficult to understand and evaluate the various reasons proposed for the male-female differences in work-trip length in the absence of information on the spatial distribution of different types of employment opportunities. To address these issues, we need disaggregated data on journey-to-work patterns for men and women as well as data on the spatial distribution of jobs. (S. Hanson and I. Johnston, 1985; page 199) By using such disaggregated data in the model runs, it is hoped that one result of the tests will be to shed light on differences between male and female urban labour markets. The 9 occupational categories are constructed as follows: 1. Managerial a) managerial, administrative and related occupations TESTING OF THE ICCM-1 MODEL / 131 2. Professional b) natural sciences, engineering and mathematics c) social sciences and related fields d) occupations in religion e) teaching and related occupations f) occupations in medicine and health g) artistic, literary, recreational and related 3. Clerical h) clerical and related occupations 4. Sales i) sales occupations 5. Service j) service occupations 6. Manufacturing k) processing occupations 1) machining and related occupations m) production, fabrication, assembly and repair 7. Construction n) construction trades occupations 8. Transportation o) transportation equipment operation occupations 9. Other p) materials handling and related occupations q) other crafts and equipment operations occ. r) farming, horticulture, animal husbandry TESTING OF THE ICCM-1 MODEL / 132 s) fishing, hunting, trapping and related occ. t) forestry and logging occupations u) mining and quarrying v) occupations not elsewhere classified The three models were run on data sets based on all occupation categories except Construction, Other, Female Manufacturing and Female Transportation. It was felt that the interpretation of usual place of work by construction workers likely would be inconsistent with the concept of a fixed place of employment, and the Other category clearly includes a very heterogeneous mix of occupations. The Female Manufacturing and Transportation categories contained an unduly large number of very small flows, so that the overall level of reliability was not acceptable. The models were also run on total male employment, total female employment and total aggregate (male and female) employment, in order to test the effect of occupational disaggregation on model performance. 2. 1971 Vancouver Commuting Data The 1971 Vancouver CMA commuting data set consists of an aggregated municipality-based journey-to-work table derived from the 1971 Census. No data are available representing disaggregated-by-occupation commuting flows for this year. The data set used here was provided by the Greater Vancouver Regional District (GVRD) transportation committee. As the information explaining data transformation methodology utilized by GVRD staff has been lost, it is not known whether the spatial aggregation procedure used to produce the municipality-based figures was similar to that used for the 1981 data; TESTING OF THE ICCM-1 MODEL / 133 specifically, the handling of Indian Reservations and other small CSD's is unknown. The data set does contain entries for workers with no usual place of work; these total 8.4% of all workers and, as the comparable figure for 1981 is 3%, may be overstated. Thus, the difference may be due to changes in definition or coding procedures as well as temporal changes in job location stability. Only workers for whom a usual (municipal) place of work is coded are utilized. The 1971 data set identifies the commuting pattern of 391,095 workers of a total workforce of 427,860. The workforce growth between 1971 and 1981 was 47.0%. The significant growth the city enjoyed during this decade was not evenly distributed across the CMA: figure VII-3 illustrates the spatial changes in residential and employment map patterns that took place. The models were tested on their ability to predict 1981 commuting flows utilizing only the full 1971 data set and the 1981 aggregated municipal origin / destination marginals. 3. The Travel Cost Matrix Both the ICCM-1 and the entropy models require an exogenous matrix defining the spatial relationship between origin and destination zones in terms of the cost of travel between zones. As each entry in the matrix reduces information concerning the spatial relation of two zones to a single number, the procedure used to establish the relationship is critical to model performance. The procedure has two aspects: first, the metric defining travel cost is established; second, the means of measuring travel cost between zones in units of the metric is determined. 1 3 4 F I G U R E V I 1 - 3 V A N C O U V E R MAP P A T T E R N A D J U S T M E N T S : 1 9 7 1 - 1 9 8 1 M u n i c i p a l i t y P l a c e o f R e s i d e n c e P l a c e o f E m p l o y m e n t 1 9 7 1 % 1 9 8 1 % 1 9 7 1 1 9 8 1 % 1 B u r n a b y 5 7 1 8 5 1 3 . 3 7 0 8 5 5 1 1 . ,3 4 0 0 3 0 1 0 . ,1 6 9 8 8 0 1 1 . ,1 2 C o q u i 11 am 1 8 6 7 0 4. 3 3 0 9 4 0 4. ,9 9 1 4 0 2, ,7 1 6 8 7 0 2. ,7 3 D e l t a 1 6 3 4 5 3. 8 3 4 1 5 0 5. .4 7 2 9 5 1, ,8 1 7 9 0 5 2. .8 4 New W e s t . 1 8 0 1 5 4. 2 1 8 8 8 5 3. .0 2 2 3 7 0 5, .6 2 3 0 7 5 3. ,7 5 N . • V a n C i t y 1 4 6 8 0 3. 4 1 9 8 8 0 3. .2 1 4 0 8 5 3, .5 2 0 7 8 0 3. .3 6 N. V a n D i s f . 2 3 4 5 5 5. 5 3 5 7 3 0 5. ,7 6 3 7 0 1. ,6 1 5 7 6 0 2. ,5 7 P o r t C o q u . 6 8 8 5 1. 6 1 3 2 3 5 2. .1 3 4 8 0 ,9 8 4 7 5 1. ,3 8 P o r t M o o d y 3 8 9 5 9 7 5 3 0 1. ,2 2 1 1 0 .5 4 0 5 0 ,6 9 R i c h m o n d 2 4 9 7 0 5. 8 4 9 9 3 5 7. .9 2 1 7 5 0 5. ,5 5 6 2 4 0 8. .9 1 0 S u r r e y 3 7 5 7 0 8. 7 7 1 6 2 5 1 1 . ,4 2 1 0 6 5 5, .3 5 3 2 3 5 8. .4 1 1 V a n c o u v e r 1 8 6 5 8 5 4 3 . 4 2 1 3 3 3 0 3 3 . .9 2 2 8 0 2 5 5 7 , .5 2 9 4 1 7 5 4 6 . ,5 1 2 W e s t V a n 1 4 9 1 0 3. 5 1 7 9 6 0 2, .9 7 5 6 0 1. .9 1 2 3 8 5 2. ,0 1 3 U E L 1 3 0 5 3 1 6 3 5 .3 3 4 2 0 .8 6 5 3 0 1. .0 1 4 Map 1e R i d g e 9 8 1 5 2. 3 1 7 5 6 5 2. .8 5 6 0 5 1. ,4 1 1 4 6 0 1. ,8 1 5 L a n g 1 e y 9 7 1 0 2. 3 2 7 3 1 0 4. .3 5 8 9 0 1. ,4 2 1 1 5 5 3. ,4 TESTING OF THE ICCM-1 MODEL / 135 a. The Travel Cost Metric The metric determining the spatial relationship of origin / destination zones is, theoretically, the disutility incurred in travelling between zones. It is often the case that Euclidean distance is the only measure that is readily available, and so is adopted as an estimate of travel cost. For example, Griffith and Jones (1980), in an investigation of commuting patterns in 24 Canadian cities, and Broughton (1981), in research based on commuting in Manchester, use straight-line distance to estimate entropy models. Broughton justifies this approach by quoting Mogridge's (1979) call for simplicity in selecting a travel cost measure. He also argues that for the purpose of investigating the properties of a model, a deliberately naive approach is favoured. However, he goes on to state: This approach assumes that the transportation network is dense, with the cost of a journey depending only on the length. There are no major barriers to travel in the Manchester conurbation, such as the large rivers present in certain comparable areas, yet the assumptions are not wholly valid. In particular, the radial nature of many major transport routes affects the cost of travel for those commuting to central jobs; this must have an effect on the calibration process. (J. Broughton, 1981; page 9) In the case of Vancouver, the use of straight-line distance to estimate cost is even more inappropriate. The urban area is dissected by an inlet and a river, so that the road network includes a number of bridge and tunnel crossings that funnel traffic on a limited number of routes. On-the-road distance would eliminate the problem of irregular routing, but would not handle the costs associated with congestion through bottlenecks. Congestion is most severe during TESTING OF THE ICCM-1 MODEL / 136 morning and evening commuting periods, and so would be an important factor in determining commuting costs. This raises the issue of whether inclusion of congestion costs in the travel cost metric is tautological, in that congestion is a result of the commuting pattern which the model is predicting. Ideally, a model should begin by assuming free-flow conditions, and iterate commuting flow predictions until the input travel cost matrix converges with a cost matrix generated through the use of an assignment model accepting as input distribution model output. For present purposes, the accuracy of the cost matrix is critical, especially in terms of the logic of the ICCM-1 procedure; thus, inclusion of congestion effects in the travel cost metric is desirable. A common approach in translating travel disutility into monetary cost is to estimate the relative importance of various factors through the use of a generalized travel cost function (see Hutchinson (1974) for a discussion of the application of such a function in the context of mode-split modelling). The initial stage is to define a measure of travel disutility (T) as an additive function of the disutility incurred through various measurable aspects of the commuting process. For example, Wilson (1969) specifies the following cost factors in determining r. .: a) in-vehicle travel time between i and j; b) excess travel time between i and j; c) distance between i and j. Other researchers have added factors relating to parking costs, fixed and variable monetary costs, a measure of mode comfort, etc. (see Daniels and Warnes, 1984). By definition, the function is mode-specific, and customarily can be calibrated only in terms of the relative costs between modes; that is, the revealed preference of commuters in selecting among available travel modes over trips of various lengths, times, etc. may be TESTING OF THE ICCM-1 MODEL / 137 used to estimate the relationship between individual cost elements. An important point is that generalized cost functions usually are calibrated as untransformed linear equations (Hutchinson, 1974; pages 61-63). This suggests that a linear relation between time spent in commuting (both in-vehicle and excess time) and monetary cost is a reasonable assumption. The spatial distribution of the incidence of excess time costs is difficult to estimate - aside from time associated with parking in central areas, it is likely that these costs are minimal for drivers and evenly distributed (except at a localized scale) for bus riders. Given these considerations, the adoption of a linear function of travel time to represent travel cost is a practical and reasonable approach. The availability of multiple modes between an origin and a destination zone suggests that any single measure of travel cost may be inadequate. The usual method of handling the mode-split of flow assignments is sequential: flow distribution is predicted by one model (usually of the gravity or entropy type), and the predicted flow is subsequently assigned to various modes by a separate procedure. If the differences between the cost matrices of travelling by two competing modes are not explainable by a scale factor (i.e., if the relative cost matrices are not essentially the same), the distribution model will be poorly specified, regardless of the choice of disutility metric. This situation can occur in urban areas in which the availability of alternative modes is not uniform. For example, in a city served by radial subway and suburban train systems, a disutility matrix based on automobile costs will not represent the costs associated with travel by the other modes. This problem can be handled to some extent by TESTING OF THE ICCM-1 MODEL / 138 defining r in terms of a composite function; in general, however, a simultaneous distribution / mode-split approach is required. In the case of the tests run on data from the Vancouver CMA, this problem is minimized by the lack of rapid transit and intra-urban train services then available, t Commuting travel was accomplished via either walking, bus or car. The road network is sufficient to describe routings for all three modes, and, in the case of bus and car travel, it is reasonable to assume that in-vehicle travel time for the bus mode could be estimated by a linear function of in-vehicle time for the car mode. As a result, the Vancouver data used in the test runs is particularly well-suited to the use of peak-hour travel time as the estimate of travel cost. b. Measuring Travel Time The travel time matrix is based on a link-to-link morning peak-hour travel time coding of the GVRD road network compiled by the GVRD planning staff in 1977. Travel times on the network were established through a combination of field samples and engineering estimates. The network is coded with directionality, so that the link times are not symmetrical. As there had been no major adjustments in the road network between 1971 and 1981, the accuracy of this data base to both 1971 and 1981 conditions should be good. The temporal discrepancy between the commuting time and commuting flow data sets will create some inaccuracy in the estimation of congestion effects; this was t This situation changed with the opening of the Skytrain rapid transit line in 1986. TESTING OF THE ICCM-1 MODEL / 139 deemed to be unavoidable. A second travel time data set based on average between-zone morning peak-hour times in 1981 for 310 traffic zones across the CMA was made available by the GVRD Development Services Department. This matrix was produced by a modelling procedure which initially inputs free-flow travel times as the exogenous cost matrix, and then iterates to a solution by matching an entropy-based distribution to sample screen-line traffic flows. The solution is reached when the adjusted travel time matrix predicts assigned flows that converge with empirical counts. Upon inspection, it was found that the synthesized travel times contained a number of anomalies and internal contradictions introduced by the generative process. Thus, the 1977 link-based data set appeared to be more reliable as an estimate of actual 1981 peak-hour automobile travel times; and was used in the test runs. Mean inter-zonal travel times were estimated by selecting points in each zone as a zone centroid, and linking the centroids to the nearest node in the road network. Shortest-time paths between centroids were determined with the use of a program developed by K.G. Denike in the Geography Department of UBC. Intra-zonal times were estimated through interpolation from the rest of the matrix. Figure VII-4 presents the travel time matrix used in the model runs. The small number of zones created severe difficulties in selecting zone centroids. The models effectively treat all flows originating in a zone to emanate from a point; similarly for trip destinations. If the zones are sufficiently small, variations in travel time between zones are handled effectively by the mean travel times; this is especially the case if the distribution of travel times between FIGURE V I I - 4 : TRAVEL TIME MATRIX 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 7 . 17 . 29. 10. 18 . 16 . 20. 15. 21 . 16 . 26 . 22. 33 . 40 . 34 2 22 . 7 . 43 . 20. 35 . 33 . 9 . 9 . 42 . 17 . 4 1 . 39 . 53 . 3 1 . 30 3 38 . 47 . 10. 33 . 49 . 47 . 50. 45 . 24 . 27 . 46 . 53 . 48 . 66 . 32 4 10 . 17 . 29 . 5 . 26 . 24 . 19 . 15 . 23 . 7 . 34 . 30. 4 1 . 39 . 32 5 2 1 . 34 . 43 . 28 . 3 . 4 . 37 . 31 . 35 . 33 . 22 . 6 . 33 . 55 . 48 6 19 . 33 . 42 . 26 . 5 . 4 . 35 . 29 . 33 . 3 1 . 25 . 8 . 35 . 52 . 45 7 23 . 7 . 46 . 20. 38 . 36 . 5 . 15 . 43 . 18 . 45 . 4 1 . 53 . 25 . 36 8 25 . 6 . 49 . 22 . 35 . 33 . 14 . 4 . 44 . 22 . 42 . 39 . 53 . 33 . 33 9 2 1 . 38 . 16 . 22 . 3 1 . 29 . 4 1 . 35 . 9 . 29 . 28 . 35 . 28 . 6 1 . 4 1 10 19 . 19 . 25. 10. 35 . 33. 22 . 20. 32 . 9 . 43 . 39 . 50. 4 1 . 24 1 1 17 . 30. 28 . 23 . 14 . 16 . 33 . 27 . 18 . • 29 . 1 1 . 14 . 18 . 5 1 . 44 12 2G . 40. 45 . 33 . 8 . 9 . 42 . 36. 35 . 38 . 22 . 7 . 32 . 59 . 52 13 28 . 42 . 3 1 . 35 . 27 . 29 . 44 . 39 . 20. 40. 20. 28 . 10. 62 . 55 14 45 . 32 . 65 . 42 . 58 . 56 . 24 . 34 . 66 . 39 . 66 . 62 . 74 . 1 2 . 53 15 40. 32 . 31 . 35 . 53 . 51 . 37 . 36 . 50. 24 . 6 1 . 57 . 69 . 55 . 1 1 TESTING OF THE ICCM-1 MODEL / 141 potential origins / destinations of two zones is normal. Generally, if the spatial pattern of origins / destinations in a zone are not clustered around the centroid, the distribution of travel times is less likely to be normal, and the travel time matrix will introduce bias in the representation of spatial structure. This is a common occurrence when large zones are employed, and, in the present case, the distribution of trip origins / destinations in one zone is distinctly bimodal (Delta, zone 3). This difficulty cannot be eliminated without disaggregating the zones. However, a high level of zonal aggregation allows for visual comparison of predicted and empirical patterns. This benefit was deemed sufficiently important for the present purpose of an illustrative test procedure to justify the use of aggregated zones. D. RESULTS OF MODEL RUNS ON 1981 DATA The two ICCM-1 model versions and the entropy model were run on 16 data sets, representing various levels of disaggregation. The overall results show that the ICCM-la model was not able to replicate the empirical flows adequately, while the ICCM-lb model and the entropy model gave acceptable results consistently. The goodness-of-fit of model predictions was estimated with the use of a series of measures - these will be described in the following section, and the results then will be presented in detail. 1. Measures of Fit The measurement of the goodness-of-fit of a model's trip predictions to the empirical pattern is a difficult procedure for two reasons. The first is that no single measure will reflect all aspects of a comparison between two distributions -TESTING OF THE ICCM-1 MODEL / 142 by definition, test statistics must utilize information selectively in order to summarize. Noting this problem, Baxter advocates the use of a range of measures to examine model performance: Openshaw and Connolly (1977; page 1067) observe that there is "an absence of any universally acceptable measure of Fit or test of model performance". There is, in fact, no reason to expect that there should be a single, preferable measure of fit. Measures that are essentially different, and also sensible, reflect on different aspects of a model's performance, and the use of several different measures for investigating how well a model performs is often desirable. (M. Baxter, 1983; page 49) As the various measures used may not be in agreement as to the level of fit, conclusions about model performance may be fragmentary or highly subjective. Three elements are subsumed in the concept of fit estimation: a) statistical inference - the probabilistic relationship between empirical and predicted patterns; b) predictive value - the relative accuracy of overall model performance: and c) goodness-of-fit - the mean relative accuracy of individual flow predictions. For the present experiment, three measures were selected to investigate these aspects of 2 2 fit: the x value; the R value; and the log-likelihood ratio test statistic ( i / / ) . The second difficulty relates to the aspatial nature of statistical measures. Each entry in the trip matrix is viewed as an independent element, so that the two-dimensional aspect of the flow pattern does not enter the analysis. This can be handled to some extent by aggregating zones to investigate the existence of spatial bias in the distribution of the fit across the matrix; in the present case, TESTING OF THE ICCM-1 MODEL / 143 the high initial level of aggregation does not allow for this procedure, and visual inspection of the predicted and empirical flows is the only usable approach. 2 a. The x Statistic 2 The x statistic used in the analysis is calculated in the following manner. Let T. . represent the empirical commuting flow from origin i to destination j, and T'. . represent the model prediction. Then, < 4 1 ) "2 " hj «*y - V 2 ' V 2 A difficulty with the x statistic is that it is sensitive to small flows, and some means must be used to handle zero empirical flows. Baxter (1983) recommends the use of an alternative approach, which has the predicted flows as the denominator. This procedure works for an entropy model, in that all cells will receive a positive flow; however, it introduces a further bias as it will reduce 2 the overall x statistic of model predictions that consistently over-allocate small flows and underallocate large flows. This situation is typical of entropy models, while the reverse is true of the ICCM approach. The ICCM model utilizes the concept of a reservation wage to construct labour-shed boundaries; as a result, many cells will be allocated a flow of 0. Furthermore, from the point of view of statistical inference, the empirical pattern should be viewed as the "expected" result of the labour market process in order to have a consistent means of 2 comparing the ICCM-1 and entropy predictions. In order to calculate the x value for the entire matrix, cells for which T = 0 were calculated by setting T = 1. Figure VII-5 presents an example of the effect of directional bias in prediction 1 4 4 F I G U R E V I I - 5 G O O D N E S S - O F - F I T MEASURES a n d D I R E C T I O N OF P R E D I C T E D ERROR E x a m p I e : Z o n e 1 Z o n e 2 T o t a l Emp i r i c a I P r e d i c t i o n A P r e d i c t i o n B 1 3 5 2 0 1 5 0 5 1 2 0 3 5 1 5 5 1 5 5 1 5 5 C o m p a r i s o n o f T e s t S t a t i s t i c V a l u e s : T e s t S t a t i s t i c P r e d i c t i o n A P r e d i c t i o n B 9 2 . 3 9 2 . 3 L i k e I i h o o d R a t i o ( B a s e d on E m p i r i c a l F l o w s ) L i k e I i h o o d R a t i o ( B a s e d on P r e d i c t e d F l o w s ) C h i - S q u a r e d V a l u e ( B a s e d on E m p i r i c a l F l o w s ) C h i - S q u a r e d V a l u e ( B a s e d on P r e d i c t e d F l o w s ) 1 7 . 7 2 7 . 0 1 2 . 9 4 6 . 5 1 0 . 9 9 . 4 1 2 . 9 1.3 TESTING OF THE ICCM-1 MODEL / 145 on various measures of model fit. Prediction A corresponds to the bias typical of 2 an ICCM model, prediction B typical of an entropy model. The x statistic calculated with empirical values as denominator weights is insensitive to the direction of bias, while using predicted values in the denominator suggests that the fit of prediction B is much superior to that of prediction A. If the logical comparison base is the empirical flow, then the first method is preferable; this is the position adopted here. The imbalanced nature of the flows in the empirical matrix and the relative 2 unreliability of extremely small- values exacerbates the problem of the x statistic's sensitivity to small values. This problem is not simply a result of the spatial aggregation nature of the data. Employment concentration due to agglomeration economies causes the map pattern of destinations typically to be highly clustered; thus, regardless of the zonal scale used, the trip matrix will be characterized by a small number of cells with large flows and a large number of cells with small flows. The imbalanced nature of urban commuting matrices is one reason why inferential tests are extremely stringent in testing the hypotheses concerning model predictions (Snickars and Weibull, 1977; page 154). Baxter states that: The statistical significance of observed values can be determined by reference to tables of chi-squared. It may be remarked that often such a procedure will lead to formal rejection of the hypothesis that the model describes the data - that is the model is misspecified in some way. In such circumstances the substantive as opposed to statistical significance of the model needs to be considered, and a model will often be acceptable despite a statistical lack of fit. (M. Baxter, 1983; TESTING OF THE ICCM-1 MODEL / 146 page 50) 2 b. The R Measure 2 The R statistic, or coefficient of determination, measures the percentage of variance explained by the model prediction as compared to a random distribution, and is a method of establishing the overall predictive value of the model. The approach used by a number of researchers (for example, Broughton (1981) and Griffith and Jones (1979)), is to partition the error variance via a linear regression approach, so that the variance explained by a random distribution is the sum of squared differences between empirical flows and the mean flow. This approach is appropriate for an unconstrained model, as the flow allocated to any cell is constrained only by the total number of trips in the system. However, doubly-constrained models must predict trip flows such that the sum of trips emanating from and arriving in any zone match the empirical data. Thus, if the marginal flows are not uniformly distributed (and, as noted above, this will rarely be the case in urban commuting data sets), the percentage of error variance explained by the constraints will be a significant portion of total explained variance. This may explain why the models tested in the two studies 2 mentioned above showed high R values, even though the measures of travel cost used were very poorly specified. An alternative approach, and the one adopted here, is to partition error variance through an analysis of variance technique. Variance explained by model prediction is compared to that explained by a constrained random distribution, so that the effect of the distribution of marginal flow magnitudes is eliminated: TESTING OF THE ICCM-1 MODEL / 147 (42) R 100 2 The R value measures absolute rather than relative error; while it is the most effective means of estimating overall accuracy of flow prediction, it is not 2 effective in measuring relative error. For this purpose, the ^ and x statistics are more useful. c. The Log-Likelihood Ratio Test Statistic The \jj statistic is a global measure of the relative fit of predicted to empirical flows, and is calculated as follows: Baxter (1983; page 49) defines this measure with the T' and T values reversed, in a manner similar to that discussed above in reference to the calculation of 2 the x statistic. The rationale for this procedure appears to be that he interprets T' as a theoretical value, and \jj then measures the information gain contained in the empirical distribution (see Snickars and Weibull (1977) for a full derivation of this statistic). However, figure VII-5 shows that this approach is extremely sensitive to under-prediction of small values. It may be noted here that the entropy approach minimizes this value, subject to the constraint set; this is one reason that the entropy model tends to over-predict small flows. The formulation of the \jj statistic used here measures the magnitude of the lack of information in the predicted trip matrix as compared to the empirical matrix - as such, it indicates the goodness-of-fit of the predicted to the actual flows, rather than that of the actual to the predicted flows. (43) </ = 2 Z. . {TV . log (T\ . / T. .)} TESTING OF THE ICCM-1 MODEL / 148 From figure VTI-5 it can be seen that the \p statistic used is still sensitive to the directional bias of error in estimating small flows. Thus, this measure can be 2 applied in conjunction with the R statistic to investigate the type of errors present in a comparison of predicted flow distributions. For example, if two 2 predicted distributions have similar R values, a lower \p value for one distribution would indicate a relatively better fit on small flows and a relatively poorer fit on large flows. In order to calculate the \jj statistic over the entire matrix, cells with no empirical flow were adjusted by setting T = 1, in the same manner as for 2 calculation of the x statistic. 2. Statistical Results: Disaggregated Data Sets Commuting flow predictions were obtained by running the three models on 12 disaggregated (by occupation and gender) and 3 aggregated data sets using the 1981 data. Figures VII-6a, -6b, and -6c present the statistical results of the disaggregated runs, and include an analysis of the constrained random distribution, obtained by distributing trips solely in accordance with zonal origin / destination magnitude, for comparison purposes. The fi parameter values fitted to each data set by the entropy model are also listed, as well as the empirical and predicted mean travel times. 2 In analyzing the results, note that the x and \p values are sensitive to sample size, so that they are comparable only within, not between, data sets. The improvement in over-all fit provided by the models, in relation to that derived F I G U R E V l l - 6 a 1 4 9 MODEL R E S U L T S : STAT 1 ST 1CAL A N A L Y S I S , 1 9 8 1 DATA M a n u f a c t u r i n g ( M a l e ) C o n s t r a i ned I C C M - 1 a I C C M - 1 b E n t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 9 . 8 5 1 9 . 8 5 1 9 . 8 5 1 9 . 8 5 P r e d . Mean T r a v e l T i m e 2 7 . 4 1 1 8 . 0 0 1 7 . 1 9 1 9 . 8 1 R - S q u a r e d (%) 0 7 1 . 0 5 7 6 . 9 3 8 6 . 3 1 Ch i - S q u a r e 7 8 1 4 8 2 2 8 4 4 1 4 6 5 7 1 0 1 4 3 F R a t i o - 3 . 4 2 5 . 3 3 7 . 7 0 L o g - L i k e l i h o o d R a t i o 3 9 6 4 2 2 4 7 6 7 1 6 4 1 1 7 7 5 7 Re 1 a t i ve L i ke1 i h o o d 0 . 3 7 5 . 5 8 6 . 8 0 4 B e t a P a r a m e t e r ( E n t r o p y ) ( . 0 7 6 ) M a n a g e r i a l ( M a l e ) C o n s t r a i n e d I C C M - 1 a I C C M - l b En t r o p y R a n d o m E m p . M e a n T r a v e l T i m e 2 0 . 6 1 2 0 . 6 1 2 0 . 6 1 2 0 . 6 1 P r e d . Mean T r a v e l T i m e 2 7 . 8 7 1 9 . 2 4 1 8 . 9 0 2 0 . 5 8 R - S q u a r e d (%) 0 5 1 . 6 1 8 8 . 0 0 9 1 . 0 1 Ch i - S q u a r e 6 8 8 8 3 2 1 2 1 2 1 1 7 7 9 7 5 6 5 F R a t i o - 3 . 2 5 5 . 8 5 9 . 1 1 L o g - L i k e l i h o o d R a t i o 3 0 5 9 6 2 0 4 0 1 1 0 3 1 0 5 2 4 5 Re 1 a t i ve L i ke1 i h o o d 0 . 3 3 3 . 6 6 3 . 8 2 9 B e t a P a r a m e t e r ( E n t r o p y ) ( . 0 8 5 ) M a n a g e r i a l ( F e m a l e ) C o n s t r a i n e d I C C M - l a 1 C C M - l b En t r o p y R a n d o m E m p . M e a n T r a v e l T i m e 1 7 . 3 2 1 7 . 3 2 1 7 . 3 2 1 7 . 3 2 P r e d . Mean T r a v e l T i m e 2 6 . 0 0 1 6 . 8 1 1 6 . 2 4 1 7 . 3 4 R - S q u a r e d (%) 0 7 2 . 8 1 9 2 . 4 3 9 3 . 2 4 Ch i - S q u a r e 7 9 0 4 0 4 1 6 9 7 1 7 0 5 7 1 8 4 7 1 F R a t i o - 1 . 9 0 4 . 6 3 4 . 2 8 L o g - L i k e l i h o o d R a t i o 1 8 9 4 5 1 0 9 0 2 4 8 8 1 3 8 7 5 Re 1 a t i ve L i ke1 i h o o d 0 . 4 2 5 . 7 4 2 . 8 0 0 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 1 5 ) T r a n s p o r t a t i o n ( M a l e ) C o n s t r a i ned I C C M - l a 1 C C M - l b E n t r o p y R a n d o m E m p . Mean T r a v e l T i m e 2 0 . 2 6 2 0 . 2 6 2 0 . 2 6 2 0 . 2 6 P r e d . Mean T r a v e l T i m e 2 7 . 4 9 1 8 . 6 5 1 8 . 2 7 2 0 . 2 9 R - S q u a r e d (%) 0 5 6 . 6 4 8 7 . 2 2 8 8 . 5 8 Ch i - S q u a r e 3 1 8 3 7 1 3 3 3 3 5 7 5 6 6 5 5 6 F R a t i o - 2 . 3 9 5 . 5 3 4 . 8 6 L o g - L i k e l i h o o d R a t i o 1 5 7 6 7 1 2 2 7 9 6 3 5 9 3 1 2 9 Re 1 a t i ve L i ke1 i h o o d 0 . 2 2 1 . 5 9 7 . 8 0 2 B e t a P a r a m e t e r ( E n t r o p y ) ( . 0 8 1 ) F I G U R E V 1 1 - 6 D 1 5 0 MODEL R E S U L T S : S T A T I S T I C A L A N A L Y S I S , 1 9 8 1 DATA S a l e s ( M a l e ) C o n s t r a i n e d I C C M - l a I C C M - l b E n t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 8 . 8 2 1 8 . 8 2 1 8 . 8 2 1 8 . 8 2 P r e d . Mean T r a v e l T i m e 2 7 . 0 1 1 8 . 6 8 1 8 . 0 5 1 8 . 8 9 R - S q u a r e d (%) 0 6 9 . 1 4 9 2 . 6 8 9 2 . 2 2 Ch i - S q u a r e 6 6 3 4 3 1 9 9 6 0 8 7 1 1 9 2 8 5 F R a t i o - 3 . 2 4 7 . 6 2 7 . 1 5 L o g - L i k e l i h o o d R a t i o 3 3 6 2 8 1 7 5 5 7 8 2 3 5 6 0 8 9 Re 1 a t i ve L i k e 1 i h o o d 0 . 4 7 8 . 7 5 5 . 8 1 9 B e t a P a r a m e t e r ( E n t r o p y ) ( . 0 9 7 ) S a l e s ( F e m a l e ) C o n s t r a i n e d 1 C C M - l a I C C M - l b E n t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 4 . 8 7 1 4 . 8 7 1 4 . 8 7 1 4 . 8 7 P r e d . Mean T r a v e l T i m e 2 6 . 8 1 1 5 . 4 1 1 4 . 5 4 1 4 . 9 5 R - S q u a r e d (%) 0 7 2 . 2 4 9 1 . 3 6 9 3 . 5 6 Ch i - S q u a r e 2 5 7 5 1 7 5 7 1 9 4 1 8 5 0 0 2 1 5 5 6 F R a t i o - 4 . 5 0 1 3 . 9 2 1 1 . 9 5 L o g - L i k e l i h o o d R a t i o 5 5 2 5 7 2 4 1 4 7 1 1 7 7 9 8 7 0 9 R e l a t i v e L i k e l i h o o d 0 . 5 6 3 . 7 8 7 . 8 4 2 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 4 7 ) C 1 e r i c a 1 (Ma 1e) C o n s t r a i n e d I C C M - l a 1 C C M - l b E n t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 9 . 0 2 1 9 . 0 2 1 9 . 0 2 1 9 . 0 2 P r e d . Mean T r a v e l T i m e 2 5 . 0 2 1 8 . 1 9 1 8 . 0 4 1 9 . 0 3 R - S q u a r e d (%) 0 7 3 . 3 4 9 0 . 72 9 2 . 4 3 Ch i - S q u a r e 2 8 1 1 5 1 3 6 5 6 1 1 4 2 2 7 1 2 8 F R a t i o - 2 . 0 6 2 . 4 6 3 . 9 4 L o g - L i k e l i h o o d R a t i o 1 6 6 7 2 1 1 1 8 9 5 5 2 6 3 7 8 5 Re 1 a t i ve L i k e 1 i h o o d 0 . 3 2 9 . 6 6 9 . 7 7 3 B e t a P a r a m e t e r ( E n t r o p y ) ( . 0 8 5 ) C l e r i c a l ( F e m a 1 e ) C o n s t r a i n e d I C C M - l a I C C M - l b En t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 7 . 1 1 1 7 . 1 1 1 7 . 1 1 1 7 . 1 1 P r e d . M e a n T r a v e l T i m e 2 6 . 1 5 1 8 . 4 1 1 8 . 0 0 1 7 . 0 9 R - S q u a r e d (%) 0 7 0 . 6 1 8 9 . 6 6 9 2 . 4 4 Ch i - S q u a r e 4 6 8 8 4 1 8 6 5 6 4 1 0 8 3 0 0 3 4 4 7 5 F R a t i o - 5 . 4 2 4 . 3 3 1 3 . 6 0 L o g - L i k e l i h o o d R a t i o 1 4 3 2 8 3 6 1 1 7 9 3 1 8 7 0 2 2 9 4 7 Re 1 a t i ve L i k e 1 i h o o d 0 . 5 7 3 . 7 7 8 . 8 4 0 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 3 1 ) 1 5 1 F I G U R E V l l - 6 c MODEL R E S U L T S : S T A T I S T I C A L A N A L Y S I S , 1 9 8 1 DATA P r o f e s s i o n a l ( M a l e ) C o n s t r a i n e d I C C M - l a 1 C C M - l b E n t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 8 . 6 4 1 8 . 6 4 1 8 . 6 4 1 8 . 6 4 P r e d . Mean T r a v e l T i m e 2 5 . 3 8 1 7 . 4 3 1 6 . 4 7 1 8 . 6 7 R - S q u a r e d (%) 0 7 1 . 3 3 7 6 . 1 7 9 1 . 5 6 Ch i - S q u a r e 6 7 4 6 9 2 1 3 7 8 1 3 9 7 7 1 1 9 2 1 F R a t i o - 3 . 1 6 4 . 8 3 5 . 6 6 L o g - L i k e l i h o o d R a t i o 3 2 8 9 2 2 4 0 3 9 1 6 3 4 5 6 3 8 8 Re 1 a t i ve L i k e 1 i h o o d 0 . 2 6 9 . 5 0 3 . 8 0 6 B e t a P a r a m e t e r ( E n t r o p y ) ( . 0 8 7 ) P r o f e s s i o n a l ( F e m a l e ) C o n s t r a i n e d I C C M - l a 1 C C M - l b En t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 6 . 3 6 1 6 . 3 6 1 6 . 3 6 1 6 . 3 6 P r e d . Mean T r a v e l T i m e 2 6 . 3 5 1 5 . 5 9 1 4 . 6 6 1 6 . 4 2 R - S q u a r e d (%) 0 8 1 . 5 3 9 2 . 0 1 9 3 . 2 5 Ch i - S q u a r e 2 0 2 7 2 3 5 1 7 5 5 3 6 2 1 5 3 3 0 6 8 F R a t i o - 3 . 9 2 5 . 6 0 6 . 1 3 L o g - L i k e l i h o o d R a t i o 6 6 9 2 2 3 1 2 5 8 1 6 8 1 9 1 8 5 7 7 Re 1 a t i ve L i k e 1 i h o o d 0 . 5 3 3 . 7 4 9 . 7 2 2 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 1 1 ) S e r v i c e ( M a l e ) C o n s t r a i n e d 1 C C M - l a 1 C C M - l b En t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 6 . 5 4 1 6 . 5 4 1 6 . 5 4 1 6 . 5 4 P r e d . Mean T r a v e l T i m e 2 5 . 2 4 1 6 . 6 1 1 5 . 7 1 1 6 . 6 2 R - S q u a r e d (%) 0 7 7 . 7 1 9 4 . 2 3 9 3 . 5 2 Ch i - S q u a r e 9 3 9 5 6 3 0 2 6 5 1 6 1 1 2 1 4 7 1 5 F R a t i o - 3 . 1 0 5 . 8 3 6 . 3 9 L o g - L i k e l i h o o d R a t i o 3 7 3 6 7 1 9 5 3 7 9 5 3 6 7 5 9 7 Re 1 a t i ve L i k e 1 i h o o d 0 . 4 7 5 . 7 4 5 . 7 9 7 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 0 9 ) S e r v i c e ( F e m a l e ) C o n s t r a i ned I C C M - l a 1 C C M - l b En t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 4 . 0 5 1 4 . 0 5 1 4 . 0 5 1 4 . 0 5 P r e d . Mean T r a v e l T i m e 2 6 . 2 6 1 5 . 4 3 1 4 . 5 3 1 4 . 1 0 R - S q u a r e d (%) 0 7 8 . 0 8 9 2 . 8 8 9 5 . 4 9 Ch i - S q u a r e 3 9 5 6 1 1 7 3 2 7 0 3 5 0 5 6 2 6 9 2 6 F R a t i o . - 5 . 4 0 1 1 . 2 9 1 4 . 6 9 L o g - L i k e l i h o o d R a t i o 8 0 8 9 5 7 0 3 7 8 1 5 4 9 8 1 1 8 6 9 Re 1 a t i v e L i k e 1 i h o o d 0 . 1 4 9 . 8 0 8 . 8 5 3 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 5 2 ) TESTING OF THE ICCM-1 MODEL / 152 from a constrained random distribution, can be obtained by comparing predicted 2 X and </ values to those obtained for the random distributions. The statistical significance of the difference between random and predicted fits 2 can be determined by taking the ratio of the random to predicted x values. This ratio follows the F distribution. The degrees of freedom available for the random distributions and ICCM-1 results are 196; for the entropy model results, 195. In both cases, the null hypothesis, i.e., that the random and predicted distributions do not differ significantly, can be rejected at the .01 probability level if the F ratio exceeds 1.10. The F statistic is listed in the tables for all model tests. The \p measure can be used to construct an index which indicates the relative gain in fit achieved by a given model as compared to that provided by a random distribution. This measure is the relative likelihood index (Wilson, 1976), and is similar to the information gain index used by Snickars and Weibull (1977). The index varies from 0, indicating that the model prediction contains no more information with regard to the empirical data than does a random * distribution, to 1, indicating that the model perfectly replicates the data. This measure is included in the tables to allow a relative comparison of goodness-of-fit both within and between data sets. Note that the index is based on the information contained in a constrained random distribution; thus it can be interpreted as indicating the improvement in fit provided by a) calibration of the fi parameter in the case of the entropy model, and b) the assumed logic of labour market processes in the case of ICCM model versions. TESTING OF THE ICCM-1 MODEL / 153 a. Comparison of Goodness-of-Fit Statistics 2 2 The predicted flows in all tests have x values well in excess of critical x values, even at a the .1 significance level (152 for 196 degrees of freedom). Thus, the null hypothesis would be rejected for all the models. This is partly 2 due to the large size of the data sets and the sensitivity of the x test to sample size, so that this test is extremely stringent. On the other hand, Batty (1978) suggests that commuting models may be considered satisfactory if they explain at least 70% of the variance in the data; all tests except three of the ICCM-la runs reach this level of explanation. From the point of view of statistical inference, the models are not successful in reproducing empirical flows; from the point of view of overall fit of predicted to empirical flow matrices, the models can be judged to provide satisfactory estimates. 2 The R values for both the ICCM-lb and the entropy models are consistently in the 90% range, indicating that both explain most of the variance in the data sets and are useful predictive tools. The ICCM-la model explains about 70% of the commuting variance of most occupational groups, falling to the 50% range for Male Manufacturing, Managerial and Transportation occupations. These results suggest that the simplistic approach to handling labour shed position (i.e., V. . = 1 for all / and j) contained in the ICCM-la algorithm is an insufficient and unrealistic assumption, whereas the approach taken by the ICCM-lb version is well supported. The ICCM-lb model has some difficulty with two occupation groups: Male Manufacturing and Professional. The heterogeneity of the Professional occupation TESTING OF THE ICCM-1 MODEL / 154 category, as well as the higher incidence of self-employment in the professions, may explain part of the problem; similarly, the potential effect of industry-based differentials in the Manufacturing sector is not accounted for. On the whole, however, the ICCM-lb results rival those of the entropy model and, considering that the ICCM-1 is a normative simulation requiring minimal data input, this level of performance is encouraging. In all tests, the i/> statistic takes on higher values for the ICCM-lb than for entropy model predictions. This indicates that, in terms of overall goodness-of-fit, 2 the entropy model is superior. The fact that R values are similar for the two models, coupled with the sensitivity of the measure to the direction of error in predicting small flows, indicates that the problem with ICCM-lb results lies in its ability to estimate accurately small values in the trip matrix. The reason for this (as discussed above) is that the ICCM-1 logic utilizes the concept of the reservation wage to construct strict labour shed boundaries. This can be seen in the distribution of the entropy and ICCM-lb prediction of male sales commuting flows as compared to the empirical data (the trip matrices 2 2 are contained in the Appendix). According to the R and x measures, the ICCM-lb prediction more accurately resembles the data than does the entropy prediction, but has a higher \j/ statistic. The ICCM-lb output trip matrix contains 100 cells (44.4% of the total number of cells) with 0 trips. The empirical distribution has 52 (23.1% of the total) and the entropy model predicts 18 (8.0% of the total). The empirical flow in the 100 cells predicted to be 0 by the ICCM-lb model is 1665, or about 17 per entry. This represents only 4.2% of TESTING OF THE ICCM-1 MODEL / 155 the total in the occupation category of 39,560. Only 7 of the 100 cells represent serious errors, in which the empirical flow exceeded 50; in only 15 did the data contain more than 25 trips. Thus, while the assumption of strict labour shed boundaries by the ICCM-1 model results in predicted trip matrices containing many empty cells which may represent large relative errors as compared to actual flows, the absolute error is small. The strict dichotomy in the ranking of job offers produced by the reservation wage concept is critical to the iteration path of the ICCM-1 model - this is the means by which spatial wage adjustments are determined. The results of test runs on the disaggregated occupation data indicate that the method allows for a high level of overall predictive accuracy while introducing an acceptable amount of error in the fitting of small flows. b. Differences Between Male and Female Commuting Patterns The disaggregated data exhibit the expected differences between male and female commuting flows; all 5 occupations analyzed show the same pattern. On average, women in managerial occupations spend 16.0% less time commuting than do men in similar occupations; in sales occupations, the difference is 21.0%; in clerical occupations, 10.0%; in professional occupations, 12.2%; and in service occupations, 15.1%. As the differences in the constrained random distribution mean times are very slight, and in some cases in the opposite direction (clerical, professional and service occupations), it would appear that empirical differences cannot be due to the map patterns. The entropy model handles the problem by fitting much higher B parameters to female data sets, indicating that women are more sensitive to TESTING OF THE ICCM-1 MODEL / 156 commuting costs than are men. This result agrees with the findings of Broughton (1981) and Hanson and Johnston (1985). However, the ICCM-lb model handles the situation equally well, without recourse to compensatory behavioural interpretation to augment the information contained in the map pattern itself. This result suggests that differences in the commuting patterns of men and women workers can be explained by analyzing labour market processes directly (and specifically in terms of choice behaviour under uncertainty) rather than by appealing to gender-based behavioural differences. The ICCM-1 approach assumes a linear relation between travel time and commuting cost; the slope of the function does not affect model results. The ability of the ICCM-lb model to predict satisfactorily both male and female flows indicates that a linear function is an adequate representation of the relationship between travel time and travel cost for both groups. While it is possible that commuting cost functions for male and female workers differ by a slope factor, this difference alone would not be sufficient to explain commuting pattern differences - the ICCM-lb results would not change even if such slope differentials were explicitly introduced into the cost matrix. The ICCM-1 model, by analyzing the map pattern of origins and destinations in terms of labour market processes rather than in the absolute space terms embedded in the gravity concept (Sayer, 1976; page 208), utilizes the economic content of the map pattern in a more realistic manner than does an analysis based either on the entropy concept or on descriptive inference. An example of the ability of the ICCM-lb version to handle both male and TESTING OF THE ICCM-1 MODEL / 157 female commuting flows is provided by the results using data describing sales occupations. Figure VII-7 illustrates the travel time cumulative frequency distributions of the empirical, constrained random and predicted flow patterns. Female commuting trips generally are much shorter than male trips, but the constrained random distributions are almost identical. The Kolmogorov-Smirnov D statistics show that the difference between the empirical and random distributions is much greater for the female than for the male sales category. In order to handle this situation, the entropy model calibrates friction factor B parameters of .097 for males and .147 for females, suggesting that women are notably more sensitive to commuting costs than are men. The ICCM-lb model fits both empirical distributions equally well, explaining over 92% of the variance in both cases, without the use of fitted parameters. c. Sensitivity of the Entropy Deterrence Function Parameter In order to test the possibility that a normative entropy model, based on a pre-selected B parameter, could perform as well as the ICCM-lb model on disparate empirical data sets, a sensitivity analysis examining the effect on entropy model predictions of B parameter adjustments was undertaken. Three data sets were used: male manufacturing, male managerial and female clerical. Figure VII-8 illustrates the results of this test. Inspection of the graphs leads to a number of conclusions. It is evident that fitting the entropy model to the 2 mean travel time provides an effective method of maximizing R values, given a negative exponential friction-of-distance function. Secondly, sensitivity of the model to B parameter adjustments is an inverse function of the magnitude of the fitted parameter value. This suggests that the effect of origin / destination constraints F I G U R E V I 1 - 7 S A L E S OCCUPATIONS - TRAVEL TIME CUMULATIVE FREQUENCY D I S T R I B U T I O N S 1 5 9 F IGURE VI 1 - 8 S E N S I T I V I T Y A N A L Y S I S OF THE ENTROPY MODEL BETA PARAMETER / s © T T T —I— 1 0 ,12 -T— ,14 I . 0 6 . 0 8 B e t a P a r a m e t e r D a t a S e t : M a l e M a n a g e r i a l M a l e M a n u f a c t u r i n g F e m a I e C l e r i c a l The c a l i b r a t e d b e t a v a l u e i s i n d i c a t e d b y : © TESTING OF THE ICCM-1 MODEL / 160 in controlling the accuracy of entropy-based prediction is more pronounced for data sets characterized by shorter commuting trips. Thirdly, the use of a /3 parameter fit to one data set to predict flows of another data set would lead to a serious level of error. Finally, no unique fi parameter exists that, if applied to all three data sets, would produce the predictive accuracy of the ICCM-lb model. This test of the relative flexibility of the ICCM-lb model is quite stringent, as the accuracy of the model in replicating male manufacturing flows is relatively poor. Thus, the analysis provides grounds for concluding that the entropy approach does not offer a suitable basis for normative modelling. d. Analysis of Rent Surfaces The interpretation of the origin / destination map pattern made by the ICCM-lb can be investigated by looking at the rent (i.e., residential location value) surfaces generated by the modelling process. Theoretically, residential rent surfaces for all occupation groups should be similar regardless of differentials in the distribution of the residences of workers in various occupations across the urban area. If this were not the case, it would suggest that there were multiple housing markets segregated by the occupation and gender of residents - this clearly is unrealistic. On the other hand, differences in wage surfaces by occupation could occur, at least to the extent that occupation groups define independent labour sub-markets. A test of the realism of modelled rent surfaces is possible by measuring the agreement of generated surfaces with each other -this test doesn't attempt to investigate the accuracy of the surfaces, but it does reflect the internal consistency of the model's logic. A similar test for wage surfaces is impossible, as there is no a priori reason to expect that wage TESTING OF THE ICCM-1 MODEL / 161 surfaces for different occupations would either be in agreement or differ significantly. The test was performed by calculating the Pearson's coefficient of correlation for residential location values of 6 selected representative occupational groups. Figure VTI-9 contains a table of the data utilized, and presents the correlation matrix of residential rent distributions. All correlation coefficients are highly significant (at the .01 level), indicating that the rent surfaces are in close agreement with each other. e. Analysis of Wage Surfaces As discussed above, there is no direct test for the reliability of wage surface derivation, as there are no a priori expectations either of coincidence or difference between data set wage distributions. However, an inspection of the zonal wages generated by the ICCM-lb model for the selected sample of data sets (figure VII-10) shows that they appear to be realistic. The correlation matrix indicates that wages vary between data sets much more than do rents - this is due to the existence of distinct labour markets, t Relatively low correlations are shown between Male Transport / Male Sales and Male Manufacturing / Female Sales groups, while Male Manufacturing / Male Transport, Female Clerical / Female Service and Male Sales / Female Clerical t Calculation of the correlation matrix of wage distributions excludes the UEL values. This is due to the poor quality of the estimation resulting from data problems. Inclusion of the high predicted UEL wages would bias correlation coefficients. This difficulty is minimal in the analysis of rent distributions, as predicted UEL rents are not out-of-range as are wages. F I G U R E V l l - 9 A N A L Y S I S OF S E L E C T E D I C C M - l b GENERATED RENT D I S T R I B U T I O N S 1 6 2 A . R E S I D E N T I A L RENTS ZONE S a I e s ( M a l e ) S a I e s : F e m a I e I C I e r i c a I ( F e m a I e 1 M a n u f a c t (Ma Ie ) S e r v i c e ( F e m a I e ) T r a n s p o r t (Ma Ie ) 1 B u r n a b y 2 8 . , 8 1 8 , , 1 2 8 , , 5 2 7 . , 5 1 9 , , 2 2 7 . , 6 2 C o q u i t . 1 8 . , 9 1 1 . . 5 1 9 . , 2 1 8 , . 0 1 4 , . 2 1 7 . , 7 3 D e l t a 2 2 . , 9 8 , . 7 2 2 , , 9 2 7 , , 7 1 7 . , 1 2 3 . , 3 4 New W e s t 2 5 . , 1 1 6 , . 5 2 4 . . 9 2 3 . , 9 1 8 , , 1 2 2 , , 2 5 N V a n C . 2 9 . , 8 2 1 . . 3 3 2 , . 7 3 7 , , 6 2 3 , . 9 3 6 , , 8 6 N V a n D. 2 7 . . 8 1 9 , . 3 3 0 , . 5 3 5 , . 1 2 1 , . 8 3 4 , . 3 7 P . C o q u i t . 1 7 . , 4 1 1 , . 5 1 7 , . 6 1 7 . , 4 1 5 . . 2 1 7 . . 5 8 P . M o o d y 1 9 . . 9 1 4 , . 2 1 9 , . 9 2 0 . . 5 1 7 , . 0 1 9 , , 2 9 R i c h m o n d 3 5 . , 2 2 1 , . 7 3 5 . . 0 3 7 , . 8 2 4 , . 3 3 5 , . 0 1 0 S u r r e y 1 9 . . 8 1 3 . . 0 2 0 , . 0 1 8 , . 8 1 4 , . 1 1 6 , . 3 1 1 V a n e . 4 8 . . 0 3 4 , . 0 4 9 , . 0 4 5 , . 7 3 4 , . 3 4 6 , . 6 1 2 W V a n 3 4 . . 7 2 3 , . 6 3 7 , . 0 3 7 , . 1 2 4 , . 9 3 7 , . 5 13 UEL 4 2 . , 1 2 7 , . 8 4 4 , . 1 4 0 , . 0 2 9 , . 4 4 0 , . 2 14 M R i d g e , 7 . 2 2 , . 4 1 , . 3 . 9 . 6 15 L a n g 1ey 2 0 . , 2 1 , . 1 2 1 . , 2 1 9 , . 7 1 9 , . 5 4 , . 6 B . C O R R E L A T I O N MATRIX S a l e s S a l e s C l e r i c a l M a n u f a c t . ( M a l e ) ( F e m a l e ) ( F e m a l e ) ( M a l e ) S e r v i c e T r a n s p o r t , ( F e m a I e ) (Ma Ie ) S a I e s ( M a l e ) 1 . 0 . 9 2 2 . 9 9 5 . 9 4 6 . 9 7 3 . 9 2 4 S a I e s ( F e m a I e 1 1 . 0 . 8 9 5 . 9 3 5 . 8 6 8 . 9 3 5 C I e r i c a I ( F e m a I e ) 1 . 0 . 9 7 4 . 9 7 9 . 9 3 2 M a n u f a c t , ( M a l e ) 1 . 0 . 9 5 6 . 9 9 3 S e r v i c e ( F e m a I e ) 1 . 0 . 9 6 9 T r a n s p o r t , ( M a l e ) 1 . 0 F I G U R E V l l - 1 0 A N A L Y S I S OF S E L E C T E D I C C M - l b GENERATED WAGE D I S T R I B U T I O N S 1 6 3 A . WAGES ZONE S a l e s S a 1 e s C l e r i c a l Manu f a c t . S e r v i c e T r a n s p c (Ma 1e) ( F e m a 1 e ) ( F e m a 1 e ) (Ma 1e) ( F e m a 1 e ) ( M a l e 1 B u r n a b y 3 3 . , 0 • 2 6 . . 8 3 1 . . 3 3 3 . 0 2 8 . 3 3 2 . 0 2 C o q u i t . 2 9 . . 0 2 2 . . 0 2 9 , . 0 3 1 . 3 2 7 . 5 2 8 . 8 3 D e l t a 4 0 , , 0 2 0 , . 8 4 1 , . 5 4 3 . 8 3 3 . 3 3 4 . 5 4 New W e s t 3 0 , :0 2 3 . . 5 2 9 , . 0 2 7 . 5 2 5 . 0 2 4 . 5 5 N V a n C . 2 0 , . 5 1 8 , . 5 2 3 . . 8 4 0 . 3 2 3 . 0 3 7 . 5 6 N V a n D. 2 0 , , 5 1 9 . . 0 2 5 , , 3 4 1 . 5 2 5 . 8 3 8 . 5 7 P . C o q u i t . 2 4 , . 8 2 4 , . 5 2 5 . . 8 2 5 . 5 2 4 . 3 2 4 . 5 8 P . M o o d y 3 3 , . 3 2 6 . . 8 3 3 , . 0 3 4 . 5 3 3 . 3 3 3 . 3 9 R i c h m o n d 4 9 , . 5 3 3 , . 3 4 9 . . 0 5 3 . 8 3 9 . 2 4 9 . 5 1 0 S u r r e y 3 0 . . 8 2 4 . . 3 3 1 . . 3 2 9 . 0 2 6 . 8 2 5 . 0 1 1 V a n e . 6 6 , . 3 5 0 . . 3 6 6 , . 3 6 7 . 0 4 9 . 3 6 6 . 3 1 2 W V a n 3 8 . . 3 3 0 . . 8 4 4 . . 3 6 1 . 3 3 8 . 2 5 7 . 5 1 3 UEL * 7 7 , . 8 7 1 . . 0 8 4 , . 5 8 3 . 5 7 4 . 7 7 7 . 2 1 4 M R i d g e 1 3 . . 0 1 2 , . 0 1 3 . . 5 1 3 . 2 1 3 . 0 1 3 . 0 1 5 L a n g l e y 4 1 , . 5 1 2 . . 3 4 5 , . 5 4 0 . 0 3 4 . 3 1 6 . 0 N o t i n c l u d e d i n c o r r e l a t i o n c a l c u l a t i o n . B . C O R R E L A T I O N MATRIX S a l e s S a l e s C l e r i c a l M a n u f a c t . ( M a l e ) ( F e m a l e ) ( F e m a l e ) ( M a l e ) S e r v i c e T r a n s p o r t ( F e m a l e ) ( M a l e ) S a I e s (Ma Ie ) 1 . 0 . 7 9 0 . 9 8 5 . 7 9 5 . 8 6 8 , 6 8 0 S a I e s ( F e m a I e ; 1 . 0 . 7 5 1 , 7 0 8 7 9 0 . 8 4 3 C I e r i c a I ( F e m a I e ) 1 . 0 . 8 6 4 . 9 7 4 7 1 5 M a n u f a c t . (Ma Ie ) 1 . 0 . 8 9 8 . 9 1 5 S e r v i c e ( F e m a I e ! 1 . 0 . 7 8 5 T r a n s p o r t (Ma Ie ) 1 . 0 TESTING OF THE ICCM-1 MODEL / 164 are highly correlated. This suggests that manufacturing and transport occupations form one distinct labour-market pattern, while clerical, sales and service occupations form another pattern. The location of these differences can be traced to North Shore municipalities (City of North Vancouver, District of North Vancouver, West Vancouver), where the model predicts high blue collar and lower white collar wage levels. This pattern can be explained by the concentration of manufacturing and trans-shipment activity centred on North Shore dock facilities. Another difference in occupation-specific wage structures is the lower overall levels of Female Sales and Service net wages. This can be explained by the residentiary nature of these industries - i.e., they are located according to a local market orientation, and thus are more evenly distributed across the urban area. The result is that locational incompatibilities creating large wage differentials are not present in these occupations. This is most notable when the Female Sales wage surface is compared to that of the Female Clerical category; clerical wage levels are much higher, due to the clustering of clerical jobs in the CBD. f. Stability of the ICCM-lb Solutions The measure of the stability of the ICCM solution is the Global Stability Index, which is the weighted mean coefficient of variation of zonal residential rents. The ICCM-lb test runs obtained values of this index consistently of approximately 30%; the lowest value was 26.8% (male service workers) and the highest value was 32.8% (male sales workers). This indicates a satisfactory homogeneity of zonal residential location values, given the degree of spatial TESTING OF THE ICCM-1 MODEL / 165 aggregation of the zonal system. 3. Predictions Using Aggregated Data Sets The three models were run on data sets containing information of the commuting pattern of a) all male workers; b) all female workers; and c) total workforce. The results of these runs are listed in figure VII-11. The relative accuracy of predictions follows the same pattern as obtained with the disaggregated data: the ICCM-la model is much poorer in terms of explained variance than the ICCM-lb and entropy models. The main difference in the statistical summary is that the 2 X and \b measures for the ICCM-lb male workers run are quite poor as compared to the entropy model. This suggests that the ability of the ICCM-lb to fit individual flows in the trip matrix suffers when the data are highly aggregated. Inspection of the predicted matrices shows that the labour shed bounding technique of the ICCM-lb approach is not able to handle small long-distance flows, as the model treats the data as if they pertained to a single labour market. The entropy model's tendency to distribute trips throughout the matrix allows the resultant prediction to more closely replicate the overall flow pattern; however, this is caused by the randomness inherent in the modelling process rather than the logic of the approach in terms of labour market operations. The fact that the ICCM-lb approach is more accurate in modelling aggregated female data than in modelling the aggregated male data suggests that the occupational boundaries defining sub-markets are less strict in female-dominated occupations than in male-dominated occupations. For example, the mass of female 1 6 6 F I G U R E V l l - 1 1 MODEL R E S U L T S : S T A T I S T I C A L A N A L Y S I S , 1 9 8 1 DATA A l l O c c u p a t i o n s ( M a l e ) C o n s t r a i n e d I C C M - l a 1 C C M - l b E n t r o p y R a n d o m E m p . M e a n T r a v e l T i m e 1 9 . 1 7 1 9 . 1 7 1 9 . 1 7 1 9 . 1 7 P r e d . Mean T r a v e l T i m e 2 6 . 9 7 1 8 . 0 3 1 6 . 6 2 1 9 . 1 5 R - S q u a r e d (%) 0 7 3 . 2 7 8 5 . 7 9 9 1 . 5 5 Ch i - S q u a r e 4 6 7 4 3 9 1 4 3 9 8 7 2 5 9 2 2 2 4 7 3 0 4 F R a t i o - 3 . 2 5 1 . 8 0 9 . 8 8 L o g - L i ke1 i h o o d R a t i o 2 4 4 5 4 2 1 4 5 4 6 2 1 2 0 7 8 6 3 8 8 5 1 Re 1 a t i v e L i ke1 i h o o d 0 . 4 0 5 . 5 0 6 . 8 4 1 B e t a P a r a m e t e r ( E n t r o p y ) ( . 0 8 7 ) A l l O c c u p a t i o n s ( F e m a l e ) C o n s t r a i n e d I C C M - l a 1 C C M - l b En f r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 6 . 3 6 1 6 . 3 6 1 6 . 3 6 1 6 . 3 6 P r e d . Mean T r a v e l T i m e 2 6 . 3 5 1 5 . 5 9 1 4 . 6 6 1 6 . 4 2 R - S q u a r e d (%) 0 8 1 . 5 3 9 2 . 0 1 9 3 . 2 5 Ch i - S q u a r e 2 0 2 7 2 3 5 1 7 5 5 3 6 2 1 5 3 3 0 6 8 F R a t i o - 3 . 9 2 5 . 6 0 6 . 1 3 L o g - L i k e l i h o o d R a t i o 6 6 9 2 2 3 1 2 5 8 1 6 8 1 9 1 8 5 7 7 Re 1 a t i ve L i k e 1 i h o o d 0 . 5 3 3 . 7 4 9 . 7 2 2 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 1 1 ) A l l O c c u p a t i o n s ( T o t a l ) C o n s t r a i n e d I C C M - l a I C C M - l b En t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 7 . 8 9 1 7 . 8 9 1 7 . 8 9 1 7 . 8 9 P r e d . Mean T r a v e l T i m e 2 6 . 6 2 1 7 . 6 1 1 7 . 3 4 1 7 . 8 8 R - S q u a r e d (%) 0 7 5 . 4 8 9 3 . 8 1 9 2 . 5 7 Ch i - S q u a r e 1 2 2 3 4 1 9 3 1 2 7 8 6 8 5 8 8 4 0 1 1 9 5 7 5 F R a t i o - 3 . 9 1 1 . 4 2 1 0 . 2 3 L o g - L i k e l i h o o d R a t i o 5 6 5 7 9 2 2 8 5 2 0 8 1 4 4 6 7 3 8 9 5 0 5 R e l a t i v e L i k e l i h o o d 0 . 4 9 6 . 7 4 4 . 8 4 2 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 0 2 ) TESTING OF THE ICCM-1 MODEL / 167 employment is in service, clerical and sales occupations - it is not unlikely that the incidence of cross-over in these occupations is high as compared to that in the male-dominated managerial, manufacturing, professional and transportation occupations. The problems associated with the ICCM-lb model's handling of male workers are 2 carried over to the fully aggregated data set. However, the high R value indicates that this model version is able to produce a satisfactory trip matrix prediction, as over 93% of the variance is explained. The indication that the model is not properly characterizing the operations of an "aggregate" labour market can be discerned from the Stability Index, which has a value of 41.2%, as compared to 30.2% for the female data and 32.8% for the male data. E. COMPARISON OF 1971 TO 1981 PROJECTIONS Transportation models are most often used for projection applications rather than for commuting flow or labour market analysis. Data describing commuting trips under initial land-use and transport network conditions are utilized for model calibration, and the model is then run with the fitted parameters in a predictive mode to provide an estimate of the commuting pattern that is likely to result from specified land-use or network changes. The premise underlying the logic of the entropy model that makes it attractive for such purposes is that the 8 parameter is associated purely with trip-makers' travel cost sensitivity, and so is independent of the map pattern and network defining the commuting situation (Gordon, 1985). If travel cost sensitivity remains relatively constant over time, then a properly specified entropy model should be able to predict flow patterns TESTING OF THE ICCM-1 MODEL / 168 under changing conditions without the need for re-calibration. The ICCM-lb model has been shown to adequately represent commuting patterns without the need for calibration; thus, use of this model for prediction to future or changed conditions does not require any adjustment to the modelling process. This section compares the ability of these two models to predict 1981 aggregate commuting flows with only the following data available: a) the full 1971 empirical commuting matrix; b) the 1981 land-use pattern, i.e., the 1981 trip matrix marginals; c) the travel time matrix relating to 1981 conditions. For purposes of comparison, the two models were First run using the 1971 data. This also provides the means of calibrating the entropy model. The two models then were run on the 1981 data - this simulates the predictive mode of both models. Finally, the entropy model was calibrated to the full 1981 data set to compare the fitting of the /3 parameter over time. The results are presented in figure VII-12. Both models explain a high proportion of the variance in the data. The appendix contains the predicted matrices -again, the ICCM-lb predicts an overly large number of empty cells in the trip matrix. The high Stability Index value of 43.6% again indicates that the ICCM-1 approach requires data based on homogeneous labour markets to obtain reliable results. The 1981 projections are both satisfactory, with the ICCM-lb explaining 2 a somewhat higher level of error variance, but also exhibiting a high x value due to poor fitting on smaller flows. The entropy model fitted to the 1981 data shows some improvement over the 1 6 9 F I G U R E V l l - 1 2 MODEL C O M P A R I S O N : I C C M - 1 B / ENTROPY P R O J E C T I O N S 1 9 7 1 - 1 9 8 1 ( A L L O C C U P A T I O N S ) 1 9 7 1 DATA C o n s t r a i n e d I C C M - l b E n t r o p y R a n d o m E m p . Mean T r a v e l T i m e 1 6 . 5 8 1 6 . 5 8 1 6 . 5 8 P r e d . Mean T r a v e l T i m e 2 5 . 2 9 1 6 . 1 3 1 6 . 5 9 R - S q u a r e d (%) 0 9 4 . 4 2 9 3 . 8 3 Ch i - S q u a r e 6 7 2 8 7 4 1 8 6 3 8 6 7 7 3 8 7 F R a t i o - 3 . 6 1 8 . 6 9 L o g - L i k e l i h o o d R a t i o 3 4 0 0 7 5 8 4 5 3 8 5 6 2 0 5 Re 1 a t i ve L i ke1 i h o o d 0 . 7 5 1 . 8 3 5 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 1 3 ) 1 9 8 1 DATA C o n s t r a i n e d 1 C C M - l b E n t r o p y R a n d o m P r o j e c t e d F i t Emp. Mean T r a v e l T i m e 1 7 . 8 9 1 7 . 8 9 1 7 . 8 9 1 7 . 8 9 P r e d . Mean T r a v e l T i m e 2 6 . 6 2 1 7 . 3 4 1 7 . 2 4 1 7 . 8 8 R - S q u a r e d (%) 0 9 3 . 8 1 9 1 . 5 5 9 2 . 5 7 Ch i - S q u a r e 1 2 2 3 4 1 9 8 5 8 8 4 0 1 2 4 0 8 5 1 1 9 5 7 5 F R a t i o - 1 . 4 2 9 . 8 6 1 0 . 2 3 L o g - L i k e l i h o o d R a t i o 5 6 5 7 9 2 1 4 4 6 7 3 9 1 9 6 7 ' 8 9 5 0 5 Re 1 a t i v e L i ke1 i h o o d 0 . 7 4 4 . 8 3 8 . 8 4 2 B e t a P a r a m e t e r ( E n t r o p y ) ( . 1 1 3 ) ( . 1 0 2 ) TESTING OF THE ICCM-1 MODEL / 170 results based on the 1971 calibration, but the difference is slight. This finding supports the use of the entropy model for purposes of projecting commuting flows, especially when aggregated data are used. However, the argument that the underlying logic of the entropy model is based on behavioural aspects of labour market processes is put in question. The important point is that the B value fitted to the 1981 data is lower than that fitted to the 1971 data, indicating a lessened sensitivity to travel cost. The expansion of employment that took place in the Vancouver CMA over this period was most concentrated in female-dominated occupations - clerical, service and sales occupations. Thus, given the findings of the 1981 disaggregated runs that female workers were more sensitive to travel cost, it is odd that the B parameter would decline during a period of rapid increase in the proportion of women in the workforce. On the other hand, the decade also was one of rapid physical expansion of the urban area; the resident workforce of the City of Vancouver grew by 23.3%, compared to much higher growth rates in suburban locations - 110.5% in Surrey, 116.1% in Richmond, and 126.1% in Delta. As a result, mean commuting times increased to reflect changes in the map pattern of employment and residence locations. It is likely that the decrease in the B parameter is due to these map pattern changes rather than to a decrease in the overall sensitivity to travel costs. This test is not conclusive, in that reliance on aggregated data prevents a full analysis of the characteristics of the models. The test does show, however, that while both approaches are useful for planning applications, the behavioural interpretation of the entropy model is suspect. TESTING OF THE ICCM-1 MODEL / 171 F. SUMMARY The relative ability of three commuting models to predict commuting patterns of disaggregated and aggregated occupational categories in Vancouver were compared using data from the 1981 Census. In addition, the abilities of the models to predict commuting patterns over time was tested using 1971 and 1981 aggregated Census data. The following conclusions were drawn: 1. The ICCM-lb normative model is able to predict disaggregated commuting patterns consistently at a level of accuracy that rivals calibrated entropy model predictions. This result provides grounds for retaining the hypothesis that labour market behaviour can be explained in terms of objective rational decision-making under uncertainty. 2. The ICCM-la model version is superior to the ICCM-lb version in all respects. Thus, the assumption of spatially naive decision-making behaviour in the urban labour market is rejected, and the hypothesis that workers make an adjustment for residence position vis a vis labour shed boundaries is accepted. 3. Generation of strict labour shed boundaries by the ICCM-1 algorithm lessens link-specific goodness-of-fit of the models; however, the problem is not significant in terms of overall predictive value. The problem is exacerbated in modelling aggregated data sets, indicating that the detailed accuracy of the model's characterization of spatial labour market behaviour requires data specification that respects homogeneity assumptions. 4. Wage and rent surfaces generated by the ICCM-lb model exhibit a high level of realism. Rent surfaces are consistent across occupational categories, indicating that the housing market is not segregated by occupation. Wage TESTING OF THE ICCM-1 MODEL / 172 surfaces show much more variation, suggesting that distinct labour sub-markets exist along blue collar / white collar lines. Specific locations of wage surface differences can be explained in terms of the distribution and relative clustering of activity in different sectors of the economy. 5. Predictive accuracy of the entropy model is highly sensitive to B parameter calibration. Furthermore, the unexpected decrease in the parameter's value over time suggests that the deterrence factor confounds map pattern and travel cost sensitivity; interpretation of the model in terms of random utility theory thus is suspect. 6. The assumption of a linear relation between travel cost and travel time is found to be a reasonable estimate for modelling purposes. CHAPTER VHI. CONCLUSION The research reported in this thesis focusses on the development and testing of a simulation approach to urban commuting modelling that explicitly defines the inter-relationship of urban location and interaction in terms of labour market processes. The theoretical framework for this methodology is based on the work of Goldner (1955), Moses (1962) and Scott (1981), which identifies the necessity of spatial wage differentials as the means by which spatial variations in labour supply and demand are reconciled. Current models relating labour market activity to commuting outcomes involve the postulate of a space economy operating in an efficient equilibrium condition. This assumption implies the necessary existence of rational decision-makers in possession of full and certain knowledge of the economic environment. The normative linear programming approach assumes that decision-makers are objectively rational, while positive models adopting the entropy approach assume that choice behaviour is based on subjective rationality. The linear programming model has been shown in the literature to be a poor predictor of empirical commuting patterns, and there is no normative interpretation of the subjective rationality assumed by the entropy model. The initial purpose of the research project was to develop a normative model structure which is able to investigate alternative characterizations of labour market behaviour by simulating spatial market processes. The solution path logic of this model, the Individual Choice Commuting Model (ICCM), is sufficiently general to allow the inclusion of a wide range of assumptions concerning choice 173 CONCLUSION / 174 behaviour and environmental conditions, including those associated with equilibrium theory. The model is constructed so that an explicit characterization of the labour market, based on the specification of the behavioural assumption set, defines a model version. The second purpose of the research project was to develop an operational version of the ICCM model that is based on the assumption that decision-makers are objectively rational, but make choices with the knowledge that choice outcomes are uncertain. This model version, termed the ICCM-1, differs from a. linear programming approach in that choice behaviour is based on an ex ante rather than ex post market market equilibrium. The theoretical basis of the model's assumption set is derived from game theory, specifically the theory of non-cooperative games under uncertainty. The complex nature of the spatial expression of this approach necessitated the use of heuristic means to extend the concept of an equilibrium mixed strategy to the urban labour market situation. Two estimation techniques were developed: a naive or non-spatial approach, defining ICCM version la; and a spatial approach, defining ICCM version lb. The third purpose of the research was to test the ability of the ICCM-1 model to explain empirical commuting distributions. The two model versions were tested on empirical commuting patterns derived from 1971 and 1981 Census data from Vancouver. In order to obtain a relative measure of the performance of the ICCM-1 model, the results were compared to those obtained from an CONCLUSION / 175 entropy-based commuting model. The tests showed that, while the ICCM-la model proved only partially successful, the ICCM-lb version generated consistently accurate predictions comparable to those of the entropy model. Considering the minimal data requirements of a normative approach, ICCM-lb model performance was superior to the entropy model in terms of efficiency of data utilization and, most significantly, interpretability of results. A. DISCUSSION The behavioural assumptions underlying the ICCM-1 version have been shown to have promise as a characterization of the urban labour market. However, many other approaches can be investigated using the simulation methodology offered by the ICCM structure. For example, inclusion of subjectivity in the evaluation of wage utility, travel disutility and risk assessment may prove to be effective in increasing our understanding both of the spatial labour market and of choice behavior in general. The model version investigated in this thesis can be seen as one of many possible simulation models that can be constructed; its predictive success is measured only in relative terms, and does not preclude the development of more realistic models utilizing more sophisticated behavioural assumptions. The model may be extended in a number of ways. As the solution path does not require a balancing technique to ensure constraint adherence, the trip matrix generated is not limited to two dimensions. For example, it is theoretically possible to model all disaggregated occupations simultaneously, by specifying estimation functions defining the probability of a worker crossing occupational CONCLUSION / 176 categories. Similarly, simultaneous trip distribution and mode split could be accomplished by treating the mixed strategy probabilities between any two destination zones as a binary choice function. In this way, binary choice mixed strategies referring to job application behaviour can be integrated with binary mode-choice behaviour established by means of a discriminant function. This would resolve the problem of integrating multiple transport networks into the analysis, and, from an applied transport planning point of view, would allow a direct means of reconciling trip distribution with trip assignment. 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Chicago: University of Chicago Press, 1929. Weber, R.J., "Noncooperative Games", in Game Theory and Its Applications, Proceedings of Symposia in Applied Mathematics, Vol. 24, Providence: American Mathematical Society, 1981. Wilson, A.G., "Statistical notes on the evaluation of calibrated gravity models". / 187 Transportation Research, Vol. 10, 1976; pages 343-345. Wilson, A.G. and M.L. Senior, "Some relationships between entropy maximizing models, linear programming models and their duals". Journal of Regional Science, Vol. 14, 1974; pages 207-215. APPENDIX: EMPIRICAL AND MODELLED COMMUTING PATTERNS 188 FEMALE M A N A G E R I A L : ICCM - l b MODEL P R E D I C T E D COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 753 . 18 . 4 . 117. 3 . 13 . 2 . 7 . 120. 1 19 . 802 . 20. 8 . 0. 0 2 65 . 190 . 0. 20. 0. 0. 15 . 2 1 . 0. 108 . 244 . 0. 3 . 0. 0 3 0. 0. 267 . 0. 0. 0. 0. 0. 2 14. 0. 285 . 0. 9 . 0. 0 4 98 . 4 . 1 . 68 . 0. 0. 1 . 1 . 2 1 . 12 1. 94 . 0. 1 . 0. 0 5 16 . 0 . 0. O. 1 20 . 63 . 0. 0 . 4 . 0. 235 . 49 . 2 . 0 . 0 6 70. 0. 0. 0. 184 . 184 . 0. 0. 18 . 0. 547 . 99 . 5 . 0. 0 7 27 . 103 . O. 1 1 . 0 . 0 . 17 . 4 . O. 48 . 79 . 0 . 2 . o. O 8 8 . 85 . 0. 4 . 0. 0. 2 . 22 . O. 1 1 . 78 . 0. 1 . 0. 0 9 72 . 0. 48 . 10. 0. 0. 0. 0. 438 . 0. 662 . 0. 1 1 . 0. 0 10 180 . 10. 21 . 190. 0. 0. 1 . 4 . 54 . 620. 29 1 . 0. 5 . 0. 0 1 1 266 . 0. 10. 10. 59 . 2 1 . 0. 0. 277 . 0. 5058 . 1 10. 36 . 0. 0 1 2 5 . 0. 0. 0. 74 . 44 . 0. 0. 10. 0. 566 . 97 . 5 . 0. 0 13 0. 0. 0. 0. 0. 0. 0. 0. 4 . 0. 35 . 0. 1 . 0. 0 14 0. 0. 0. 0. 0. 0. 97 . 0. 0. 0. O. O. 8 . 185 . 0 15 0. 0. 9 . 0. 0. 0. 0. 0. 0. 1 59 . 0. 0. 12 . 0. 500 E M P I R I C A L COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 1 4 15 1 675 . 40. 20. 70. 25 . 5 . 10. 0. 75 . 40. 995 . 10. 20. 0. 0 2 145 . 170. 0. 50. 0. 10. 40. 10 . 5 . O. 230. O. O. 5 . 0 3 40. 5 . 2 15. 10. 0. 10. 0. 0. 85 . 125. 270. 0. • 5 . 5 . 5 4 75 . 25 . 5 . 155 . 0. 0. 10. 0. 5 . 50. 85 . 0. 0. 0. 0 5 20. 0. 10. 0. 100. 25 . 0. 0. 5 . 10. 280. 40. 0. 0 . 0 6 80. 5 . 0. 0. 200. 190. 0. 0. 15 . 10. ' 545. 55 . 5 . 0. 0 7 10. 40 . 0. 10. 0 . 5 . 40. 0. 5 . 20. 135 . 5 . 0. 15 . 5 8 35 . 20. 5 . 10. 10. 0. 10. 25 . 0. 10. 80. 0. 0 . 0 . 5 9 45 . 0. 10. 20. 0. 0. 0. 0. 585 . 15 . 555 . 5 . 5 . 0. 0 IO GO. 40. 55 . 70. 10. IO. 5 . 10. 50. 7 IO. 270. 5 . O. 15 . 65 1 1 285 . 45 . 30. 15 . 40 . 55 . 10. 5 . 250 . 40 . 4980 . 20 . 75 . 0 . 0 12 30. 0. 0. 0. 55 . 15 . 0. 0. 35 . 0. 435 . 230. 0. 0. 0 13 0 . 0. 0. 0. 0 . O. 0 . 0. 20 . O. 20 . O . O . 0 . O 14 40. 20 . 0. 20. 0 . 0. 5 . 10. 10. 5 . 30 . 5 . 0. 145 . 0 15 20. 0. 10. 0. 0. 0. 5 . 0. 1 5 . 150. 60. 0. 0. 0. 420 FEMALE M A N A G E R I A L : ENTROPY MODEL 1 2 3 4 1 542 . 38 . 12 . 88 2 105 . 130. 3 . 30 3 33 . 3. 222 . 13 4 107 . 1 1 . 3 . 44 5 24 . 1 . 1 . 2 6 84 . 4 . 2 . 9 7 46 . 64 . 1 . 15 8 28 . 54 . 0 . 9 9 89 . 3. 43 . 18 10 245 . 54 . 33 . 158 1 1 177 . 9 . 14 . 20 12 25 . 1 . 1 . 3 13 1 . 0. 0. 0 14 20. 20. 1 . 7 15 35 . 19 . 27 . 14 1 2 3 4 1 675 . 40. 20. 70 2 145 . 170. 0. 50 3 40. 5 . 2 15. 10 4 75 . 25 . 5 . 155 5 20. 0. 10. 0 6 80. 5 . 0. 0 7 10. 40. 0. 10 8 35 . 20. 5. 10 9 45 . 0. 10. 20 10 GO. 40. 55 . 70 1 1 285 . 45 . 30. 15 12 30. 0. 0 . 0 13 o . 0. 0. 0 14 40. 20. 0. 20 15 20. 0. 10. 0 PREDICTED COMMUTING FLOWS 5 6 7 8 9 50. 45 . 10. 8 . 135 8 . 7 . 39. 17 . 13 3 . 3 . 1 . 1 . 205 6. 5 . 3 . 2 . 30 61 . 39 . 0. 0. 6 137 . 111. 1 . 1 . 2 1 3 . 2 . 31 . 4 . 6 3 . 3 . 8 . 1 1 . 4 9 . 8 . 1 . 1 . 443 13 . 1 1 . 15. 8 . 69 81 . 46 . 2. 2 . 198 64 . 4 1 . 0 . 0 . 1 1 0. 0. 0. 0 . 3 1 . 1 . 19 . 3 . 2 3 . 2 . 4 . 2 . 14 EMPIRICAL COMMUTING FLOWS 5 6 7 8 9 25 . 5 . 10. 0. 75 0. 10. 40. 10. 5 0. 10. 0. 0. 85 0. 0 . 10. O. 5 100. 25 . 0. 0 . 5 200. 190. 0. 0. 15 0. 5 . 40. 0. 5 10. 0. 10. 25 . 0 0. 0. 0. 0. 585 10. 10. 5 . 10. 50 40. 55 . 10. 5 . 250 55 . 15 . 0. 0. 35 0. 0 . 0. O. 20 0. 0 . 5. 10. 10 o. O. 5 . O. 15 10 1 1 12 13 14 15 119. 884 . 33 . 10. 2 . 9 115. 17 1. 5 . 1 . 7 . 16 72 . 190. 2 . 4 . 0. 25 93 . 98 . 4 . 1 . 1 . 3 4 . 305 . 45 . 2 . 0. 0 13 . 6 13. 103 . 5 . O. 2 51 . 53 . 2 . 1 . 7 . 4 24 . 57 . 2 . 0 . 2 . 4 22 . 578 . 6 . 15 . 0. 3 478 . 225 . 8 . 3 . 3 . 52 28 . 5124 . 86 . 60. 1 . 3 4 . 570. 76 . 5 . 0. 0 0. 32 . 0 . 3 . 0 . 0 25. 26 . 1 . 0 . 16 1. 3 137 . 45 . 2 . 0 . 1 . 374 10 1 1 12 13 14 15 40. 995 . 10. 20. 0 . 0 0. 230. 0 . 0. 5 . 0 125. 270. 0 . 5 . 5 . 5 50. 85 . O . O. O. 0 10. 280. 40 . 0 . 0 . 0 10. 545 . 55 . 5 . 0 . 0 20. 135 . 5 . 0. 15 . 5 10. 80. 0 . 0 . 0. 5 15 . 555 . 5 . 5 . 0 . 0 7 10. 270. 5 . 0 . 15 . 65 40. 4980. 20 . 75 . O. 0 0. 435 . 230. 0 . 0 . 0 0. 20. 0 . O. 0 . 0 5 . 30. 5 . 0 . 145 . 0 150. 60. O . O. 0 . 420 FEMALE P R O F E S S I O N A L : I C C M - l b MODEL P R E D I C T E D COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1894 . 168 . 3 . 664 . 16 . 49 . 25 . 33 . 197 . 4 12. 1538 . 48 . 86 . 0 . 0 2 2 3 0 . 979 . 0 . 164 . 0 . 0 . 233 . 9 1 . 0 . 4 3 3 . 39 1 . 0 . 34 . 0 . 10 3 0 . 0 . 1 3 1 9 . 0 . 0 . 0 . 0 . 0 . 4 7 0 . 9 9 . 2 9 9 . 0 . 129 . 0 . 0 4 366 . 54 . 1 . 553 . 0 . 0 . 10. 1 1 . 5 0 . 5 16 . 24 1 . 0 . 18 . 0 . O 5 66 . 0 . 0 . 0 . 592 . 250 . 0 . 0 . 3 . 0 . 6 2 7 . 157 . 25 . 0 . 0 6 232 . 0 . 0 . 8 . 785 . 634 . 0 . 2 . 2 1 . 0 . 1238 . 272 . 57 . 0 . 0 7 8 0 . 4 17. 0 . 70 . 0 . 0 . 224 . 17 . O. 159 . 68 . 0 . 15 . O. 0 8 25 . 269 . 0 . 24 . O. O. 23 . 56 . O. 42 . 74 . O. 8 . O . O 9 3 18. 0 . 237 . 12 1. 0 . 0 . 0 . 0 . 1 1 4 2 . 0 . 1837 . 0 . 159 . 0 . 0 10 574 . 2 20 . 66 . 12 18. O. 0 . 23 . 32 . 68 . 2 0 7 7 . 4 0 3 . O . 62 . O . 8 9 1 1 1 1 29 . 0 . 29 . 173 . 352 . 130 . 0 . 15 . 7 3 0 . 0 . 1 6 3 1 8 . 4 10. 5 6 9 . 0 . 0 12 36 . 0 . 0 . 0 . 287 . 143 . 0 . 0 . 6 . 0 . 1 186 . 243 . 5 0 . 0 . 0 1 3 1 . 0 . 0 . 0 . O. O. O. 0 . 13 . O. 1 5 0 . O. 26 . O. 0 14 0 . 0 . 0 . 0 . 0 . 0 . 243 . 0 . 0 . 0 . 0 . 0 . 1 3 0 . 1 1 7 0 . 0 15 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 4 2 9 . 0 . 0 . 26 . 0 . 1274 E M P I R I C A L COMMUTING FLOWS 1 2 3 4 • 5 6 7 8 9 10 1 1 12 13 14 15 1 1885 . 225 . 4 0 . 5 0 0 . 9 0 . 7 0 . 6 0 . 25 . 105 . 1 4 0 . 1 8 9 0 . 15 . 25 . 25 . 4 0 2 4 3 0 . 645 . 25 . 3 6 0 . 2 0 . 2 0 . 195 . 8 0 . 10. 65 . 5 9 5 . 10. 15. 75 . 2 0 3 85 . 45 . 9 2 0 . 185 . 0 . 10. 5 . 10. 155 . 4 2 5 . 4 5 5 . 0 . 15 . 0 . 5 4 3 2 5 . 100 . 35 . 785 . 0 . 0 . 4 0 . 0 . 15 . 145 . 3 4 5 . 0 . 10. 5 . 15 5 7 0 . 15. 10. 10. 6 4 5 . 160 . 10 . 0 . 15 . 2 0 . 6 3 0 . 1 3 0 . 5 . O. O 6 2 15. 10. 10. 25 . 725 . 695 . 15 . 5 . 5 . 4 0 . 1285 . 175 . 4 0 . 0 . 5 7 1 2 0 . 2 3 0 . 15 . 105 . 15 . 0 . 2 15. 10. 5 . 10. 2 0 5 . 0 . 15 . 105 . O 8 6 0 . 105 . O. 75 . 10. 5 . 25 . 75 . O. 10. 135 . O. 5 . 15 . O 9 135 . 2 0 . 1 4 0 . 25 . 0 . 5. 10. 5 . 1 5 9 0 . 1 0 0 . 1 6 6 0 . 5 . 1 10. 0 . 10 10 2 6 0 . 180 . 255 . 455 . 15 . 5. 15 . 5 . 75 . 26 10. 6 6 0 . 0 . 5. 25 . 2 6 5 1 1 1 105 . 185 . 1 6 0 . 255 . 2 8 0 . 115 . 40 . 3 0 . 6 9 5 . 3 0 5 . 1 5 3 7 5 . 175 . 1080 . 3 0 . 25 12 95 . 15 . 5 . 4 0 . 2 15. 120 . 0 . 5 . 10. 2 0 . 7 3 5 . 6 10. 75 . 5. 0 13 5 . 0 . 0 . 10. 5 . 0 . 0 . 10. 0 . 5 . 14 5. 10. 0 . 0 . 0 14 8 0 . 275 . 5 . 9 0 . 5 . 0 . 1 10. 0 . 15 . 0 . 8 0 . O. 0 . 8 6 5 . 10 15 8 0 . 55 . 35 . 75 . 5 . 5 . 15 . 0 . 10. 2 7 0 . 1 7 5 . 0 . 5 . 2 0 . 9 8 0 FEMALE P R O F E S S I O N A L : ENTROPY MODEL P R E D I C T E D COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1428 . 195 . 53 . 542 . 178 . 132 . 57 . 34 . 246 . 3 9 9 . 1655 . 77 . 9 0 . 15 . 33 2 3 2 0 . 702 . 13. 2 12. 32 . 24 . 228 . 77 . 28 . 4 2 3 . 37 1 . 14 . 12 . 48 . 6 1 3 98 . 15. 933 . 9 1 . 12 . 9 . 4 . 3 . 3 7 9 . 253 . 386 . 5 . 37 . 2 . 88 4 4 1 1 . 78 . 21 . 3 8 0 . 29 . 22 . 25 . 13 . 79 . 435 . 274 . 13 . 15 . 7 . 16 5 101 . 10. 4 . 25 . 3 16. 168 . 3 . 2 . 17 . 2 0 . 8 6 6 . 154 . 3 0 . 1 . 2 6 2 9 0 . 25 . 10. 7 1 . 5 8 0 . 385 . 8 . 5. 5 0 . 58 . 142 1. 28 1 . 55 . 3 . 7 7 124 . 303 . 4 . 9 1 . 10. 7 . 153 . 17 . 1 1 . 163 . 103 . 5 . 5 . 4 0 . 13 8 54 . 184 . 2 . 4 0 . 8 . 6 . 3 1 . 32 . 5 . 57 . 78 . 3 . 3 . 9 . 10 9 348 . 22 . 257 . 165 . 48 . 36 . 6 . 4 . 1074 . 109 . 1526 . 2 1 . 1 8 0 . 2 . 17 10 7 0 0 . 2 9 0 . 153 . 1008 . 5 0 . 37 . 84 . 36 . 135 . 1 6 1 3 . 4 6 6 . 22 . 25 . 25 . 185 1 1 8 4 9 . 83 . 106 . 231 . 502 . 239 . 24 . 16 . 6 2 0 . 1 7 0 . 1 5 7 9 0 . 3 4 0 . 8 5 8 . 8 . 2 0 12 7 9 . 7 . 4 . 19 . 248 . 132 . 2 . 1 . 24 . 16 . 1 1 8 1 . 187 . 46 . 1 . 2 1 3 5 . O . 2. 1 . 2 . 1 . 0 . 0 . 10 . 1 . 12 1. 1 . 43 . 0 . 0 14 63 . 1 1 1 . 3 . 47 . 6 . 5 . 109 . 12 . 5 . 93 . 59 . 3 . 3 . 1004 . 12 15 79 . 79 . 91 . 73 . 8 . 6 . 19 . 7 . 2 1 . 354 . 73 . 3 . 4 . 6 . 9 0 8 E M P I R I C A L COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 . 14 15 1 1885 . 225 . 4 0 . 5 0 0 . 9 0 . 7 0 . 6 0 . 2 5 . 105 . 140 . 1 8 9 0 . 15 . 25 . 25 . 40 2 4 3 0 . 645 . 25 . 3 6 0 . 2 0 . 2 0 . 195 . 8 0 . 10. 65 . 5 9 5 . 10 . 15 . 75 . 20 3 85 . 45 . 9 2 0 . 185 . 0 . 10. 5 . 10. 155 . 425 . 4 5 5 . 0 . 15 . 0 . 5 4 325 . 1 0 0 . 35 . 7 8 5 . 0 . 0 . 4 0 . 0 . 15 . 145 . 345 . 0 . 10 . 5 . 15 5 7 0 . 15. 10. 10. 6 4 5 . 160 . 10. 0 . 15. 2 0 . 6 3 0 . 130 . 5 . 0 . 0 e 2 15. 10. 10. 2 5 . 725 . 6 9 5 . 15. 5 . 5 . 4 0 . 1285 . 175 . 4 0 . 0 . 5 7 1 2 0 . 2 3 0 . 15 . 105 . 15 . 0 . 2 1 5 . 10. 5 . 10 . 2 0 5 . 0 . 15 . 105 . 0 8 6 0 . 105 . O. 75 . 10. 5 . 25 . 75 . 0 . 10. 135 . 0 . 5 . 15 . 0 9 135 . 2 0 . 140 . 25 . 0 . 5 . 10. 5 . 1 5 9 0 . 1 0 0 . 1 6 6 0 . 5 . 1 10 . 0 . 10 10 2 6 0 . 180 . 255 . 4 5 5 . 15 . 5 . 15 . 5 . 75 . 2 6 1 0 . 6 6 0 . 0 . 5 . 25 . 265 1 1 1 105 . 185 . 160 . 255 . 2 8 0 . 115 . 4 0 . 3 0 . 6 9 5 . 305 . 1 5 3 7 5 . 175 . 1 0 8 0 . 3 0 . 25 12 95 . 15 . 5 . 4 0 . 2 15. 120 . 0 . 5 . 10 . 2 0 . 735 . 6 10 . 75 . 5 . 0 13 5 . 0 . 0 . 10 . 5 . 0 . 0 . 10. O. 5 . 145 . 10 . 0 . O. 0 14 8 0 . 275 . 5. 9 0 . 5 . 0 . 1 10. 0 . 15 . 0 . 8 0 . 0 . 0 . 8 6 5 . 10 15 8 0 . 55 . 35 . 75 . 5 . 5 . 15 . 0 . 10. 2 7 0 . 175 . 0 . 5 . 2 0 . 9 8 0 FEMALE S E R V I C E : I C C M - l b MODEL P R E D I C T E D COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1387 . 138 . 19 . 342 . 65 . 66 17 . 26 . 308 . 367 . 1 2 7 0 . 78 . 26 . 0 . 1 . 2 127 . 638 . 0 . 56 . O. 0 13 1. 56 . 0 . 307 . 2 8 0 . 0 . 8 . O 2 1 . 3 0 . 0 . 783 . 0 . 0 . 0 0 . 0 . 58 1 . 0 . 2 5 9 . 0 . 3 0 . 0 . 28 . 4 224 . 37 . 5 . 245 . 0 . 2 6 . 7 . 67 . 4 14. 17 1. 8 . 5 . 0 . 3 . 5 47 . 0 . 0 . 0 . 4 6 0 . 173 0 . O. 1 3 . O. 5 19 . 169 . a . 0 . 0 . 6 102 . 0 . 0 . 0 . 388 . 27 1 0 . 1 . 31 . 0 . 6 3 5 . 186 . 1 1 . 0 . 0 . 7 67 . 4 16. O. 36 . 0 . 0 194 . 17 . 0 . 169 . 86 . 0 . 5 . o . O . a 14 . 19 1. 0 . 7 . 0 . 0 14 . 37 . 0 . 28 . 59 . 0 . 2 . 0 . 1 . 9 186 . 0 . 166 . 37 . 0 . 0 0 . 0 . 1403 . 0 . 1 2 7 1 . 12 . 4 1 . 0 . 0 . 10 4 9 6 . 222 . 96 . 762 . 0 . 0 18 . 3 1 . 179 . 2 3 9 7 . 4 6 0 . O. 23 . 0 . 147 . 1 1 8 13. 0 . 66 . 53 . 404 . 177 0 . 13 . 1 133 0 . 1 3 2 8 3 . 498 . 179 . 0 . 0 . 1 2 12 . 0 . 0 . 0 . 132 . 6 1 0 . 0 . 13 . 0 . 5 3 0 . 143 . 8 . 0 . 0 . 13 0 . O . 0 . 0 . O. 0 0 . 0 . 5 . O . 28 . 1 . 2 . o . 0 . 14 0 . 0 . 0 . 0 . 0 . 0 163 .! 0 . 0 . 0 . O. 0 . 63 . 8 8 5 . 0 . 15 0 . 0 . 35 . 0 . 0 . 0 0 . 0 . 0 . 233 . 0 . 0 . 6 . 0 . 12 15. E M P I R I C A L COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1 8 1 0 . 1 10. 10. 2 0 0 . 35 . 25 15 . 10. 1 0 0 . 75 . 1 6 3 0 . 4 0 . 25 . 10 . 15 . 2 2 7 0 . •675 . 5 . 180 . 5 . 0 8 0 . 3 0 . 25 . 45 . 2 8 5 . 10 . 0 . 10. 5 . 3 5 0 . 0 . 805 . 45 . 10. 5 5 . . 0 . 2 15. 2 6 0 . 2 5 5 . 5 . 5 . 0 . 2 0 . 4 285 . 5 0 . 25 . 595 . 0 . 5 10. 5 . 2 0 . 3 0 . 165 . 5 . 0 . 0 . 0 . 5 O. 10. 0 . 10. 735 . 135 0 . 0 . 15 . 15 . 3 0 0 . 165 . 5 . 0 . 0 . S 35 . 0 . 0 . 0 . 4 5 0 . 4 10 O. 0 . 6 0 . 0 . 4 4 0 . 2 3 0 . 0 . O. 0 . 7 5 0 . 3 7 0 . 0 . 4 0 . 5 . 0 335 . 3 5 . 5 . 5 . 1 10 . 0 . 0 . 35 . 0 . 8 7 0 . 9 0 . O. 3 0 . 0 . 0 15 . 85 . 5 . 15 . 45 . 0 . 0 . 0 . 0 . 9 15 . 10. 5 0 . 2 0 . 0 . 0 0 . 0 . 2 3 2 0 . 35 . 6 30 . 15 . 2 0 . 0 . 0 . 10 195 . 130 . 170. 275 . 2 0 . 5 5 . 15 . 125 . 3 155 . 4 7 0 . 15 . 15 . 0 . 235 . 1 1 6 2 5 . 75 . 100 . 9 0 . 105 . 145 15 . 10. 7 6 0 . 75 . 14 1 7 5 . 75 . 345 . O. 25 . 12 15 . 5 . 0 . 0 . 8 0 . 10 0 . 0 . 45 . 0 . 2 15. 5 2 5 . 5 . 0 . 0 . 13 0 . 0 . 0 . 0 . 0 . 0 0 . 0 . 0 . 0 . 35 . 0 . 0 . 0 . 0 . 1 4 30 . 1 10 . 5 . 3 0 . 0 . 0 30 . 0 . 5 . 15 . 45 . 5 . 0 . 8 2 0 . 5 . 15 25 . 10. 0 . 25 . 5 . 10 5 . 0 . 35 . 2 0 0 . 5 0 . 0 . 0 . 10. 1 1 1 5 . FEMALE S E R V I C E : ENTROPY MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 144 1 . 107 . 21 . 277 . 143 . 1 0 0 . 23 . 2 1 . 348 . 28 1 . 1229 . 78 . 26 . 2 . ' 3 2 182 . 603 . 3 . 75 . 13 . 9 . 154 . 66 . 18 . 298 . 155 . 7 . 2 . 10 . 30 3 29 . 3 . 83 1 . 19 . 3 . 2 . 1 . 1 . 494 . 118 . 132 . 2 . 6 . 0 . 4 1 4 3 0 9 . 36 . 7 . 2 0 0 . 14 . 10. 9 . 7 . 87 . 374 . 123 . 8 . 3 . 1. 6 5 44 . 2 . 1 . 5 . 355 . 158 . 0 . 0 . 1 1 . 5 . 5 7 6 . 226 . 7 . o . O 6 92 . 4 . 1 . 10. 406 . 245 . 1 . 1 . 22 . 1 1 . 5 6 5 . 258 . 8 . 0 . 1 7 99 . 383 . 1 . 47 . 5. 4 . 179 . 17 . 10 . 163 . 54 . 3 . 1 . 16 . 8 8 29 . 174 . 0 . 14 . 3 . 2 . 18 . 35 . 3 . 35 . 33 . 2 . 0 . 2 . 5 9 149 . 4 . 1 3 0 . 39 . 17 . 12 . 1 . 1 . 1877 . 34 . 7 9 0 . 9 . 48 . 0 . 4 10 6 5 5 . 222 . 107 . 7 7 9 . 3 0 . 2 1 . 49 . 28 . 184 . 2 2 9 5 . 26 1 . 17 . 5 . 5 . 172 1 1 377 . 18 . 29 . 46 . 3 1 3 . 1 2 0 . 4 . 4 . 6 5 7 . 47 . 1 4 3 8 6 . 3 14. 302 . 0 . 4 12 17 . 1 . O. 2 . 142 . 63 . 0 . 0 . 9 . 2 . 4 9 1 . 165 . 7 . O. O 13 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 3 . 0 . 24 . 0 . 7 . 0 . 0 14 25 . 62 . 0 . 12 . 2 . 1 . 7 2 . 7 . 2 . 48 . 16 . 1 . 0 . 8 4 7 . 4 15 24 . 28 . 39 . 16 . 2 . 1 . 5 . 2. 1 1 . 2 14. 15 . 1 . 0 . 1 . 113 1 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 18 10. 1 10. 10. 2 0 0 . 35 . 25 . 15 . 10. 1 0 0 . 75 . 1 6 3 0 . 4 0 . 25 . 10. 15 2 2 7 0 . 6 7 5 . 5 . 1 8 0 . 5 . 0 . 8 0 . 3 0 . 25 . 45 . 2 8 5 . 10 . 0 . 10. 5 3 5 0 . 0 . 805 . 45 . 10. 5 . 5 . 0 . 2 15. 2 6 0 . 2 5 5 . 5 . 5. 0 . 20 4 285 . 5 0 . 25 . 595 . 0 . 5 . 10. 5 . 2 0 . 3 0 . 165 . 5 . 0 . O . 0 5 0 . 10. 0 . 10 . 735 . 135 . 0 . 0 . 15 . 15 . 3 0 0 . 165 . 5 . 0 . 0 6 35 . 0 . 0 . 0 . 4 5 0 . 4 10. 0 . 0 . 6 0 . 0 . 4 4 0 . 2 3 0 . 0 . 0 . 0 7 5 0 . 3 7 0 . 0 . 4 0 . 5 . 0 . 335 . 35 . 5 . 5 . 1 10. 0 . 0 . 35 . 0 8 70 . 9 0 . 0 . 30 . 0 . 0 . 15 . 85 . 5 . 15 . 45 . 0 . 0 . 0 . O 9 15 . 10. 5 0 . 2 0 . 0 . 0 . 0 . 0 . 2 3 2 0 . 35 . 6 3 0 . 15 . 2 0 . 0 . 0 10 195 . 130 . 170 . 275 . 2 0 . 5 . 5 . 15. 125 . 3155 . 4 7 0 . 15 . 15 . 0 . 235 1 1 6 2 5 . 75 . 1 0 0 . 9 0 . 105 . 145 . 15. 10. 7 6 0 . 75 . 14 1 7 5 . 75 . 345 . O . 25 12 15 . 5 . 0 . O. 8 0 . 10. O. 0 . 45 . 0 . 2 15. 525 . 5 . O . 0 13 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 35 . 0 . 0 . 0 . 0 14 3 0 . 1 10. 5 . 3 0 . 0 . 0 . 3 0 . 0 . 5 . 15 . 45 . 5 . 0 . 8 2 0 . 5 15 25 . 10 . 0 . 25 . 5 . 10. 5. 0 . 35 . 2 0 0 . 5 0 . 0 . 0 . 10. 1115 FEMALE C L E R I C A L : I C C M - l b MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 a 9 10 1 1 12 13 1 A 1 5 1 5 4 3 9 . 2 10. 97 . 9 18. 158 . 208 . 46 . 58 . 8 5 8 . 1007 . 5 152 . 173 . 72 . 0 . 8 0 2 7 3 0 . 1033 . 0 . 2 4 0 . 0 . 0 . 327 . 14 1. 118 . 9 7 9 . 2 2 6 5 . 25 . 3 0 . o. 136 3 0 . 0 . 1346 . 0 . 0 . 0 . 0 . 0 . 1 2 7 0 . 3 16. 2 7 8 2 . 0 . 66 . 0 . 190 4 824 . 52 . 24 . 573 . 0 . 5 . 14 . 14 . 1 8 0 . 9 0 3 . 7 7 9 . 2 0 . 12 . 0 . 25 5 2 3 0 . 0 . 0 . 5 . 1026 . 5 7 0 . 0 . 1 . 77 . 0 . 2 0 0 9 . 326 . 22 . 0 . 0 6 552 . 0 . 0 . 3 0 . 1 0 1 0 . 1036 . 0 . 4 . 169 . 5. 2 9 9 3 . 4 2 0 . 35 . 0 . 1 7 226 . 375 . 0 . 87 . 0 . 0 . 265 . 23 . 36 . 3 12. 6 3 0 . 5 . 1 1 . o. 25 8 119 . 345 . 0 . 47 . 0 . 0 . 4 1 . 105 . 22 . 135 . 5 8 3 . 7 . 8 . 0 . 27 9 778 . 0 . 364 . 156 . 0 . 0 . 0 . 0 . 2 8 9 3 . 83 . 46 15. 46 . 94 . 0 . 26 to 1544 . 240 . 227 . 1405 . O. 0 . 4 1 . 48 . 523 . 3 9 8 3 . 2 7 4 5 . 35 . 49 . 0 . 325 I 1 2394 . 0 . 197 . 2 2 9 . 6 2 9 . 378 . 0 . 19. 2 0 1 6 . 149 . 3 0 4 0 2 . 672 . 3 10. 0 . 15 12 86 . 0 . 0 . 0 . 265 . 176 . 0 . 0 . 7 0 . ' 0 . 182 1. 247 . 2 1 . 0 . 0 13 3 . 0 . 1 . O. 0 . 0 . 0 . 0 . 17 . O. 1 4 0 . 2 . 6 . o. 0 14 0 . 0 . 0 . 0 . 0 . 0 . 3 4 6 . 0 . 0 . 0 . 3 7 6 . 0 . 257 . 12 5 0 . 0 15 0 . 0 . 203 . 0 . 0 . 0 . 0 . 0 . 0 . 8 4 9 . 762 . 0 . 27 . 0 . 1979 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 5 4 3 5 . 125 . 6 0 . 5 5 5 . 105 . 95 . 7 0 . 15. 3 7 0 . 185 . 7 3 6 5 . 45 . 3 0 . 5 . 15 2 1 8 2 0 . 1 135 . 25 . 5 2 0 . 25 . 2 0 . 195 . 155 . 7 0 . 1 5 0 . 1845 . 10. 15 . 25 . 15 3 3 8 0 . 15 . 1 6 5 0 . 2 3 0 . 5 . 15 . 2 0 . 0 . 745 . 1 155 . 1685 . 2 0 . 5 . 0 . 45 4 855 . 105 . 55 . 1 100 . 3 0 . 15. 2 0 . 0 . 145 . 150 . 9 10. 10. 0 . 15 . 15 5 140 . 5 . 0 . 5 . 1 0 6 0 . 5 0 0 . 5 . O. 3 0 . 15 . 2 1 7 5 . 305 . 25 . 0 . 0 6 265 . 10. 0 . 35 . 1 3 4 0 . 1265 . 5 . 5 . 8 0 . 25 . 2 7 9 0 . 4 2 0 . 15 . 0 . 0 7 335 . 3 1 0 . 5 . 1 2 0 . 10. 15 . 4 8 0 . 35 . 2 0 . 5 0 . 5 7 0 . 5 . 0 . 35 . 5 a 385 . 130. 5 . 9 0 . 10 . 15 . 5 0 . 1 4 0 . 15 . 35 . 5 3 5 . 10 . O . 10 . 10 9 130 . 10. 95 . 3 0 . 15 . 15 . 0 . 0 . 4 9 0 0 . 1 0 0 . 3 5 7 5 . 25 . 105 . 0 . 5 10 805 . 125 . 4 4 0 . 6 0 0 . 25 . 45 . 4 0 . 10. 3 8 0 . 5 8 7 5 . 2 3 2 5 . 4 0 . 3 0 . 15 . 4 10 1 1 1820 . 7 0 . 9 0 . 2 2 0 . 2 5 0 . 125 . 25 . 10. 14 15. 2 10. 3 2 2 6 5 . 105 . 7 7 0 . 1 5 . 20 1 2 8 0 . 0 . 0 . 5 . 1 9 0 . 2 4 0 . 5 . 0 . 10. 0 . 1 1 5 0 . 9 8 0 . 2 0 . 5 . 0 13 5 . 0 . 0 . 0 . 0 . 0 . o.. 0 . 0 . 0 . 165 . 0 . 0 . 0 . 0 1 4 255 . 190 . 5 . 95 . 5 . 5 . 140 . 40 . 5 . 5 0 . 3 0 O . 5 . O. 1 120 . 1 5 15 165 . 3 0 . 3 0 . 65 . 15 . 5 . 25 . 5 . 7 0 . 7 2 0 . 4 0 0 . 0 . 5 . 5 . 2 2 8 0 1J1 MALE S E R V I C E : I C C M - l b MODEL P R E D I C T E D COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 104 8 . 8 1 . 14 . 256 . 73 . 72 . 10 . 23 . 235 . 184 . 1 198 . 79 . 18 . 0 . O 2 149 . 447 . 0 . 65 . 0 . O. 89 . 59 . 1 1 . 1 9 0 . 42 1. 12 . 7 . 0 . O 3 0 . 0 . 688 . 0 . 0 . 0 . 0 . 0 . 527 . 0 . 5 5 7 . 0 . 23 . 0 . 0 4 2 16. 27 . 5 . 222 . 1 . 7 . 4 . 8 . 66 . 263 . 2 2 0 . 1 2 . 4 . O. 0 5 42 . 0 . 0 . 0 . 345 . 157 . 0 . 1 . 16 . 0 . 4 7 0 . 145 . 5 . 0 . 0 6 104 . 0 . 0 . 5 . 356 . 295 . 0 . 2 . 39 . 0.- 703 . 194 . 9 . 0 . 0 7 72 . 255 . 0 . 36 . 0 . 0 . 1 1 1 . 15 . 2 . 9 1 . 149 . 4 . 4 . 0 . 0 8 22 . 138 . 0 . 1 1 . 0 . 0 . 10 . 4 0 . 0 . 18 . 96 . 3 . 2 . 0 . 0 9 169 . 0 . 127 . 43 . 0 . 2 . 0 . 0 . 9 9 0 . 0 . 12 17. 24 . 27 . 0 . 0 10 4 10. 129 . 7 0 . 548 . 0 . 0 . 11 . 27 . 17 1. 1 1 6 7 . 6 0 2 . 22 . 15 . 0 . 0 1 1 743 . 0 . 54 . 85 . 407 . 2 13. 0 . 14 . 9 2 5 . 0 . 1 2 9 7 0 . 5 0 3 . 131 . 0 . 0 12 1 3 . 0 . 0 . 0 . 8 1 . 45 . 0 . 0 . 13 . 0 . 3 7 3 . 96 . 4 . 0 . 0 1 3 1 . 0 . 0 . O. 0 . 0 . 0 . 0 . 5 . 0 . 37 . 1 . 2 . 0 . 0 14 0 . 0 . 0 . 0 . 0 . O. 176 . 26 . 0 . 0 . 0 . 0 . 7 1 . 4 9 0 . 0 15 0 . 0 . 93 . 0 . 0 . 0 . 0 . 0 . 0 . 4 16 . O. 0 . 15 . 0 . 70 1 E M P I R I C A L COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 1 4 15 1 1025 . 8 0 . 25 . 1 6 0 . 4 0 . 6 0 . 35 . 5 . 1 1 5 . 95 . 1595 . 35 . 15 . 5 . 0 2 245 . 445 . 0 . 130 . 5 . O. 55 . 6 0 . 15 . 20 . 4 6 0 . 15 . 0 . 0 . 0 3 1 6 0 . 10. 6 1 0 . 2 0 . 0 . 15 . 0 . 0 . 2 3 5 . 2 4 0 . 4 8 0 . 5 . 15 . 0 . 5 4 185 . 55 . 10. 445 . 5 . 0 . 10. 5 . 45 . 45 . 245 . 0 . 5 . 0 . 0 5 G5 . 0 . 0 . 10. 4 10. 140. 0 . 5 . 15 . 10 . 3 7 0 . 1 4 0 . 5 . 10. 0 e 5 0 . 10. 10. 15 . 4 6 0 . 3 1 0 . 0 . 10. 45 . 0 . 5 9 5 . 195 . 0 . 0 . 5 7 65 . 1 4 0 . 0 . 55 . 25 . 0 . 1 9 0 . 5 . 5 . 10. 2 15 . 0 . 0 . 25 . 5 8 45 . 50 . 0 . 10 . 0 . 5 . 15 . 75 . 5 . 10 . 1 2 0 . O . 0 . 5 . O 9 6 0 . 15 . 4 0 . 5 0 . 15 . 5 . 0 . 0 . 1435 . 15 . 9 0 5 . 10. 5 0 . 0 . 0 10 2 10. 65 . 205 . 160. 35 . 2 0 . 15 . 0 . 155 . 1 5 4 0 . 6 3 0 . 10. 5 . 5 . 1 15 1 1 755 . 6 0 . 115. 145 . 235 . 2 0 0 . 40 . 3 0 . 8 9 0 . 1 30 . 1 2 7 9 0 . 3 8 5 . 2 3 0 . 15 . 25 12 25 . 0 . 0 . 0 . 3 0 . 3 0 . 0 . 0 . 10. 15. 2 2 0 . 2 9 5 . 0 . 0 . 0 1 3 0 . O. 0 . 10 . 0 . 0 . O. 0 . O. 0 . 35 . O. O . 0 . 0 14 5 0 . 105 . • 0 . 25 . 5 . 0 . 4 0 . 15 . 0 . 5 . 95 . 5 . 0 . 4 10 . 5 15 5 0 . 4 0 . 4 0 . 35 . 0 . 5 . 10. 5 . 3 0 . 2 0 0 . 2 4 5 . 0 . 5 . 15 . 5 4 5 MALE S E R V I C E : ENTROPY MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 807 . 87 . 31 . 204 . 130. 101 . 27 . 25 . 273 . 189 . 1288 . 89 . 2 1 . 6 . 12 2 197 . 322 . 8 . 85 . 25 . 20. 1 10. 59 . 35 . 2 12. 3 14. 17 . 3 . 19 . 24 3 68 . 8 . 602 . 4 1 . 1 1 . 9 . 3 . 2 . 490. 142 . 362 . 8 . 10. 1 . 38 4 243 . 36 . 13 . 14G . 23 . 18 . 12 . 10. 92 . 2 10. 224 . 16 . 4 . 3 . 6 5 53 . 4 . 2 . 9 . 202 . 113. 1 . 1 . 18 . 9 . 604 . 155 . 6 . 0. 1 6 117. 8 . 4 . 19 . 288 . 200. 3 . 3 . 40. 20. 77 1 . 220. 9 . 1 . 2 7 99 . 182 . 3 . 48 . 10. 8 . 96 . 17 . 17 . 107 . 114. 8 . 2 . 20. 7 8 38 . 96 . 1 . 18 . 7 . 5 . 17 . 27 . 7 . 33 . 75 . 5 . 1 . 4 . 5 9 177 . 9 . 127 . 55 . 32 . 25 . 3 . 3 . 1017 . 46 . 1042 . 22 . 37 . 1 . 6 10 509 . 163 . 1 10. 475 . 47 . 37 . 50. 33 . 192 . 947 . 47 1 . 33 . 8 . 12 . 84 1 1 54 1 . 42 . 68. 98. 400. 201 . 13. 13. 756 . 92 . 13167. 425 . 2 18. 3. 8 12 19 . 1 . 1 . 3 . 73 . 4 1 . 0. O. 1 1 . 3 . 378 . 87 . 5 . 0 . O 13 1 . 0 . 0. 0. 1 . 0 . 0. 0. 4 . 0 . 34 . 1 . 4 . 0 . 0 14 45 . 60. 2 . 22 . 6 . 5 . 6 1 . 1 1 . 7 . 55 . 58 . 4 . 1 . 4 18. 5 15 75 . 57 . 83 . 45. 10. 8 . 14 . 8 . 39 . 269 . 96 . 7 . 1 . 4 . 507 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1025 . 80. 25 . 160. 40. 60. 35. 5. 115. 95 . 1595 . 35 . 15 . 5 . 0 2 245 . 445 . 0 . 130. 5 . 0. 55. 60 . . 15 . 20. 460. 15 . 0 . 0. 0 3 160. 10. 6 10. 20. O. 15 . 0. 0. 235 . 240. 480. 5 . 15. 0. 5 4 185 . 55 . 10. 445 . 5 . 0. 10. 5. 45 . 45 . 245 . 0 . 5 . 0. 0 5 65 . 0 . 0 . 10. 4 10. 140. 0. 5 . 15 . 10. 370. 140 . 5 . 10. 0 6 50. 10. 10. 15 . 460. 310. 0. IO. 45 . 0 . 595 . 195 . 0. 0. 5 7 65 . 140. 0 . 55 . 25 . 0 . 190. 5 . 5 . 10. 2 15. O. 0. 25 . 5 8 45 . 50. 0. 10. 0. 5 . 15 . 75 . 5 . 10. 120. 0 . 0. 5 . 0 9 60. 15 . 40. 50. 15 . 5 . 0. 0. 1435 . 15 . 905 . 10. 50. 0 . 0 10 2 10. 65 . 205 . 160. 35 . 20 . 15 . O. 155 . 1540. 630. 10. 5 . 5 . 1 15 1 1 755 . 60. 115. 145 . 235. 200. 40. 30. 890. 130. 12790. 385 . 230. 15 . 25 12 25 . 0 . 0. 0. 30. 30. 0. 0 . 10. 15 . 220. 295 . 0. 0 . 0 13 0. 0. 0 . 10. 0. 0. 0 . 0. 0 . 0 . 35 . 0. 0 . 0 . 0 14 50. 105 . 0. 25 . 5 . 0. 40. 15 . 0. 5 . 95 . 5 . 0. 4 10 . 5 15 50 . 40. 40. 35 . 0. 5 . 10. 5 . 30. 200. 245 . 0 . 5 . 15 . 545 MALE S A L E S : I C C M - l b MODEL PREDICTED COMMUTING FLOWS 10 12 13 15 1 1743 . 7,0. 3 0 . 2 3 9 . 24 . 49 . 13 . 25 . 3 2 5 . 305 . 1436 . 63 . 2 . 0 . 2 1 2 296 . 393 . 0 . 75 . 0 . 0 . 118 . 69 . 54 . 338 . 7 2 0 . 0 . 1 . 0 . 46 3 0 . 0 . 56 1 . 0 . 0 . 0 . 0 . 0 . 594 . 107 . 954 . 0 . 2 . 0 . 65 4 2 6 0 . 17 . 7 . 144 . 0 . O. 4 . 6 . 66 . 27 1 . 2 12. 6 . 0 . O. 7 5 8 0 . 0 . 0 . 3 . 294 . 2 13. 0 . 0 . 3 0 . 0 . 5 5 7 . 137 . 1 . 0 . 0 6 298 . 0 . 0 . 16 . 45 1 . 6 16. 0 . 3 . 103 . 0 . 1 3 1 9 . 278 . 2 . 0 . 0 7 105 . 160 . 0 . 3 1 . 0 . 0 . 109 . 13 . 19 . 12 1. 2 2 5 . 0 . 0 . 0 . 7 8 59 . 148 . 0 . 17 . 0 . 0 . 16 . 58 . 12 . 51 . 2 0 9 . 0 . 0 . 0 . 9 9 332 . 0 . 149 . 54 . 0 . 0 . 0 . 0 . 1334 . 25 . 157 1 . 10 . 4 . 0 . 1 10 675 . 102 . 94 . 466 . 0 . 0 . 1 3 . 26 . 255 . 1 562 . 9 8 0 . 0 . 2 . • 0 . 1 3 4 1 1 8 5 9 . 0 . 67 . 7 1 . 153 . 95 . 0 . 9 . 8 15. 39 . 9 0 6 4 . 288 . 1 1 . 0 . 0 12 75 . 0 . 0 . 0 . 158 . 132 . 0 . 0 . 59 . 0 . 1 1 1 2 . 227 . 1 . 0 . 0 1 3 1 . 0 . 0 . 0 . 0 . 0 . O. 0 . 5 . 0 . 28 . 0 . O. 0 . o 14 0 . 0 . 0 . 0 . 0 . 0 . 152 . 0 . 0 . 0 . 3 18. 0 . 8 . 4 5 0 . 0 15 0 . 0 . 8 1 . 0 . 0 . 0 . 0 . 0 . 0 . 326 . 2 8 0 . 0 . 1 . 0 . 9 17 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 1 4 15 1 1 4 0 0 . 85 . 4 0 . 155 . 55 . 45 . 4 0 . 5 . 3 10. 1 10 . 2 0 3 0 . 2 0 . 0 . 5 . 45 2 475 . 425 . 10. 140 . 35 . 35 . 75 . 45 . 9 0 . 1 0 0 . 6 10. 3 0 . 0 . 2 0 . 20 3 1 7 0 . 15 . 545 . 85 . 15 . 5 . 0 . 5. 3 1 0 . 3 7 0 . 7 6 0 . 0 . 0 . 0 . 5 4 245 . 15 . 15 . 2 5 0 . 2 0 . 15 . 0 . 10. 55 . 1 0 0 . 2 6 5 . 0 . 0 . 0 . 10 5 130. 0 . 0 . 5 . 305 . 140 . 0 . 0 . 50 . 10 . 5 7 0 . 1 0 0 . O. O. 5 6 2 5 0 . 5 . 25 . 3 0 . 365 . 5 7 0 . 15 . 10. 125 . 25 . 1475 . 1 7 0 . 10 . 0 . 10 7 160 . 9 0 . 0 . 3 5 . 15 . 10. 145 . 5 . 3 0 . 5 0 . 2 3 0 . 0 . 5 . 5 . 10 8 90 . 4 0 . 0 . 35 . 0 . 10. 10. 9 0 . 5 0 . 45 . 2 0 5 . 5 . 0 . 0 . O 9 2 2 0 . 25 . 4 0 . 5 0 . 2 0 . 3 0 . 5 . 0 . 1 5 7 0 . 65 . 1425 . 10. 5 . 0 . 15 10 435 . 6 0 . 235 . 160 . 25 . 35 . 2 0 . 5 . 2 8 0 . 1880 . 9 4 0 . 2 0 . 0 . 15 . 2 0 0 1 1 8 0 0 . 45 . 55 . 95 . 125 . 135 . 40 . 15 . 6 8 5 . 1 1 5 . 9 185 . 100 . 2 0 . 15 . 4 0 12 105 . 0 . 0 . 25 . 7 0 . 55 . 0 . 5 . 25 . 2 0 . 9 1 0 . 5 5 0 . 0 . 0 . 0 13 0 . 0 . 0 . 0 . 0 . 0 . O. 0 . O. 5 . 3 0 . 0 . 0 . 0 . 0 14 155 . 8 0 . 0 . 5 . 0 . 5 . 55 . 10. 3 0 . 3 0 . 1 5 0 . 5 . 0 . 3 9 0 . IO 15 1 5 0 . 5 . 25 . 45 . 3C*. 15 . 5 . 5 . 6 0 . 2 2 0 . 2 0 5 . 0 . 0 . 0 . 8 4 0 >o CD M A L E . S A L E S : ENTROPY MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 a 9 10 1 1 12 13 14 15 1 1221 . 74 . 36 . 178 . 1 1 0 . 133 . 29 . 2 3 . 3 8 0 . 273 . 176 1 . 85 . 4 . 7 . 3 1 2 397 . 27 1 . 13 . 94 . 3 0 . 36 . 1 15. 5 8 . 6 9 . 344 . 572 . 23 . 1 . 23 . 64 3 138 . 9 . 523 . 44 . 12 . 15 . 4 . 3 . 647 . 2 14. 577 . 10 . 2 . 1 . 86 4 2 7 3 . 22 . 11 . 87 . 15 . 18 . 9 . 7 . 94 . 195 . 243 . 12 . 0 . 2 . 1 1 5 93 . 4 . 3 . 9 . 1 4 0 . 126 . 2 . 1 . 29 . 16 . 7 6 9 . 1 1 9 . 1 . 0 . 2 6 3 14. 13 . 8 . 31 . 32 1 . 35 1 . 5 . 5. 98 . 52 . 1 5 9 9 . 273 . 2 . 2 . 9 7 154 . 116 . 4 . 4 0 . 9 . 1 1 . 73 . 14 . 27 . 134 . 166 . 8 . 0 . 18 . 15 8 96 . 97 . 2 . 25 . 10. 12 . 23 . 3 1 . 18 . 69 . 168 . 7 . O. 6 . 15 9 324 . 10. 132 . 57 . 32 . 39 . 4 . 3 . 1254 . 8 0 . 1496 . 25 . 6 . 1 . 16 10 8 7 9 . 1 4 0 . 123 . 4 10. 49 . 59 . 54 . 33 . 301 . 1238 . 7 8 0 . 38 . 2 . 15 . 189 1 1 5 7 5 . 26 . 5 0 . 6 3 . 202 . 165 . 10 . 9 . 631 . 96 . 9 3 7 6 . 2 30 . 19 . 3 . 15 12 88 . 4 . 3 . 9 . 132 . 119 . 2 . 1 . 44 . 15 . 1 1 7 8 . 166 . 2 . 1 . 2 13 1 . 0 . 0 . 0 . 0 . 0 . O. 0 . 4 . 0 . 28 . • 0 . 0 . 0 . 0 14 105 . 59 . 4 . 28 . 8 . 9 . 67 . 13 . 17 . 101 . 125 . 6 . 0 . 366 . 17 15 127 . 44 . 76 . 4 0 . 9 . 1 1 . 14 . 8 . 58 . 3 19. 1 5 0 . 7 . 0 . 4 . 737 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1 4 0 0 . 85 . 4 0 . 155 . 5 5 . 45 . 4 0 . 5 . 3 10. 1 10. 2 0 3 0 . 2 0 . 0 . 5 . 45 2 4 7 5 . 425 . 10. 1 4 0 . 3 5 . 35 . 7 5 . 45 . 9 0 . 1 0 0 . 6 1 0 : 3 0 . 0 . 2 0 . 20 3 1 7 0 . 15 . 5 4 5 . 8 5 . 15 . 5 . 0 . 5. 3 1 0 . 3 7 0 . 7 6 0 . 0 . 0 . 0 . 5 4 2 4 5 . 15 . 15 . 2 5 0 . 2 0 . 15 . 0 . 10 . 55 . 1 0 0 . 2 6 5 . 0 . 0 . 0 . 10 5 1 3 0 . 0 . 0 . 5 . 305 . 140 . 0 . 0 . 5 0 . 10 . 5 7 0 . 1 0 0 . 0 . 0 . 5 6 2 5 0 . 5 . 25 . 3 0 . 365 . 5 7 0 . 15 . 10. 125 . 25 . 1475 . 1 7 0 . 10. O. 10 7 1 6 0 . 9 0 . 0 . 35 . 15 . 10 . 145 . 5 . 3 0 . 5 0 . 2 3 0 . 0 . 5 . 5 . 10 8 9 0 . 4 0 . 0 . 35 . 0 . 10. 10. 9 0 . 5 0 . 45 . 205 . 5 . 0 . 0 . 0 9 2 2 0 . 25 . 4 0 . 5 0 . 2 0 . 3 0 . 5 . 0 . 1 5 7 0 . 65 . 1425 . 10. 5 . 0 . 15 10 4 3 5 . 6 0 . 235 . 1 6 0 . 25 . 35 . 2 0 . 5 . 2 8 0 . 1 8 8 0 . 9 4 0 . 2 0 . 0 . 15 . 2 0 0 1 1 8 0 0 . 45 . 55 . 95 . 125 . 135 . 4 0 . 15 . 685 . 1 1 5 . 9 1 8 5 . 100 . 2 0 . 15 . 40 12 105 . 0 . 0 . 25 . 7 0 . 55 . 0 . 5. 25 . 2 0 . 9 10 . 5 5 0 . 0 . 0 . 0 13 0 . 0 . 0 . 0 . 0 . O. 0 . 0 . 0 . 5 . 3 0 . 0 . 0 . 0 . 0 14 155 . 8 0 . 0 . 5. 0 . 5 . 5 5 . 10. 3 0 . 3 0 . 150 . 5 . 0 . 3 9 0 . 10 15 1 5 0 . 5 . 25 . 45 . 3 0 . 15 . 5 . 5 . 6 0 . 2 2 0 . 205 . 0 . O. 0 . 8 4 0 MALE M A N A G E R I A L : I C C M - l b MODEL P R E D I C T E D COMMUTING FLOWS 1 2 3 4 5 6 7 . 8 9 10 1 1 12 13 14 15 1 1800. 7 1 . 16 . 260. 0. 4 . 12 . 32 . 354 . 304 . 1757 . 27 . 9 . 0 . 18 2 3 16. 393 . 0. 88 . 0. 0. 119. 90. 37 . 336 . 878 . 0. 4 . 0 . 45 3 0. 0. 849 . 0. 0 . 0. 0. 0. 938 . 179 . 17 10. 0. 13 . 0 . 9 1 4 266 . 17 . 4 . 152 . 0 . 0. 4 . 8 . 7 1 . 261 . 256 . 0. 1 . 0. 6 5 73 . 0. O. 4 . 265 . 183 . 0. 1 . 24 . 0 . 576 . 82 . 2 . 0 . 0 G 452 . 0. 0. 32 . 661 . 888 . 0. 5 . 146 . 6 . 2289 . 27 1 . 10. 0. 0 7 132 . 188 . 0. 42 .' 0 . 0. 129 . 19. 13 . 14 1. 322 . 0 . 2 . 0. 7 8 72 . 167 . 0. 23 . 0 . 0 . 18 . 85 . 7 . 59 . 288 . 0 . 1 . 0 . 10 9 387 . 0. 155 . 7 1 . 0. 0. 0. 0. 1650. 37 . 2095 . 0 . 16 . 0. 0 10 683 . 99 . 66 . 487 . 0 . 0. 1 1 . 33 . 245 . 1464 . 1147. 0. 6 . 0. 134 1 1 848 . 0. 36 . 82 . 87 . 7 . 0. 12 . 845 . 49 . 10325. 164 . 39 . 0 . 0 12 132 . 0. 0. 0. 236 . 178 . 0 . 0 . 89 . 0 . 2 128 . 247 . 9 . 0. 0 13 3 . 0. 0. 0. 0. 0. O. 0. 1 3 . 0 . 87 . 0 . 1 . 0. O 14 0. 0. 0. 0. 0 . 0. 149 . 0. 0 . 0 . 283 . 0 . 49 . 495 . 0 15 0. 0. 30. 0. 0 . 0. 0 . 0. 0 . 374 . 373 . 0. 3 . 0 . 1090 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 14 10. 125 . 65 . 145 . 50. 95 . 35 . 10. 2 15. 150. 2280 . 35 . 15 . 15 . 20 2 425 . 240. 30. 155 . 45 . 40. 100. 55 . 75. 105 . 940. 10. 0. 20. 65 3 365 . 45 . 540. 135 . 15 . 40. 15 . 5 . 7.10. 400. 1 460 . 10. 0 . 5 . 35 4 230. 15 . 30. 270. 10. 10. 5 . 5. 50. 60 . 340. 0 . 0 . 5 . 15 5 75 . 20. 5 . 15 . 265 . 70. 0 . 10. 40. 5 . 625 . 65 . 10. O. 5 6 445 . 35 . 20 . 45 . 425 . 590 . O. 5 . 130. 60 . 2820. 130. 30. O. 25 7 190. 90. 5 . 50. 15 . 15 . 125 . 25 . 40. 45 . 365 . 5 . 0 . 15 . 10 8 120. 30. 0 . 45 . 5 . 10. 25 . 115. 15 . 75 . 255 . 20. O. 5 . 10 9 285 . 15 . 85 . 50. 45 . 35 . 0. 0. 1660. 105 . 2065 . 20. 5 . O. 40 10 375 . 100. 190. 140. 40. 40. 25 . 15 . 375 . 1700. 1 120. 0. 5 . 30. 220 1 1 755 . 90. 1 10. 100. 165 . 125 . 25 . 15 . 930. 125 . 9790. 95 . 90. 10. 70 12 230. 30 . 30. 40 . 145 . 180. 10. 5 . 1 10. 5 . 1840. 385 . 10. 0 . 0 13 0. 0. 0. 0. 0 . 0 . 0. 0. 10. 5 . 9 0 : 0. 0. 0 . 0 14 150. 70. 5 . 5 . 10. 5 . 40. 15 . 20 . 40. 2 10 . 10. 0. 385 . 10 15 1 10 . 30. 40. 45 . 15 . 5 . 35 . 5 . 50. 330. 3 15. O. 0. 5 . 885 MALE M A N A G E R I A L : ENTROPY MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1110. 82 . 4 1 . 184 . 112. 130. 34 . 32 . 407 . 279 . 2125. 59 . 14 . 13 . 44 2 394 . 244 . 16 . 100. 34 . 39 . 109 . 67 . 87 . 326 . 755 . 18 . 3 . 35 . 78 3 248 . 20. 638 . 8 1 . 25 . 29 . 8 . 8 . 982 . 34 1 . 1209 . 13 . 12 . 4 . 16 1 4 251 . 24 . 12 . 82 . 17 . 19 . 1 1 . 9 . 100. 175 . 314 . 9 . 2 . 4 . 15 5 88 . 5 . 3 . 10. 105 . 94 . 2 . 2 . 32 . 17 . 78 1 . 61 . 4 . 1 . 3 6 480. 25. 16 . 57 . 407 . 432 . 1 1 . 12 . 176 . 94 . 2776 . 234 . 14 . 6 . 2 1 7 180. 12 1. 6 . 50. 13 . 15 . 76 . 20. 40. 148 . 266 . 7 . 2 . 29 . 23 8 111. 97 . 3 . 31 . 12 . 14 . 26 . 37 . 27 . 78 . 253 . 6 . 1 . 1 1 . 22 9 401 . 16 . 146 . 79. 44 . 51 . 7 . 7 . 1340. 1 10. 2129. 23 . 25 . 3 . 29 10 828 . 143 . 118. 381 . 55 . 63 . 59 . 43 . 330. 1046 . 1036 . 29 . 7 . 24 . 2 12 1 1 635 . 36 . 59 . 82 . 2 11. 174 . 15 . 15 . 703 . 124 . 10185. 157 . 66 . 7 . 25 12 160. 8 . 8 . 19 . 190. 17 1. 4 . 4 . 90. 31 . 2162. 154 . 1 1 . 2 . 7 13 4 . 0. 1 . 1 . 1 . 1 . 0. O. 10. 1 . 82 . 1 . 2 . 0. 0 14 1 10. 58 . 5. 30. 9 . 1 1 . 60. 16 . 22 . 99 . 178 . 5 . 1 . 348 . 22 15 163 . 55 . B3 . 53 . 14 . 16 . 19 . 13 . 84 . 342 . 263 . 7 . 2 . 9 . 748 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 ' 10 1 1 12 13 14 15 1 14 10. 125 . 65 . 145 . 50. 95. 35 . 10. 2 15. 150. 2280. 35 . 15 . 15 . 20 2 425 . 240. 30. 155 . 45 . 40. 100. 55 . 75 . 105 . 940. 10. 0. 20. 65 3 365 . 45 . 540. 135. 15. 40. 15 . 5 . 7 10. 400. 1460. 10. 0. 5 . 35 4 230. 15 . 30. 270. 10. 10. 5 . 5. 50. 60. 340. 0. 0. 5 . 15 5 75 . 20. 5 . 15 . 265 . 70. 0. 10. 40. 5 . 625 . 65 . 10. 0. 5 6 445 . 35. 20. 45 . 425 . 590. 0. 5. 130. 60. 2820. 130. 30. 0. 25 7 190 . 90. 5 . 50. 15 . 15 . 125. 25 . 40. 45 . 365 . 5 . 0. 15 . 10 8 120. 30. 0. 45 . 5. 10. 25 . 115. 15 . 75 . 255 . 20. 0. 5 . 10 9 285 . 15 . 85 . 50. 45 . 35 . O. 0. 1660. 105 . 2065 . 20. 5 . 0 . 40 10 375 . 100. 190. 140. 40. 40. 25 . 15 . 375 . 1700. 1 120. 0. 5 . 30. 220 1 1 755 . 90. 1 IO. 100. 165 . 125 . 25 . 15 . 930. 125 . 9790 . 95 . 90. 10 . 70 12 230. 30. 30. 40. 145 . 180. 10. 5 . 1 10. 5 . 1840. 385 . 10. 0. 0 13 0. 0. 0. 0. 0. 0. 0. 0. 10. 5 . 90. 0. 0. 0. 0 14 150. 70. 5 . 5 . 10. 5 . 40. 15 . 20. 40. 2 10. 10. 0. 385 . 10 15 1 10. 30. 40. 45. 15 . 5 . 35 . 5 . 50. 330. 315 . 0. 0. 5 . 885 r o O MALE MANUFACTURING: I C C M - l b MODEL P R E D I C T E D COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 . 1 1 12 13 14 15 1 2 7 6 7 . 115 . 9 5 . 698 . 2 6 9 . 2 7 9 . 51 . 104 . 8 8 7 . 55 1 . 1598 . 31 . 7 . 0 . 34 2 4 2 0 . 53 1 . 4 . 168 . 37 . 51 . 3 8 5 . 256 . 179 . 537 . 72 1 . 1 1 . 3 . 0 . 72 3 0 . 0 . 1 194 . 0 . 0 . 0 . 0 . 0 . 1337 . 95 . 8 7 2 . 5 . 6 . 0 . 9 1 4 5 0 0 . 33 . 28 . 523 . 35 . 37 . 19. 3 0 . 222 . 621 . 2 8 7 . 5 . 1 . 0 . 14 5 101 . 0 . 1 . 0 . 6 3 9 . 388 . 0 . 2 . 72 . 0 . 4 8 9 . 37 . 2 . 0 . 0 6 186 . 0 . 3 . 8 . 52 1 . 557 . 0 . 5 . 122 . O. 5 7 7 . 39 . 2 . 0 . O 7 166 . 24 1 . 0 . 76 . 7 . 14 . 391 . 53 . 7 3 . 2 11 . 252 . 4 . 1 . 0 . 1 1 8 6 1 . 147 . 0 . 26 . 8 . 12 . 3 9 . 157 . 32 . 5 5 . 154 . 3 . 1 . 0 . 10 9 404 O. 308 . 95 . 62 . 72 . O. O. 2 6 8 5 . O. 1334 . 12 . 9 . 0 . 0 10 1218 . 195 . 3 11. 1549 . 72 . 100 . 6 6 . 129 . 8 5 8 . 3 3 3 2 . 1251 . 22 . 7 . 0 . 284 1 1 1592 . 13. 239 . 177 . 927 . 633 . 0 . 51 . 2 5 6 5 . 0 . 1 1 6 5 6 . 13 1. 38 . 0 . 0 12 15 . 0 . 0 . 0 . 68 . 49 . 0 . O. 23 . O. 159 . 10 . 1 . O. 0 13 1 . 0 . 1 . 0 . 1 . 1 . 0 . 0 . 8 . 0 . 19 . 0 . 0 . 0 . 0 14 0 . 0 . 0 . 0 . 0 . 0 . 48 1 . 47 . 0 . 0 . 6 2 8 . 0 . 24 . 1 180 . 0 15 0 . 0 . 273 . 0 . 0 . 0 . 0 . 0 . 152 . 5 0 0 . 353 . 5 . 3 . 0 . 1799 E M P I R I C A L COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1895 . 130. 2 15. 4 2 0 . 3 2 0 . 2 2 0 . 100 . 8 0 . 755 . 3 4 0 . 2 8 6 0 . 25 . 5 . 40 . 8 0 2 745 . 2 6 0 . 85 . 3 8 0 . 75 . 95 . 195 . 2 10. 2 3 5 . 2 4 5 . 735 . 0 . 5 . 5 0 . 6 0 3 3 5 0 . 65 . 5 6 0 . 165 . 5 0 . 35 . 5 0 . 10 . 9 4 0 . 5 15. 7 8 5 . 0 . 0 . 0 . 75 4 305 . 95 . 195 . 6 2 5 . 6 0 . 3 0 . 8 0 . 35 . 2 0 5 . 185 . 5 0 5 . 5 . O. 5 . 25 5 140 . 5 . 3 0 . 4 0 . 5 6 0 . 445 . 10. 0 . 65 . 45 . 345 . 35 . 0 . 5 5 6 2 2 0 . 25 . 3 0 . 10. 445 . . 4 8 0 . 10. 15 . 1 1 5 . 35 . 5 8 0 . 55 . 0 . 0 . 0 7 285 . 120. 25 . 125 . 3 0 . 5 0 . 295 . 65 . 6 0 . 9 0 . 275 . O. 0 . 6 0 . 20 8 8 0 . 55 . 10. 5 0 . 3 0 . 3 0 . 5 5 . 1 2 0 . 4 0 . 3 0 . 2 0 0 . 5 . 0 . 0 . 0 9 285 . 45 . 2 10. 115 . 55 . 45 . 10. 0 . 2 6 0 5 . 95 . 1 4 4 0 . 10. 25 . 5 . 35 10 9 0 0 . 205 . 6 5 5 . 7 4 0 . 135 . 1 6 0 . 105 . 45 . 102 5 . 3 0 1 5 . 1685 . 2 0 . 10. 50 . 6 4 5 1 1 1765 . 105 . 335 . 385 . 7 7 0 . 5 0 0 . 170 . 1 0 0 . 2 8 6 5 . 5 4 5 . 1 0 2 3 5 . 55 . 65 . 25 . 100 12 25 . 0 . 0 . 5 . 45 . 40 . 5 . 0 . 2 0 . 5 . 105 . 75 . O. O . 0 13 2 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . O. . o . 10 . O. 0 . 0 . 0 14 2 0 0 . 140 . 3 0 . 140 . 2 0 . 3 0 . 305 . 1 2 0 . 55 . 7 0 . 2 8 0 . 25 . 0 . 9 2 5 . 20 15 2 15. 3 0 . 8 0 . 120 . 5 0 . 3 0 . 4 0 . 35 . 2 3 0 . 6 7 5 . 305 . 5 . 5 . 15 . 1250 MALE MANUFACTURING: ENTROPY MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1689 . 137 . 15 1. 524 . 374 . 353. 132 . 111. 1119. 588 . 2 112. 39 . 12 . 39. 105 2 522 . 283 . 50. 237 . 99 . 94 . 295 . 169 . 2 19. 527 . 653 . 10. 2 . 75 . 137 3 2 10. 18 . 840. 120. 47 . 44 . 18 . 15 . 1 170. 335 . 606 . 5 . 5 . 7 . 160 4 486 . 50. 55 . 277 . 74 . 70. 52 . 40. 348 . 42 1 . 4 15. 8 . 2 . 15 . 44 5 155 . 10. 14 . 35 . 310. 233 . 10. 9. 102 . 43 . 759 . 35 . 3 . 3 . 10 6 229 . 14 . 19 . 52 . 339 . 296 . 14 . 13 . 152 . 63 . 768 . 38 . 3 . 5 . 15 7 235. 137 . 19. 115. 38. 36. 194. 52 . 99. 236. 233. 4 . 1 . 58 . 42 8 100. 73 . 8 . 49 . 24 . 23 . 48 . 59 . 45 . 86 . 145 . 2 . 1 . 16 . 26 9 449 . 2 1 . 312. 162 . 107 . 101 . 21 . 19 . 2 145. 168 . 1396 . 1 1 . 13 . 6 . 47 10 1499 . 260. 452 . 1 156 . 227 . 2 14. 251 . 167 . 1072 . 22 10 . 128 1 . 24 . 7 . 8 1 . 495 1 1 1349 . 87 . 278 . 333 . 866 . 603 . 84 . 76 . 2401 . 374 . 1 1273. 122 . 62 . 29 . 84 12 23 . 1 . 3 . 5 . 47 . 35 . 1 . 1 . 23. 6 . 168 . 7 . 1 . 1 . 2 13 2 . 0. 1 . 0. 1 . 1 . 0. 0. 6 . 1 . 18 . 0 . 0 . 0 . 0 14 234 . 109 . 24 . 114. 44 . 42 . 243 . 65. 91 . 254 . 251 . 5 . 1 . 822 . 6 1 15 248 . 79 . 234 . 14 1. 47 . 45 . 66 . 40. 223 . 578 . 267 . 5 . 1 . 23 . 1087 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1895 . 130. 2 15. 420. 320. 220. 100. 80. 755 . 340. 2860 . 25 . 5 . 40. 80 2 745 . 260 . 85 . 380. 75 . 95 . 195 . 2 10. 235 . 245 . 735 . 0 . 5 . 50. 60 3 350 . 65 . 560. 165 . 50. 35 . 50. 10. 940. 515. 785 . 0 . 0 . 0 . 75 4 305 . 95 . 195 . 625 . 60. 30. 80. 35 . 205 . 185 . 505 . 5 . 0 . 5 . 25 5 140. 5 . 30. 40. 560. 445 . 10. 0. 65 . 45 . 345 . 35. 0. 5 . 5 G 220. 25 . 30. 10. 445 . 480. 10. 15 . 115. 35 . 580. 55 . 0 . 0 . 0 7 285 . 120. 25 . 125 . 30. 50. 295 . 65 . 60. 90. 275 . 0. 0 . 60. 20 8 80. 55 . 10. 50. 30. 30. 55 . 120. 40. 30. 200. 5 . 0. 0 . 0 9 285 . 45 . 2 10. 115. 55 . 45 . 10. 0. 2605 . 95 . 1440 . 10. 25 . 5 . 35 10 900. 205 . 655 . 740. 135 . 160. 105 . 45 . 1025 . 3015. 1685 . 20. 10. 50. 645 1 1 1765 . 105 . 335 . 385 . 770. 500. 170. 100. 2865 . 545 . 10235. 55 . 65 . 25 . 100 12 25 . 0 . 0 . 5 . 45 . 40. 5 . 0. 20. 5 . 105 . 75 . 0. 0 . 0 13 20. 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0. 10. 0. 0 . 0 . 0 14 200. 140. 30. 140. 20. 30. 305. 120. 55 . 70. 280 . 25 . 0. 925 . 20 15 2 15. 30. 80 . 120. 50. 30. 40. 35 . 230. 675 . 305 . 5. 5 . 15 . 1250 t o o U J MALE C L E R I C A L : I C C M - l b MODEL P R E D I C T E D C O M M U T I N G F L O W S 1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 14 15 1 1508 . 48 . 30. 225 . 63 . 58 . 10. 16 . 329 . 198 . 1386 . 32 . 12 . 0 . 20 2 187 . 192 . 5 . 54 . 0. 0. 72 . 36 . 54 . 175 . 546 . 6 . 5 . O . 28 3 0. 0. 272 . 0 . 0 . 0. 0 . 0 . 393. 53 . 598 . 0 . 9 . 0 . 36 4 222 . 12 . 7 . 135 . 3 . 4 . 3 . 4 . 67 . 169 . 203 . 4 . 2 . 0 . 6 5 5 1 . 0. 1 . 1 . 209 . 99 . 0 . 0 . 25 . 0 . 420. 44 . 3 . o . 0 6 1 38 . 0. 3 . 8 . 240. 205 . 0 . 1 . 62 . 2 . 7 14. 65 . 5 . 0. 2 7 62 . 74 . 1 . 2 1 . O . 0. 62 . 6 . 18 . 59 . 16 1. 2 . 2 . o . 6 8 33 . 67 . 0. 1 1 . 0. 0. 9 . 28 . 12 . 26 . 148. 2 . 1 . 0 . 6 9 205 . 0. 88 . 37 . 0. 3 . 0 . 0 . 991 . 18 . 1 146 . 9 . 15 . 0 . 8 10 447 . 59 . 67 . 352 . 0 . 0. 9 . 14 . 2 18. 790. 755 . 10. 8 . o . 7 1 1 1 733 . 8 . 66 . 65 . 2 19. 114. O . 7 . 832 . 38 . 8900. 130. 58 . o . 10 12 23 . 0. 1 . 0. 67 . 38 . 0 . 0. 27 . 0 . 446 . 39 . 3 . 0 . 0 13 1 . 0. 1 . 0. 0 . 0 . O . 0. 8 . 0 . 48 . O . 1 . o . 0 14 0 . 19 . 0. O . O . 0. 104 . 5 . O . O . 153 . 0 . 45 . 195 . O 15 0. 1 . 63 . 0 . 0. 0. 0 . 0 . 25. 160. 191 . 0 . 4 . 0. 375 E M P I R I C A L C O M M U T I N G F L O W S 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1 195 55 . 35 . 160. 50. 70. 5 . 25 . 220. 1 10. 1960. 25 . 0 . 5 . 20 2 335 . 120. 15 . 125. 20. 15 . 40. 30. 55 . 35 . 530. 0. 5 . 5 . 30 3 135 . 10. 205 . 10. 25 . 0 . 15 . 0 . 295 . 180. 4 50. 0 . 10. 5 . 20 4 195 . 30. 50. 180. 15 . 5 . 20. 0. 35 . 35. 260. 0 . 0 . 5 . 10 5 50. 15 . 0 . 10. 170 . 85 . 0 . 0. 35 . 20. 4 30. 35 . 5 . 0 . O 6 95 . 0. 5 . 25 . 240. 130. 0 . 0. 45 . 10. 835 . 60 . 0 . O . O 7 85 . 25 . 10. 20. 5 . 5 . 95 . 0. 5 . 15 . 170. 5 . 0 . 25 . 10 8 80. 0. 5 . 15 . 5 . 10. 5 . 35 . 15. 10. 1 55 . 5 . O . 5 . 0 9 200. 0. 60. 15 . 35 . 5 . 0. 5 . 1145. 40. 995 . 10. 5 . 0 . 5 10 245 . 80. 130. 165 . 25. 15. 20. 10. 255 . 935 . 755 . 15 . 10. 5 . 135 1 1 775 . 80. 50. 1 20 . 125 . 160 . 20. 0 . 875 . 150 . 8600 . 50. 145. 10 . 20 12 30. 5 . 0. 10. 50. 20. 0 . 0 . 20. 5 . 365 . 140. 0 . 0 . 0 1 3 5 . 0 . 0. 0 . 0. 0. 0. 0. 0. 0 . 55 . 0. 0. 0. 0 14 80. 55 . 5 . 20. 15 . o. 45 . I O . 15 . 20. 1 10. O . O . 130. 15 15 105 . 5. 35 . 35 . 20. 0. 0. 0 . 45 . 125 . 145 . 0. 0 . 0. 305 MALE C L E R I C A L : ENTROPY MODEL 1 2 3 4 1 1005 . 58 . 38 . 178 2 2G3 . 128 . 1 1 . 7 1 3 93 . 6 . 246 . 32 4 2 17. 16 . 1 1 . 76 5 G9 . 3 . 3 . 9 6 160. 7 . 6 . 20 7 97 . 51 . 3 . 28 8 60. 4 1 . 2 . 18 9 237 . 8 . 89 . 50 10 583 . 79 . 86 . 286 1 1 625 . 28 . 60. 86 12 38 . 2 . 2 . 5 13 3 . 0. 1 . 0 14 75. 31 . 3 . 22 15 86 . 23 . 45. 30 1 2 3 4 1 1 195 . 55 . 35 . 160 2 335 . 120. 15 .. 125 3 135 . 10. 205 . 10 4 195 . 30. 50. 180 5 50. 15 . 0. 10 6 95. 0. 5 . 25 7 85 . 25 . 10. 20 8 80. 0. 5 . 15 9 200. 0. 60. 15 10 245 . 80. 130. 165 1 1 775 . 80. 50. 120 1 2 30. 5 . 0. 10 13 5 . O. 0. 0 14 80. 55 . 5 . 20 15 105 . 5 . 35 . 35 PREDICTED COMMUTING FLOWS 5 6 7 8 9 114. 86 . 28 . 17 . 402 25. 19. 67 . 27 . 63 1 1 . 8 . 3 . 2 . 401 16 . 12 . 9. 5 . 95 92 . 54 . 1 . 1 . 28 153 . 106 . 3. 2 . 64 8 . 6 . 38 . 7 . 23 7 . 6 . 13 . 12 . 16 29 . 22 . 4 . 2 . 864 43 . 33 . 38 . 18 . 254 234 . 126 . 14 . 9 . 756 5 1 . 30. 1 . 1 . 23 1 . 0. 0. 0. 7 7 . 5 . 38 . 7 . 17 8 . 6 . 9 . 4 . 48 EMPIRICAL COMMUTING FLOWS 5 6 7 8 9 50. 70. 5 . 25. 220 20. 15 . 40. 30. 55 25. 0. 15. 0. 295 15 . 5 . 20. 0. 35 170. 85 . 0. 0. 35 240. 130. 0. 0. 45 5 . 5 . 95 . O. 5 5. 10. 5. 35 . 15 35 . 5 . 0. 5 . 1 145 25. 15 . 20. 10. 255 125 . 160. 20. 0. 875 50. 20. 0. 0. 20 0. 0. 0. 0. 0 15 . 0. 45 . 10. 15 20. 0. 0. 0. 45 10 1 1 12 13 14 15 204 . 1709 . 42 . 19 . 7 . 28 175 . 447 . 9 . 3 . 14 . 37 103 . 402 . 4 . 7 . 1 . 43 122 . 242 . 6 . 3 . 2 . 9 1 1 . 54 1 . 37 . 4 . O . 2 25 . 824 . 6 1 . 7 . 1 . 5 64 . 127 . 3 . 1 . 9 . 9 34 . 12 1. 3 . 1 . 3 . 8 52 . 1117. 1 1 . 22 . 1 . 12 594 . 648 . 16 . 7 . 10. 106 ' 98 . 8905 . 120. 98 . 4 . 17 6 . 455 . 28 . 4 . 0. 1 0. 45 . 0. 2 . 0. 0 54 . 108 . 3 . 1 . 139 . 1 1 146 . 124 . 3 . 1 . 3 . 282 10 1 1 12 13 14 15 1 10. 1960. 25 . 0. 5 . 20 35 . 530. 0. 5 . 5 . 30 180. 450. 0. 10. 5 . 20 35 . 260. 0. 0. 5 . 10 20. 430. 35 . 5 . O. 0 10. 835 . 60. 0. 0. 0 15 . 170 . 5 . 0. 25 . 10 10. 155 . 5 . 0. 5 . 0 40. 995 . 10. 5. 0. 5 935 . 755 . 15 . 10. 5 . 1 35 150. 8600 . 50. 145 . 10 . 20 5 . 365 . 140. O. 0. 0 0. 55 . 0. 0. 0. 0 20. 1 10. 0. 0. 130. 15 125 . 145 . O. 0. 0 . 305 MALE P R O F E S S I O N A L : I C C M - l b MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 2085 . 93 . 13 . 376 . 8 . 38. 15. 39 . 247 . 300. 1706 . 44 . 116. 0 . 0 2 3 12. 58 1 . 0. 1 19 . 0 . 0. 154 . 114. 0 . 34 1 . 609 . 0 . 49 . 0 . 12 3 O. O. 894 . 0. 0. 0. O. 0. 605 . 132 . 77 1 . O. 168 . O. 9 4 295 . 22 . 3 . 218. 0. 0. 4 . 9 . 47 . 258 . 210. 0. 18 . 0 . 1 5 69 . 0 . 0. 3 . 357 . 2 14. 0 . 1 . 12 . 0 . 563 . 129 . 28 . 0 . O 6 348 . O. 0. 24 . 707 . 8 11. O. 4 . 62 . 0 . 1695 . 333 . 96 . 0. O 7 119. 262 . 0. 54 . 0. 0. 158 . 23. 0 . 134 . 163 . 0 . 22 . 0 . 0 8 42 . 167 . 0. 20. 0 . 0. 16 . 73 . 0 . 39 . 128 . 0 . 12 . 0 . 0 9 330 . O. 146 . 77 . O. 0. 0. 0. 1063 . 17 . 17 19. O. 179 . 0 . 0 10 664 . 119. 61 . 658 . 0 . 0. 13. 38 . 136 . 1393 . 647 . 0. 8 1 . 0 . 6 1 1 1 1252 . 0. 45 . 138 . 232 . 96 . O. 19 . 836 . 37 . 16006. 373 . 73 1 . 0. O 12 74 . 0. 0. 0. 256 . 176. 0. 0. 34 . 0. 1593 . 297 . 84 . 0. 0 13 4 . 0. 0. 0. 0 . 0. 0. 0. 19 . 0 . 194, 0. 42 . 0. 0 14 0 . 0. 0. 0. O. 0. 160. 0 . 0 . 0 . 0 . 0 . 355 . 580 . 0 15 0. 0. 18 . 0. 0 . 0. 0. 0 . 0 . 383 . 0 . 0 . 36 . 0. 808 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1600. 135 . 95 . 215 . 105. 105 . 55 . 55 . 175 . 135 . 2210. 75 . 45 . 35 . 40 2 555 . 355 . 35 . 205. 55 . 40. 90. 60. 60 . 100. 675 . 5. 10. 35 . 10 3 2 15. 20. 565 . 70. 10. 15. 0 . 5 . 340. 345 . 900. 5 . 65 . 0 . 25 4 155 . 45 . 40. 365 . 25 . 0. 5 . 25 . 35 . 85 . 275 . 15. 5 . 5 . 5 5 90. 5 . 5. 15. 300. 170. 5 . 5 . 35 . 15 . 605 . 90. 25 . 10. 0 6 425 . 30. 5 . 45 . 380. 585 . 5 . 20. 60 . 80. 2 150. 220. 75 . 0 . O 7 135 . 145 . 15 . 65 . 5. 10. 155 . 30. 50. 25 . 250. 10. 5 . 25 . 10 8 70. 35 . 5 . 35. 20. 20. 25 . 35 . 5 . 15 . 185 . 10. 10. 25 . 0 9 2 15. 15 . 75 . 45 . 10. 30. 5 . 0. 1 185 . 55 . 1735 . 0. 150. 5 . 5 10 395 . 125 . 145 . 250. 15 . 15. 30. 10. 160. 1555 . 925 . 20. 70. 10. 145 1 1 1275 . 125 . 140. 270. 345 . 225 . 30. 40. 885 . 400. 14240. 2 15. 1475 . 20. 80 12 195 . 10. 0. 40. 270. 95 . 5 . 5 . 15 . 15 . 1310 : 495 . -~ 60. • 0. O 13 20. 0. 5 . 5 . 0. 0. 0. 10. 10. 5 . 200. 0 . 0 . 0. 5 14 170. 160. 5 . 30. 5 . 10. 70. 20. 10. 25 . 180. 5 . 0 . 390. 5 15 80. 40. 45 . 30. 15 . 15 . 20. 0 . 40. 180 . 165 . 10. 15 . 20. 570 MALE P R O F E S S I O N A L : ENTROPY MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1319. 130. 54 . 292 . 145. 147 . 44 . 42 . 310. 322 . 1984 . 90. 148 . 16 . 38 2 394 . 342 . 17 . 135 . 37 . 37 . 128 . 78 . 55 . 325 . 593 . 23 . 29 . 38 . 59 3 184 . 20. 579 . 82 . 20. 20. 7 . 6. 494 . 256 . 720. 13 . 83 . 3 . 92 4 274 . 35 . 14 . 122 . 20. 20. 13 . 1 1 . 70. 190. 267 . 12 . 20. 5 . 1 2 5 108 . 8 . 4 . 17 . 148 . 115. 3 . 3 . 25 . 20. 777 . 100. 4 1 . 1 . 3 6 442 . 3 1 . 16 . 69 . 428 . 397 . 1 1 . 12 . 104 . 83 . 2059 . 289 . 118. 5 . 14 7 165 . 156 . 6 . 6 1 . 13 . 13 . 82 . 2 1 . 23 . 136 . 191 . 9 . 13 . 29 . 16 8 75 . 92 . 3. 28 . 9 . 9. 20. 30. 1 1 . 52 . 134 . 6 . 7 . 8 . 1 1 9 370. 20. 158 . 97 . 44 . 45 . 7 . 7 . 836 . 99 . 158 1 . 28 . 2 17. 2 . 19 10 751 . 176 . 123 . 472 . 54 . 54 . 60. 44 . 193 . 957 . 73 1 . 33 . 55 . 23 . 145 1 1 1 126 . 85 . 119. 192 . 4 19. 299 . 29 . 30. 82 1 . 2 12. 14907. 368 . 1113. 12 . 32 12 144 . 10. 8 . 23 . 198 . 154 . 4 . 4 . 52 . 27 . 1604 . 189 . 92 . 2 . 4 13 IO. 1 . 2 . 2. 3 . 2 . 0. 0. 17 . 2 . 164 . 3 . 54 . 0. 0 14 115. 83 . 6 . 43 . 1 1 . 1 1 . 74 . 19 . 15. 103 . 145 . 7 . 10. 428 . 17 15 118. 55 . 7 1 . 52 . 1 1 . 1 1 . 16 . 1 1 . 39 . 252 . 148 . 7 . 10. 7 . 437 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1600. 135. 95 . 2 15. 105. 105 . 55 . 55. 175. 135 . 2210. 75 . 45 . 35 . 40 2 555 . 355 . 35 . 205 . 55 . 40. 90. 60. 60. 100. 675 . 5 . 10. 35 . 10 3 2 15. 20. 565 . 70. 10. 15. 0. 5 . 340. 345 . • 900. 5 . 65 . 0. 25 4 155 . 45 . 40. 365 . 25. 0. 5 . 25 . 35 . 85 . 275 . 15 . 5 . 5 . 5 5 90. 5 . 5 . 15 . 300. 170. 5 . 5 . 35 . 15 . 605 . 90. 25 . 10. 0 6 425 . 30. 5 . 45 . 380. 585 . 5 . 20. 60. 80. 2 150. 220. 75 . 0. 0 7 135 . 145 . 15 . 65. 5 . 10. 155 . 30. 50. 25 . 250. 10. 5 . 25 . 10 8 70. 35 . 5 . 35 . 20. 20. 25 . 35 . 5 . 15 . 185 . 10. 10. 25 . 0 9 2 15. 15 . 75 . 45 . 10. 30. 5 . 0. 1 185. 55 . 1735 . 0. 150. 5 . 5 IO 395 . 125 . 145 . 250. 15. 15 . 30. 10. 160. 1555 . 925 . 20. 70. 10. 145 1 1 1275 . 125 . 140 . 270. 345 . 225. 30. 40. 885 . 400. 14240. 2 15. 1475 . 20. 80 12 195 . 10. 0. 40. 270. 95 . 5 . 5 . 15 . 15 . 1310. 495 . 60. O. 0 1 3 20. 0. 5 . 5 . 0. 0. 0. 10. 10. 5 . 200 . 0. 0. 0. 5 14 170 . 160 . 5 . 30. 5 . 10. 70. 20. 10. 25 . ISO. 5 . 0. 390. 5 15 80. 40 . 45 . 30. 15 . 15 . 20. 0. 40 . 180. 165 . 10. 15 . 20. 570 MALE AGGREGATED O C C U P A T I O N S : I C C M - l b MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 14 15 1 15866. 338 . 1 22 . 2704 . 599 . 784 . 59 . 293 . 3138 . 2209 . 123 14 . 46 1 . 198 . 0. 0 2 2330 . 408 1 . 0. 634 . 0 . 0. 6 19. 104 2 . 193 . 2574 . 5909 . 47 . 91 . O . 0 3 0. 0 . 6918. 0. 0 . 0 . 0 . 0 . 6357 . 0 . 7 126 . 0 . 230. 0 . 0 4 26 18 . 92 . 33 . 1964 . 0 . 16 . 2 1 . 80 . 7 13. 2831 . 1933 . 54 . 35 . 0. 0 5 483 . 0 . O . 0 . 3096 . 1806 . 0 . 0 . 18 1. O . 4023 . 757 . 49 . O . O 6 1660. 0. 0 . 0. 42 10. 4522 . 0. 0 . 589 . 0. 8 138 . 1343 . 109 . 0 . 0 7 1004 . 2 1 39 . 0 . 328 . 0 . 0. 656 . 2 19. 49 . 1085 . 2 142. 5 . 49 . 0. 0 a 3 16. 1 339 . 0 . 9 1 . 0 . 0 . 73 . 785 . O . 175 . 1435 . 12 . 24 . O . 0 9 2 153 . 0 . 1303 . 270. 0 . 0. 0 . 0 . 12232 . 0 . 1 1784. 106 . 287 . 0. 0 to 6254 . 35 1 . 770 . 6280. 0. 0. 65 . 298 . 2495 . 16630. 8247 . 77 . 177 . 0. 0 1 1 7620 . 0 . 359 . 332 . 2720. 1666 . 0 . 0. 8766 . 0 . 90144. 2 165. 1048 O . O 12 236 . 0. 0. 0 . 1362 . 94 1 . 0 . 0 . 279 . 0. 6 192 . 97 1 . 79 . 0. 0 1 3 6 . O . 1 . 0 . 0 . O . O . 0. 90. 0 . 495 . 7 . 27 . O . 0 14 O . O . 0. 0. 0 . 0. 3537 . 0. O . 0. 0. 0. 892 . 5 120. 0 15 0 . 0 . 1 163 . 0 . 0 . 0 . 0 . 0 . 0 . 2758 . 0 . 0 . 374 . 0 . 10650 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1 1880 850. 640. 1650. 865 . 815 . 4 15. 245 . 2305 . 1315. 17275. 285 . 100. 145 300 2 3645 . 2600. 245 . 14 15. 325 . 325 . 785 . 570. 710. 885 . 54 15 . 90. 30. 225 . 255 3 1900 . 265 . 4 305 . 625 . 200. 165 . 1 10. 25 . 3625 . 2750 . 6250. 25 . 105 . 30. 250 4 1915 . 4 10. 530. 2805 . 200. 105 . 145 . 1 10. 615. 720. 2625 . 20. 15. 50. 125 5 770 . 55 . 55 . 130. 2890 . 1480 . 25 . 50 . 330. 145 . 37 15 . 655 . 45 . 30. 20 G 1725 . 105 . 1 10 . 195 . 3060 . 3445 . 40. 80 . 625 . 270. 9670 . 1055 . 130. 5 . 55 7 1210. 855 . 75 . 480. 150. 135 . 1535 . 230. 250. 300. 2045 . 35 . 10. 245 . 120 8 7 15. 260. 45 . 2 20. l O O . 120. 220. 600. 190. 2 15. 14 15. 65 . 15 . 50 20 9 1565 . 150. 690 . 4 15. 250. 210. 20. 10. 12905. 535 . 10920. 80. 260. 25 . too 10 3930. 1000. 2235 . 2255 . 440 . 4 30. 350. 145 . 3140. 16120. 9020. 140. 150. 230. 2060 1 1 8390 . 790. 1 190. 1440. 2505 . 1785 . 455 . 3 10. 9215. 2060. 82615. 1 185 . 2 300 . 150 430 12 630 . 45 . 35 . 135 . 705 . 575 . 25 . 30. 275 . 90. 5 185. 2250. 80. 0 . 0 13 45 . 0. 5 . 15 . 0 . 0. 0. 5 . 35 . 15 . 500. 0. 0. 0 . 5 1 4 1090. 735 100 . 295 . 90 . 70 . 765 . 280 . 160 . 370 . 1580. 85 . 5 . 3840 90 15 1 170 . 270. 4 10 . 455 . 210. 75 . 140. 60. 655 . 2535 . 1995 . 30 . 25 . 95 6820 O 00 MALE AGGREGATED O C C U P A T I O N S : ENTROPY MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 97 1 1 . 814 . 491 . 1995 . 1392 . 1317 . 4 17. 344 . 3825 . 2644 . 14775. 587 . 291 . 126 . 358 2 29 10. 2 148 . 160. 924 . 350. 332 . 1 199. 640. 680 . 2679 . 4428 . 148 . 56 . 304 . 56 1 3 1239 . 113. 4852 . 511. 178 . 168 . 58 . 48 . 5577 . 1922 . 4909 . 75 . 149 . 25 . 807 4 247G . 270. 163 . 102 1 . 230. 2 17. 150. 114. 1064 . 19 t5 . 2439 . 97 . 48 . 45 . 14 1 5 783 . 51 . 40. 114. 1399 . 1020. 26 . 23 . 309 . 164 . 5706 . 643 . 79. 9 . 29 6 2 122 . 126 . 98 . 308 . 2677 . 2323 . 70. 63 . 836 . 445 . 10006. 1231 . 152 . 27 . 85 7 1304 . 1050. 60. 452 . 132 . 125 . 830. 186 . 305 . 1200. 1529. 6 1 . 28 . 250. 163 8 628 . 657 . 27 . 2 18. 98 . 93 . 2 18. 277 . 160. 486 . 1 138 . 4 1 . 16 . 72 . 12 1 9 2589 . 118. 137 1 . 633 . 405 . 383 . 60. 54 . 9793 . 769 . 11190. 17 1. 405 . 18 . 176 10 7553 . 1511. 1536 . 4409 . 701 . 663 . 773 . 492 . 3246 . 10739. 7438 . 295 . 146 . 255 . 1889 1 1 6505 . 420. 856 . 1030. 3151 . 2 106 . 215. 193 . 794 1 . 1364 . 87 124. 188 1 . 17 15. 77 . 240 12 563 . 33 . 37 . 82 . 1005 . 733 . 19 . 17 . 342 . 1 18 . 6332 . 654 . 96 . 7 . 23 13 28 . 2 . 7 . 4 . 1 1 . 7 . 1 . 1 . 74 . 6 . 439 . 6 . 38 . 0 . 1 14 949 . 588 . 57 . 329 . 114. 108 . 784 . 175 . 203 . 953 . 12 13. 48 . 22 . 3828 . 183 15 12 19. 489 . 9 14. 503 . 147 . 139. 2 IO. 123 . 680 . 2922 . 1559 . 62 . 28 . 76 . 5874 EMPIRICAL COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1 1880. 850. 640. 1650. 865. 8 15. 415. 245 . 2 305 . 1315. 17275 . 285 . 100. 145 . 300 2 3645 . 2600. 245 . 14 15. 325 . 325 . 785. 570. 7 10. 885 . 54 15 . 90. 30. 225 . 255 3 1900. 265 . 4305 . 625 . 200. 165 . 1 10. 25 . 3625 . 2750. 6250. 25 . 105 . 30. 250 4 19 15. 4 10. 530. 2805 . 200. 105 . 145 . 1 10. 6 15. 720. 2625 . 20 . 15 . 50. 1 25 5 770. 55 . 55 . 130. 2890. 1480. 25 . 50. 330. 145 . 37 15 . 655 . 45 . 30. 20 6 1725 . 105 . 1 10. 195 . 3060. 3445 . 40. 80. 625 . 270. 9670. 1055 . 130. 5 . 55 7 1210. 855 . 75 . 480. 150. 135 . 1535 . 230. 250 . 300. 2045 . 35 . 10. 245 . 1 20 8 7 15. 260. 45 . 220. 100. 120. 220. 600. 190 . 2 15. 14 15. 65 . 15 . 50. 20 9 1 565 . 150. 690. 4 15. 250. 2 10. 20. 10. 1 2905. 535 . 10920. 80 . 260 . 25 . l O O 10 3930 . 1000. 2235 . 2255 . 440. 430. 350. 145 . 3 140. 16120. 9020. 140. 150. 230. 2060 1 1 8390. 790. 1 190 . 1440. 2505 . 1785 . 455 . 3 10. 92 15 . 2060. 826 15. 1185. 2300. 150. 430 12 630. 45 . 35 . 135 . 705 . 575 . 25. 30. 275 . 90. 5185. 2250. 80. 0 . 0 13 45 . 0. 5 . 15 . 0. 0. O. 5 . 35 . 15 . 500. 0. 0. 0 . 5 14 1090. 735 . 100. 295 . 90. 70. 765 . 280 . 160. 370. 1580. 85 . 5 . 3840 . 90 15 1 170. 270. 4 10. 455 . 2 10. 75 . 140. 60. 655 . 2535 . 1995 . 30. 25 . 95 . 6820 F E M A L E AGGREGATED O C C U P A T I O N S I C C M - l b MODEL P R E D I C T E D COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1 177 1 . 5 4 5 . 124 . 2 3 9 4 . 3 0 6 . 4 17. 86 . 148 . 1882 . 24 13 . 1 0 2 5 7 . 4 6 3 . 2 0 0 . 0 . 0 2 1377 . 3 7 2 6 . 0 . 53 1 . 0 . 0 . 7 5 9 . 4 12 . 0 . 2 4 0 7 . 3 7 0 7 . 0 . 76 . 0 . 20 3 0 . 0 . 4 6 6 3 . O. O. O. O. 0 . 3 4 9 2 . 2 2 7 . 4 4 7 1. 0 . 2 17 . 0 . 0 4 1845 . 14 1. 32 . 1632 . O. 0 . 29 . 39 . 403 . 2 5 2 9 . 1475 . 45 . 34 . 0 . 0 5 393 . 0 . 0 . 0 . 2 5 7 0 . 1265 . 0 . 0 . 1 0 0 . 0 . 3 8 7 6 . 9 3 5 . 56 . 0 . 0 6 112 1. 0 . 0 . 0 . 2 8 2 9 . 2 5 9 8 . 0 . 2 . 285 . 0 . 6 3 6 0 . 135 1. 104 . 0 . 0 7 5 15. 1707 . 0 . 243 . O. 0 . 7 5 7 . 8 1 . 0 . 9 0 8 . 1099 . 0 . 35 . 0 . O 8 182 . 1 2 4 1 . 0 . 87 . 0 . 0 . 9 0 . 3 10. 0 . 2 7 2 . 9 0 8 . 0 . 2 0 . 0 . 0 9 1598 . 0 . 9 9 8 . 3 11 . O. 0 . O. 0 . 7 9 5 7 . O . 1 0 1 3 4 . 7 0 . 302 . O. 0 10 3614 . 6 5 6 . 505 . 445 1 . 0 . 0 . 8 0 . 137 . 1063 . 1272 1 . 4 7 7 1 . 0 . 144 . 0 . 5 1 4 1 1 5 5 8 3 . 0 . 335 . 4 16 . 1736 . 8 8 9 . 0 . 33 . 537 1 . 0 . 7 8 4 4 0 . 2 3 0 4 . 1068 . 0 . 0 12 14 1. 0 . 0 . 0 . 8 8 9 . 52 1 . 0 . 0 . 1 26 . O . 4 8 5 7 . 977 . 74 . 0 . 0 13 3 . 0 . 2 . 0 . 0 . 0 . 0 . 0 . 51 . 0 . 3 9 8 . 6 . 26 . 0 . 0 1 4 0 . 0 . 0 . 0 . 0 . 0 . 1295 . O. 0 . O. O . 0 . 6 9 4 . 4 505 . 0 15 0 . 0 . 266 . O . 0 . O. 0 . 0 . 0 . 2 1 2 3 . O. 0 . 48 . O . 7 4 8 8 E M P I R I C A L COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1 1 9 4 0 . 6 4 5 . 170 . 1 5 2 0 . 3 0 0 . 2 4 0 . 1 8 0 . 50 . 9 10. 5 5 0 . 1 4 1 0 0 . 150 . 1 1 0 . 45 . 95 2 3 3 7 0 . 3 2 5 5 . 85 . 1 2 5 0 . 65 . 6 0 . 6 10. 3 2 5 . 135 . 3 10. 33 15 . 3 0 . 3 0 . 1 2 0 . 55 3 655 . 95 . 43 10. 5 2 5 . 25 . 45 . 35 . 15 . 1 5 1 5 . 2 5 4 5 . 3 1 1 5 . 3 0 . 25 . 25 . 1 10 4 1 7 9 0 . 3 4 5 . 1 8 0 . 3 160 . 45 . 25 . 85 . 10. 2 5 0 . 4 6 0 . 17 3 0 . 4 0 . 10. 35 . 40 5 265 . 25 . 2 0 . 25 . 3 0 5 5 . 9 7 0 . 15 . 0 . 6 0 . 7 0 . 38 15 . 8 4 5 . 3 0 . 0 . 0 6 700 . 25 . 10. 85 . 3 2 0 0 . 3 1 4 5 . 25 . 10 . 2 3 0 . 8 0 . 5 7 4 0 . 1325 . 7 0 . 0 . 5 7 6 4 0 . 1 2 5 5 . 25 . 3 10. 4 0 . 25 . 1 3 5 0 . 9 0 . 3 0 . 125 . 1 160 . 10 . 10 . 2 4 0 . 35 8 7 2 0 . 4 5 5 . 2 0 . 2 3 5 . 35 . 35 . 1 10 . 4 2 0 . 35 . 105 . 8 8 5 . 10 . 5 . 25 . 15 9 5 0 0 . 5 0 . 3 7 0 . 1 35 . 4 0 . 35 . 10. 2 0 . 12 2 0 0 . 2 9 5 . 7 3 7 5 . 6 0 . 2 5 5 . 5 . 20 10 1 8 1 0 . 5 4 0 . 1 160. 16 10 . 75 . 85 . 85 . 55 . 9 15. 1 6 4 0 5 . 4 4 7 0 . 6 0 . 5 0 . 70 1265 1 1 46 15 . 4 4 5 . 4 5 0 . 6 9 5 . 7 9 0 . 5 7 5 . 165 . 8 0 . 4 0 9 5 . 8 0 0 . 8 0 4 4 0 . 5 0 5 . 2 3 7 0 . 6 0 . 9 0 12 255 . 15 . 5 . 55 . 6 0 0 . 4 2 5 . 5 . 5 . 130. 30 . 2 8 8 5 . 3 0 6 0 . 105 . 10. O 1 3 20 . 0 . 0 . 10 . 5 . O. 0 . 5 . 15 . 5 . 4 15. 10 . O . 0 . 0 14 495 . 7 3 5 . 25 . 245 . 25 . 5 . 345 . 65 . 5 0 . 140 . 4 7 5 . 15 . 0 . 3 8 2 5 . 55 15 375 . 1 20 . 95 . 195 . 3 0 . 20 . 75 . 10 . 160 . 1 6 8 0 . 8 30 . 0 . 10 . 45 . 6 2 8 0 FEMALE AGGREGATED O C C U P A T I O N S : ENTROPY MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 10048. 574 . 157. 1831 . 912. 787 , 17 1. 127. 2083. 1928 . 1 1480. 522 . 252 . 24 . 1 10 2 2083 . 2992 . 37 . 724 . 146 . 126 . 1015. 396 . 199 . 2433 . 2380. 84 . 27 . 109 . 264 3 473 . 31 . 4683 . 242 . 43 . 37 . 9 . 7 . 3632 . 1 197 . 2233 . 25 . 93 . 2 . 364 4 2307 . 194 . 53 . 1 180. 1 10. 95 . 66 . 43 . 544 . 208 1 . 1383 . 63 . 30. 9 . 48 5 420. 16 . 7 . 46 . 1607 . 942 . 5 . 4 . 87 . 55 . 4892 . 1046 . 64 . 1 . 5 6 1 107 . 38 . 15 . 120. 2528 . 19 18. 13 . 1 1 . 229 . 144 . 6765 . 1645 . 101 . 3 . 14 7 828 . 1353. 1 1 . 327 . 45. 39 . 769 . 83 . 79 . 967 . 642 . 29 . 12 . 107 . 55 8 405 . 976 . 5 . 160. 42 . 36 . 153 . 2 16. 44 . 366 . 600. 24 . 8 . 24 . 5 1 9 1540. 36 . 785 . 363 . 159 . 137 . 1 1 . 9 . 9 138 . 336 . 8274 . 9 1 . 447 . 1 . 4 1 10 5057 . 1049. 623 . 4332 . 24 1 . 208 . 313. 158 . 1193. 11254. 303 1 . 138 . 66 . 50. 942 1 1 2894 . 112. 187 . 358 . 1599 . 824 . 33.! 28 . 3209 . 377 . 83160. 1533 . 1823 . 6 . 32 12 220. 7 . 5 . 24 . 84 1 . 493 . 3 . 2 . 87 . 29 . 488 1 . 9 17. 73 . 1 . 3 13 10. 0. 2 . 1 . 4 . 2 . 0. 0. 34 . 1 . 357 . 3 . 70. 0. 0 14 352 . 391 . 7 . 139. 25 . 2 1 . 48 1 . 52 . 30. 468 . 3 11. 14 . 6 . 4 159 . 45 15 407 . 237 . 348 . 208 . 29 . 25 . 55 . 24 . 14 1. 1965 . 360. 16 . 7 . 10. 6093 EMPIRICAL COMMUTING FLOWS 1 ' 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 1 1940. 645 . 170. 1520. 300. 240. 180. 50. 9 10 . 550. 14100. 150. 1 10. 45 . 95 2 3370. 3255 . 85. 1250. 65 . 60. 6 10. 325 . 135 . 3 10. 33 15 . 30. 30. 120. 55 3 655 . 95 . 43 10. 525 . 25 . 45 . 35. 15. 15 15. 2545 . 3 115. 30. 25 . 25 . 1 10 4 1790. 345 . 180. 3160. 45 . 25 . 85. 10. 250. 460. 1730. 40. 10. 35 . 40 5 265 . 25 . 20. 25 . 3055 . 970. 15 . 0. 60. 70. 3815. 845 . 30. 0. 0 6 700. 25 . 10. 85 . 3200. 3145. 25 . 10. 230. 80. 5740 . 1325 . 70. 0. 5 7 640 . 1255. 25. 3 10. 40. 25 . 1350. 90. 30. 125 . 1 160. 10. 10. 240 . 35 8 720. 455. 20. 235 . 35 . 35 . 1 10. 420. 35 . 105 . 885 . 10. 5 . 25 . 15 9 500. 50. 370. 135 . 40. 35 . 10. 20. 12200. 295. 7375 . 60. 255 . 5 . 20 10 18 10. 540. 1 160. 16 10. 75 . 85 . 85 . 55. 9 15. 16405. 4470. 60. 50. 70. 1 265 1 1 46 15 . 445 . 450. 695 . 790. 575 . 165 . 80. 4095 . 800. 80440. 505 . 2370. 60. 90 12 255 . 15. 5 . 55. 600. 425 . 5 . 5. 130. 30. 2885 . 3060. 105 . 10. 0 13 20. 0. 0. 10. 5 . 0. 0. 5 . 15 . 5 . 4 15. 10. 0. 0. 0 14 495 . 735 . 25 . 245 . 25 . 5 . 345 . 65 . 50. 140. 475 . 15 . 0. 3825 . 55 15 375 . 120. 95 . 195 . 30. 20 . 75 . IO. 160. 1680 . 8 30. 0 . 10. 45 . 6280 1 9 8 1 AGGREGATED D A T A : I C C M - l b MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 2 5 5 0 1 . 1457 . 634 . 4 9 3 7 . 1 1 4 5 . 1327 . 247 . 4 8 5 . 5 1 3 6 . 5 2 0 6 . 2 2 3 3 6 . 9 2 9 . 3 6 7 . 0 . 38 1 2 3 6 6 3 . 6251 . 0 . 1 3 0 0 . 0 . 0 . 2 0 5 2 . 1 1 4 7 . 8 0 0 . 4 8 8 5 . 9 4 8 4 . 146 . 137 . 0 . 67 1 3 0 . 0 . 92 14. 0 . 0 . 0 . 0 . 0 . 8 3 8 0 . 1789 . 1 2 9 4 1 . 0 . 3 4 8 . 0 . 1028 4 4224 . 3 9 0 . 1 7 0 . 3 3 1 0 . 0 . 72 . 8 6 . 1 3 0 . 1 175 . 5 0 6 2 . 3 6 5 9 . 1 19 . 6 5 . 0 . 133 5 1 0 7 5 . 0 . 0 . 43 . 54 14 . 2 9 7 3 . 0 . 1 1 . 4 4 3 . 0 . 7 9 4 9 . 1583 . 1 0 0 . 0 . 0 6 3 106 . O. 10. 22 1 . 6 6 4 5 . 666 1 . 0 . 42 . 1 188 . 27 . 1 4 5 9 4 . 2 5 2 4 . 202 . 0 . 0 7 1448 . 2842 . 0 . 591 . 0 . 0 . 2 1 1 2 . 238 . 313 . 1 9 5 0 . 3 2 8 6 . 42 . 62 . 0 . 137 8 626 . 2 0 2 5 . 0 . 251 . 0 . 0 . 2 4 0 . 8 2 9 . 156 . 654 . 2 3 7 8 . 39 . 37 . 0 . 126 9 4 3 4 3 . 0 . 2 4 9 8 . 975 . 0 . 0 . 0 . O. 1 8 5 9 9 . 4 6 9 . 2 1 7 3 2 . 28 1 . 5 3 0 . O. 78 10 954 1 . 2 1 3 6 . 1799 . 93 12 . 0 . 0 . 237 . 5 1 1 . 401 1 . 2 5 4 6 6 . 1 4 5 8 7 . 258 . 2 8 5 . 0 . 2 157 1 1 • 1 4 4 8 2 . 273 . 1581 . 1644 . 495 1 . 2 9 7 5 . 0 . 227 . 14744 . 9 5 0 . 1 6 2 7 5 3 . 44 12 . 2 0 0 5 . 0 . 0 12 6 9 0 . O. 0 . 0 . 2 165 . 14 17. 0 . 0 . 6 0 3 . O. 1 0 8 2 4 . 1802 . 144 . 0 . 0 13 32 . 0 . 13 . 0 . 0 . 0 . 0 . 0 . 143 . 0 . 861 . 14 . 47 . 0 . 0 14 0 . 9 8 0 . 0 . 0 . 0 . 0 . 3 188 . 362 . 0 . 0 . 0 . 0 . 19 17. 9 6 2 5 . 0 15 0 . 42 . 1675 . 0 . 0 . 0 . 0 . 0 . 12 1. 5 4 6 8 . 3 5 8 9 . 0 . 114 . 0 . 1386 1 E M P I R I C A L COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 2 3 8 2 0 . 1495 . 8 10. 3 170 . 1 165 . 1055 . 5 9 5 . 295 . 32 15 . 1865 . 3 1 3 7 5 . 4 3 5 . 2 10. 190 . 395 2 7 0 1 5 . 5 8 5 5 . 3 3 0 . 2 6 6 5 . 3 9 0 . 385 . 1395 . 8 9 5 . 8 4 5 . 1 195 . 8 7 3 0 . 1 2 0 . 6 0 . 345 . 3 10 3 2 5 5 5 . 3 6 0 . 8 6 1 5 . 1 150 . 2 2 5 . 2 1 0 . 145 . 4 0 . 5 1 4 0 . 5 2 9 5 . 9 3 6 5 . 55 . 1 3 0 . 55 . 3 6 0 4 3 7 0 5 . 755 . 7 10. 5 9 6 5 . 245 . 130 . 2 3 0 . 1 2 0 . 8 6 5 . 1 180 . 4 3 5 5 . 6 0 . 25 . 85 . 165 5 1035 . 8 0 . 75 . 155 . 5 9 4 5 . 2 4 5 0 . 4 0 . 5 0 . 3 9 0 . 215 . 7 5 3 0 . 1 5 0 0 . 75 . 3 0 . 20 6 2 4 2 5 . 1 3 0 . 1 2 0 . 2 8 0 . 6 2 6 0 . 6 5 9 0 . 65 . 9 0 . 8 5 5 . 3 5 0 . 1 5 4 1 0 . 2 3 8 0 . 2 0 0 . 5 . 6 0 7 1 8 5 0 . 2 110 . 100 . 7 9 0 . 190 . 160 . 2 8 8 5 . 3 2 0 . 2 8 0 . 425 . 3 2 0 5 . 45 . 2 0 . 4 8 5 . 155 8 1435 . 7 15. 65 . 455 . 135 . 155 . 3 3 0 . 1 0 2 0 . 225 . 3 2 0 . 2 3 0 0 . 75 . 2 0 . 75 . 35 9 2 0 6 5 . 2 0 0 . 1 0 6 0 . 5 5 0 . 2 9 0 . 245 . 3 0 . 3 0 . 2 5 1 0 5 . 8 3 0 . 1 8 2 9 5 . 1 4 0 . 5 1 5 . 3 0 . 120 10 5 7 4 0 . 1 5 4 0 . 3 3 9 5 . 3 8 6 5 . 5 15. 5 1 5 . 4 3 5 . 2 0 0 . 4 0 5 5 . 3 2 5 2 5 . 1 3 4 9 0 . 2 0 0 . 2 0 0 . 3 0 0 . 3 3 2 5 1 1 1 3 0 0 5 . 1235 . 1 6 4 0 . 2 135 . 3295 . 2 3 6 0 . 6 2 0 . 3 9 0 . 133 10.' 2 8 6 0 . 1 6 3 0 5 5 . 1 6 9 0 . 4 6 7 0 . 2 10. 5 2 0 12 8 8 5 . 6 0 . 4 0 . 190 . 1305 . 1 0 0 0 . 3 0 . 35 . 4 0 5 . 1 2 0 . 8 0 7 0 . 53 10 . 185 . 10 . 0 13 65 . O. 5 . 25 . 5 . 0 . 0 . 10 . 5 0 . 2 0 . 9 1 5 . 10 . O. O. 5 14 1585 . 1 4 7 0 . 125 . 5 4 0 . 115. 75 . 1 1 1 0 . 345 . 2 10. 5 1 0 . 2 0 5 5 . 1 0 0 . 5 . 7 6 6 5 . 145 15 1545 . 3 9 0 . 505 . 6 5 0 . 2 4 0 . 9 5 . 2 15. 7 0 . 8 15. 42 15 . 2 8 2 5 . 3 0 . 3 5 . 1 4 0 . 13 100 1 9 8 1 AGGREGATED D A T A : ENTROPY MODEL 1 2 3 4 1 19206. 1468 . 652 . 3830 2 5028 . 492 1 . 189 . 1670 3 1696 . 144 . 9443 . 765 4 4735 . 49 1 . 2 18. 2 1 35 5 1247 . 70. 42 . 165 6 3284 . 167 . 101 . 435 7 2 157 . 2337 . 66 . 793 8 1048 . 1543 . 29 . 386 9 4 191. 157 . 2235 . 1025 10 12755. 2703 . 22 15 . 8650 1 1 9643 . 543 . 1005 . 14 16 12 788 . 40. 36 . 104 13 37 . 2 . 9 . 5 14 1294 . 1033 . 54 . 476 15 162 1 . 777 . 1300. 73 1 • 1 2 3 4 1 23820. 1495 . 8 10. 3 170 2 7015 . 5855 . 330. 2665 3 2555 . 360. 86 15. 1 150 4 3705 . 755 . 710. 5965 5 1035 . 80. 75 . 155 6 2425 . 130. 120. 280 7 1850. 21 10. 100. 790 8 1435 . 7 15. 65 . 455 9 2065 . 200. 1060. 550 10 5740. 1540. 3395 . 3865 1 1 13005. 1235 . 1640. 2135 12 885 . 60. 40. 190 13 65 . 0. 5 . 25 14 1585 . 1470. 125 . 540 15 1545. 390. 505 . 650 PREDICTED COMMUTING FLOWS 5 6 7 8 9 10 2341 . 2 135 . 598 . 480. 6030. 4685 500. 456 . 2221 . 1069 . 856 . 5115 207 . 188 . 58 . 47 . 9260. 318 1 347 . 3 16. 222 . 16 1. 1646 . 3927 2927 . 1965 . 29 . 25 . 39 1 . 224 5127 . 422 1 . 75 . 67 . 103 1 . 590 175 . 159. 1587 . 275 . 367 . 2 194 142 . 129. 378. 504 . 198. 870 566 . 516. 64 . 57 . 18663. 1 132 934 . 851 . 1 102 . 650. 4435 . 2 1606 4902 . 2972 . 221 . 196 . 11401. 1732 1849 . 1242 . 18 . 16 . 4 12. 14 1 16 . 9 . 1 . 1 . 111. 7 129 . 117. 1293 . 224 . 199 . 1458 161 . 147 . 258 . 138 . 766 . 5065 EMPIRICAL COMMUTING FLOWS 5 6 7 8 9 10 1 165 . 1055 . 595 . 295 . 3215. 1865 390. 385. 1395. 895 . 845 . 1 195 225 . 2 10. 145 . 40. 5 140. 5295 245. 130. 230. 120. 865. 1 180 5945 . 2450. 40. 50. 390. 2 15 6260. 6590. 65. 90. 855 . 350 190. 160. 2885 . 320. 280. 425 135 . 155 . 330. 1020. 225 . 320 290. 245. 30. 30. 25105. 830 5 15. 5 15. 435 . 200. 4055 . 32525 3295 . 2360. 620. 390. 13310. 2860 1305 . 1000. 30. 35 . 405 . 120 5 . 0. 0. 10. SO. 20 115. 75 . 1 1 10. 345 . 2 10. 5 10 240. 95 . 215. 70. 8 15. 42 15 1 1 12 13 14 15 26367. 1 142 . 549 . 142 . 466 6903 . 244 . 86 . 4 30. 848 7 150. 101 . 248 . 2 1 . 1 192 3903 . 169 . 8 1 . 53 . 19 1 10735. 158 1 . 149 . 8 . 30 16979. 2769 . 260 . 24 . 88 2 180. 94 . 4 1 . 376 . 2 18 1765. 69 . 24 . 99 . 177 19569. 276 . 832 . 15 . 208 10516 . 455 . 2 19. 289 . 2920 169540. 3595 . 3530. 64 . 234 1 1295. 1502 . 173 . 6 . 2 1 806 . 10. 95 . 0. 1 1449 . 63 . 27 . 8022 . 2 18 18 15. 79 . 34 . 75 . 1 1903 1 1 12 13 14 15 31375. 435. 2 10. 190. 395 8730. 120. 60. 345 . 310 9365 . 55 . 130. 55 . 360 4355. 60. 25 . 85 . 165 7530. 1500. 75 . 30 20 15410. 2380. 200. 5 . 60 3205 . 45 . 20. 485 . 155 2300. 75 . 20. 75 . 35 18295. 140. 5 15. 30. 120 13490. 200. 200. 300. 3325 163050. 1690. 4670 . 2 10. 520 8070 . 53 10. 185 . 10. 0 9 15. 10. 0. O. 5 2055 . 100. 5 . 7665 . 145 2825 . 30. 35 . 140. 13 100 1 9 8 1 AGGREGATED DATA : PROJECTED ENTROPY MODEL 1 2 3 4 1 20853. 1287 . 52 1 . 38 12 2 5180. 5393 . 145 . 1666 3 1483 . 103 . 10537. 669 4 5075 . 440. 178 . 2291 5 1 104 . 49 . 28 . 128 6 3049 . 120. 68 . 355 7 2200. 2564 . 49 . 792 8 1053 . 1722 . 2 1 . 379 9 3886 . 109 . 2053 . 890 10 13 179. 25 19. 2009 . 9348 1 1 8297 . 365. 7 19. 108 1 12 673 . 26 . 24 . 78 13 3 1 . 1 . 7 . 4 14 1223 . 1016 . 38 . 440 15 1444 . 68 1 . 1 199 . 652 1 2 3 4 1 23820. 1495 . 8 10. 3170 2 7015 . 5855 . 330. 2665 3 2555 . 360. 86 15 . 1 150 4 3705 . 755 . 7 10. 5965 5 1035 . 80. 75 . 155 6 2425 . 130. 120. 280 7 1850 . 2 1 10. 100. 7 90 8 1435 . 7 15. 65 . 455 9 2065 . 200. 1060. 550 10 5740. 1540. 3395. 3865 1 1 13005. 1235 . 1640. 2135 12 885 . 60. 40. 190 13 65. 0. 5. 25 1 4 1585 . 1470 . 125 . 540 15 1545 . 390. 505 . 650 PREDICTED COMMUTING FLOWS 5 6 7 8 9 10 2292 . 2 128 . 506 . 443 . 5853 . 4292 454 . 422. 2372. 1 180. 738. 5187 163 . 15 1. 40. 35 . 9850. 2925 317 . 294 . 193 . 151 . 1595 . 4054 32 16. 2128 . 19 . 19 . 3 10. 162 5651 . 4687 . 53 . 52 . 856 . 447 154 . 143 . 1772 . 285 . 313. 2203 130. 120. 385 . 592 . 168 . 84 1 478 . 444 . 43. 42 . 20592. 896 823 . 764 . 990. 6 17. 4 142 . 232 17 4436 . 262 1 . 143. 14 1. 10118. 12 17 1959. 1296 . 12 . 1 1 . 332 . 99 13 . 8 . 1 . 0. 104 . 5 107 . 100. 1383 . 222 . 156 . 137 1 127 . 118. 2 14. 119. 637 . 5010 EMPIRICAL COMMUTING FLOWS 5 6 7 8 9 10 1 165 . 1055 . 595 . 295 . 32 15 . 1865 390. 385 . 1395 . 895 . 845 . 1 195 225 . 210. 145. 40. 5 140. 5295 245 . 130. 230. 120. 865 . 1 180 5945 . 2450. 40. 50. 390 . 2 15 6260. 6590. 65 . 90. 855 . 350 190. 160. 2885 . 320. 280. 425 135 . 155 . 330. 1020. 225 . 320 290. 245 . 30. 30. 25105. 830 515. 515. 435. 200. 4055 . 32525 3295. 2360. 620 . 390. 13310. 2860 1305 . 1000. 30. 35 . 405 . 120 o . 0. 0. 10. 50. 20 115. 75 . 1110. 345 . 2 10. 510 240. 95 . 2 15. 70. 8 15. 42 15 1 1 12 13 14 15 26049. 1095 . 538 . 84 . 338 647 1 . 2 17. 76 . 3 14. 7 18 642 1 . 78 . 233 . 1 1 . 10O0 3603 . 151 . 74 . 32 . 145 10546. 1721 . 139 . 4 . 18 16550. 3023 . 243 . 12 . 55 1958 . 82 . 36 . 294 . 173 1649 . 62 . 22 . 7 1 . 146 18839. 229 . 858 . 7 . 139 9357 . 393 . 193 . 184 . 2565 174754. 3331 . 3608 . 30. 134 11305. 1647 . 166 . 3 . 12 813. 9 . 115. 0. 0 12 19. 5 1 . 22 . 8535 . 170 1439 . 60. 27 . 44 . 13 100 1 1 12 13 14 15 31375 . 435 . 2 10. 190. 395 8730. 120. 60. 345 . 3 10 9365 . 55 . 130. 55 . 360 4355 . 60. 25 . 85 . 165 7530. 1500. 75 . 30. 20 15410. 2380. 200. 5 . 60 3205 . 45 . 20. 485 . 155 2300. 75 . 20. 75 . 35 18295. 140. 5 15. 30. 120 13490. 200. 200. 300. 3325 163050. 1690. 4670 . 2 10 . 520 8070. 53 10. 1B5 . 10. 0 915. 10. 0. 0. 5 2055 . 100. 5 . 7665 . 145 2825 . 30. 35 . 140. 1 3 lOO 1 9 7 1 AGGREGATED D A T A : I C C M - l b MODEL PREDICTED COMMUTING FLOWS 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 16476. 913. 260. 5339 . 617 . 509 . 116. 288 . 2232 . 2296 . 17859. 57 1 . 257 . 0. 28 2 1797 . 3933 . 0. 1 153 . 0. 0. 952 . 647 . 263 . 2060. 5818. 0. 86 . 0 . 120 3 0. 0. 3954 . 0. 0. 0. 0 . 0. 3329 . 523 . 6449 . 0 . 2 17. 0. 134 4 3505. 322 . 9 1 . 4829 . 0 . 4 . 53 . 101 . 665 . 3045 . 3526 . 76 . 59 . 0. 25 5 520. 0. 0 . 0. 3856 . 1254 . 0. 6 . 163 . 0 . 599 1 . 1032 . 64 . O. 0 6 1527 . 0 . 0. 80. 4472 . 2697 . 0. 22 . 428 . 0 . 10337 . 1562 . 124 . 0. 0 7 604 . 1593 . O. 468 . 0. 0. 876 . 120. 87 . 727 . 1575 . 0. 35 . 0. 0 8 2 18. 1101. 0. 174 . 0 . 0. 95 . 401 . 39 . 222 . 1220. 0. 20. 0. 1 1 9 17 18. 0 . 852 . 611. 0 . 0. 0 . 0. 6191 . 57 . 12783 . 63 . 280. 0. 0 10 4 309 . 1065 . 626 . 8121 . 0. 0 . 85 . 243 . 1333 . 9370. 7 138 . 0 . 154 . 0. 466 1 1 8496 . 63 . 663 . 1289 . 3291 . 1148. 0. 136 . 6494 . 156 . 142359. 2991 . 1439 . 0 . 0 12 263 . 0. 0. 0. 1644 . 639 . 0. 0 . 248 . 0 . 91 10. 1303 . 103 . 0. 0 13 8 . 0. 6 . 0. O. 0. 0. O. 74 . 0 . 844 . 9 . 40. 0 . O 14 0. 0. 0. 0. 0. 0 . 1602 . 91 . 0. 0 . O. O. 102 1 . 5605 . 0 15 0. 0 . 703 . 0. 0 . 0 . 0. 0. 0 . 2260. 0. 0 . 60. 0. 5012 EMPIRICAL COMMUTING FLOWS 1 2 3 4 .5 6 7 8 9 10 1 1 12 13 14 15 1 15955. 765 . 395 . 3185. 960. 265 . 235. 185. 1 195 . 730. 23350. 235 . 125. 60. 120 2 3055 . 3965 . 150 . 2455 . 295 . 60. 405 . 395 . 275 . 500. 5030. 50. 30. 125 . 40 3 1205 . 170. 3975 . 985 . 120. 50. 35 . 45. 1440. 1 335 . 5065 . 25 . 75 . 10. 70 4 2235. 720. 320. 7555 . 155 . 40. 155 . 115. 460. 910. 3485 . 30. 15 . 30. 75 5 520 . 45 . 15 . 125 . 4395 . 1325 . 30. 15 . 115. 55 . 5 195 . 985 . 50 . • 5 . 10 6 1245 . 95 . 30. 180. 3915. 2965 . 50. 35 . 260. 90. 10870. 1350. 125 . 20. 20 7 840. 750. 45 . 510. 120. 40. 1550. 180. 85 . 140. 1630. 15 . 10. 155 . 15 8 640. 3 10. 40 . 340 . 65 . 40. 80. 555 . 60. 90. 12 15. 20. 10. SO. 5 9 8 10. 60. 295 . 400. 130. 55 . 35 . 15 . 10180. 300. 9975 . 35 . 250. 0 15 10 32 10. 725 . 1035 . 3560. 330. 105 . 175. 130. 1115. 14295. 72 15 . 105 . 70. 70. 770 1 1 8245 . 600. 7 10. 1890. 2420. 835 . 335 . 200. 5975 . 1200. 141800. 1060. 3000 1 IO. 145 12 550. 50. 30. 150. 795 . 4 15. 20. 20. 165 . 60. 72 15. 3675 . 150. 10. 5 13 20. 0. 5 . 25 . 10. 0. 0. 5. 15 . 5 . 895 . 0 . 0. 0. 0 14 465 . 595 . 35 . 3 15. 90. 30. 285 . 135. 40. 1 IO. 1035 . 5 . 10. 4935 . 40 15 445 . 155 . 75 . 390. 80. 25 . 50. 25. 155 . 895 . 1 105 . 10. 20. 45 . 4560 1 9 7 1 AGGREGATED D A T A : ENTROPY MODEL 1 2 3 4 1 13103. 890. 274 . 436*5 2 27 12. 3 107 . 63 . 1590 3 7 15. 54 . 4245 . 588 4 4 155 . 396 . 122 . 34 19 5 656 . 32 . 14 . 139 6 1699 . 73 . 32 . 360 7 983 . 1261 . 18 . 645 8 465 . 838 . 8 . 305 9 1829 . 56 . 807 . 764 10 5922 . 1246 . 754 . 7656 1 1 5663. 274 . 4 10. 1344 12 449 . 19 . 13 . 95 13 24 . 1 . 4 . 5 14 518. 474 . 14 . 340 15 546 . 283 . 378 . 449 1 2 3 4 1 15955. 765. 395 . 3 185 2 3055. 3965. 150. 2455 3 1205 . 170. 3975 . 985 4 2235 . 720. 320. 7555 5 520. 45 . 15. 125 6 1245 . 95 . 30. 180 7 840. 750. 45. 510 8 640. 3 10. 40. 340 9 8 10. 60. 295 . 400 10 32 10. 725 . 1035 . 3560 1 1 8245 . 600. 7 10. 1890 1 2 550 . 50. 30. 150 13 20 . 0. 5 . 25 14 465 . 595 . 35 . 315 15 445 . 155 . 75 . 390 PREDICTED COMMUTING FLOWS 5 6 7 8 9 10 1634 . 9 12. 275. 285 . 2634 . 2 143 270. 151 . 1074 . 633 . 277 . 2 157 89 . 50. 17 . 17 . 3403 . 112 1 295 . 164 . 137 . 127 . 935 . 2637 2168. 862 . 10. 1 1 . 132 . 76 3572 . 1781 . 25. 29. 342. 198 78 . 44 . 685 . 130. 100. 782 65 . 36 . 147 . 268 . 53 . 295 255 . 143 . 17 . 20. 6943 . 335 420. 234 . 385 . 284 . 1333 . 8289 3435. 1220. 85. 98. 4946 . 660 1482 . 589 . 7 . 8 . 159 . 52 12 . 4 . 0. 0. 58 . 3 52 . 29 . 507 . 96 . 47 . 462 54 . 30. 70. 46 . 172 . 1505 EMPIRICAL COMMUTING FLOWS 5 6 7 8 9 10 960. 265. 235. 185 . 1 195 . 730 295 . 60. 405. 395. 275 . 500 120. 50. 35 . 45 . 1440. 1335 155 . 40. 155 . 115. 460 . 9 10 4395. 1325 . 30. 15. 115. 55 3915. 2965 . 50. 35 . 260. 90 120. 40. 1550. 180. 85. 140 65 . 40. 80. 555 . 60. 90 130. 55 . 35. 15 . 10180. 300 330. 105 . 175. 130. 1115. 14295 2420. 835 . 335. 200. 5975 . 1200 795 . 4 15. 20. 20. 165 . 60 10. 0. 0. 5 . 15 . 5 90. 30. 285 . 135. 4.0. 1 10 80. 25 . 50. 25 . 155 . 895 1 1 12 13 14 15 19973. 703 . 336 . 7 1 . 162 4 134 . 116. 39 . 22 1 . 287 3779. 38 . 112. 7 . 368 3600. 127 . 60. 35 . 90 7647 . 1044 . 82 . 3 . 8 1 1251 . 1720. 135 . 9 . 24 1068 . 38 . 16 . 177 . 59 889 . 28 . 10 . 43 . 49 10820. 1 10. 401 . 4 . 50 5 130. 18 1 . 86 . 111. 880 145531. 2322 . 2445. 27 . 70 9 196 . 1122. 1 IO. 2 . 6 772. 7 . 89 . 0. 0 630. 22 . 9 . 4870 . 55 664 . 23 . 10. 23 . 378 1 1 1 12 13 14 15 23350. 235 . 125 . 60. 1 20 5030. 50. 30. 125. 40 5065 . 25 . 75 . 10. 70 3485. 30. 15 . 30. 75 5195. 985 . 50. 5 . 10 10870. 1350. 125 . 20. 20 1630. 15. 10. 155 . 15 12 15. 20. 10. 30 . 5 9975 . 35 . 250 . 0. 15 72 15. 105 . 70. 70. 770 141800. 1060. 3000. 1 10. 145 72 15 . 3675 . 150. 10 . 5 895 . 0. O . 0. 0 1035 . 5 . 10. 4935 . 40 1 105 . 10. 20 . 45 . 4560 

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