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A multi-regional computable general equilibrium model in the Haritage of A. Anas and I. Kim

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A multi-regional computable general equilibrium model in the Haritage of A. Anas and I. Kim

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A MULTI-REGIONAL COMPUTABLE GENERAL EQUILIBRIUM MODEL
IN THE HARITAGE OF A.ANAS AND I.KIM

by

Sungbin Cho

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PLANNING)

December 2006

Copyright 2006 Sungbin Cho

ii
ACKNOWLEDGEMENTS

I have listed a few names here not because I do not express my gratitude to the
people who are not mentioned, but there are too many I should express my thanks. First, I
would like to express my deepest gratitude to my supervisor, Professor James E. Moore for
his continuous support and encouragement over the last decade. Also, I would like to share
my heart to Professor Peter Gordon, and Professor Harry Richardson, who always have been
in my side with thoughtful advices. I also thanks to Professor Maged Dessouky for his
constructive advice.
I am very grateful to friends at Shan-Sem Church, including pastor Seok-Hyung
Yoon, and deacon Hong, who prayed for me for long times as much as I has been prodigal.
Their prayer was not in vain, I hope. I know it is my turn to pray for them now. Also many
thanks should go to friends at ImageCat – Ron and Charlie. I also thank to Professor Kyu-
Young Cho who abruptly appeared and drag me to setup the oral exam for this study, and
Shubharoop Gosh who was helping me to the end.
Finally, I would like to thank my wife Seiyoung who has been with me even in
hardest time, and my children Eunseo, Eunjin who have been my eternal joy.

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TABLE OF CONTENTS

ACKNOWLEDGEMENT ································································································ ii

LIST OF TABLES ·············································································································· vi

LIST OF FIGURES ··········································································································· vii

ABSTRACT ······················································································································· ix

1. INTRODUCTION ········································································································· 1
1.1 BACKGROUND ····································································································· 1
1.2 OBJECTIVES OF THE STUDY ············································································ 8
1.3 ORGANIZATION OF THE THESIS ····································································· 9

2. OVERVIEW ON THE SPATIAL INTERACTION MODELS ······························· 11
2.1 INTRODUCTION ·································································································· 11
2.2 LOWRY MODEL AND PUTMAN’S DRAM/EMPAL ······································· 11
2.3 INTEGRATED MODELS BASED ON URBAN ECONOMIC THEORIES ······· 13
2.4 SPATIAL MODELS BASED ON INPUT-OUTPUT FRAMEWORK ················ 14
2.4.1 MEPLAN ······································································································ 16
2.4.2 SCPM ············································································································ 17
2.5 CGE AS AN ALTERNATIVE MODEL ································································ 20

3. FORMULATION OF URBAN CGE MODEL ··························································· 23
3.1 OVERVIEW OF THE MODEL ············································································ 23
3.2 HOUSEHOLD MODEL ························································································ 26
3.2.1 Basic Household Model ················································································ 26
3.2.2 Extended Household Model for Multi-Modal Transportation System ········· 33
3.3 FIRM MODEL ······································································································· 35
3.4 DEVELOPER MODEL ························································································· 41
3.5 LOCAL GOVERNMENT MODEL ······································································ 44
3.6 EQUILIBRIUM CONDITIONS ············································································ 49
3.6.1 Equilibrium in the Market for Land ······························································ 50
3.6.2 Equilibrium in the Market for Housing Floor Space ···································· 51
3.6.3 Equilibrium in the Market for Industrial Floor Space ·································· 52
3.6.4 Equilibrium in the Market for Labor ···························································· 53
3.6.5 Equilibrium in the Market for Commodities ················································ 54
3.6.6 Equilibrium in the Transportation Network System ····································· 56

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3.7 VARIABLES ········································································································ 60
3.7.1 Model Solution ····························································································· 60
3.7.2 Existence and Uniqueness of the Solution ···················································· 61
3.8 SOLUTION ALGORITHM ··················································································· 64

4. CHARACTERISTICS OF THE URBAN CGE MODEL ········································· 75
4.1 OVERVIEW ·········································································································· 75
4.2 THE SETTING FOR LINEAR-CITY APPLICATIONS ······································ 77
4.3 BASELINE OF THE LINEAR-CITY ··································································· 83
4.3.1 Land Use Pattern and Floor Development ···················································· 83
4.3.2 Household Allocation Pattern ······································································· 87
4.3.3 Industrial Activity Allocation Pattern ··························································· 92
4.3.4 Travel Demand and Congestion in the Transportation System ···················· 98
4.4 SENSITIVITY ANALYSES FOR SELECTIVE PARAMETERS ······················ 103
4.4.1 Effect of Elasticity of Substitution in Regional Production ························· 104
4.4.2 Effect of Distance Decay in Intermediate Goods Movement ······················ 109
4.4.3 Effect of Growing Population in Urban Structure ······································· 117
4.4.4 Effect of Network Configuration in Travel Time to CBD ··························· 124
4.5 CONVERGENCE AND RESILIENCE OF THE ALGORITHM ······················ 129
4.6 REVIEW ON THE CES PRODUCTION FUNCTIONS IN
MULTI-REGIONAL CONTEXT ······································································· 134

5. MODEL APPLICATION ··························································································· 141
5.1 OVERVIEW ········································································································141
5.2 SETTING FOR SCAG APPLICATIONS ··························································· 144
5.3 SCAG BASELINE ······························································································ 153
5.4 POLICY OPTION ANALYSES ········································································· 160
5.4.1 Land Use Control ························································································· 160
5.4.2 Impacts of Unexpected Events ···································································· 164
5.4.3 Transportation System Management Options ·············································· 167

6. CONCLUSION ············································································································ 172
6.1 SUMMARY AND CONCLUSION ·····································································172
6.2 SUGGESTIONS FOR FUTURE RESEARCH ··················································· 174

BIBLIOGRAPHY ············································································································· 176

APPENDICES ··················································································································· 181
Appendix A. PARAMETERS FOR “LINEAR-CITY” APPLICATION ···················· 181
A.1 Parameters for Households ············································································· 181
A.2 Parameters for Firms ······················································································· 181
A.3 Parameters for Developers and Local Governments ······································ 182
A.4 Parameters for Transportation Market ···························································· 182
Appendix B. PARAMETERS FOR SCAG APPLICATION ····································· 183
B.1 Parameters for Households ············································································· 183
B.2 Parameters for Firms ······················································································· 183

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B.3 Parameters for Developers and Local Governments ······································· 184
B.4 Parameters for Transportation Market ···························································· 184
Appendix C. PRICE WITH ELASTICITY OF SUBSTITUTION,
j
θ =1 ··············· 185
Appendix D. SCAG BASELINE ················································································ 188
D.1 Land Use ·········································································································188
D.2 Households ·····································································································191
D.3 Industrial Activity ···························································································195
D.4 Transportation ·································································································201
Appendix E. RESULTS FROM SCAG POLICY OPTION ANALYSES ················· 206
E.1 Land Use – Housing Development Option ····················································· 206
E.2 Land Use – Downtown Development Option ················································· 208
E.3 Land Use – No Land Use Plan Option ···························································· 209
E.4 Unexpected Event – Coastal Zone ·································································· 210
E.5 Unexpected Event – Puente-Hill Earthquake Scenario ··································· 211
E.6 Transportation Investment – Speed 5% Increase Option ································ 212
E.7 Transportation Investment – Capacity 5% Increase Option ···························· 213
E.8 Transportation Investment – Completion of I-710 ·········································· 214
Appendix F. NOTATIONS ························································································· 225

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LIST OF TABLES

Table 3.1 Number of Variables to be Solved ······························································· 61
Table 4.1 Baseline Transportation Network Performance in the Linear-City ············· 99
Table 4.2 Regional Production of the Linear-City by Elasticity of
Substitution of Industry 3 ··········································································· 105
Table 4.3 Regional Production in the Linear-City by the Distance Decay
Coefficients for Intermediate Inputs. ·························································· 111
Table 4.4 Regional Production of the Linear-City by Number of Households ··········· 119
Table 4.5 Network Configuration Scenarios for the Linear-City ······························· 125
Table 5.1 Land Use Included in the SCAG Zone System ··········································· 147
Table 5.2 Free Flow Speed and Capacity for SCAG Applications ····························· 150
Table 5.3 Distribution of Links by the Attributes ······················································· 151
Table 5.4 SCAG Baseline Estimation ·········································································· 154
Table 5.5 Land Use Options for SCAG Application ··················································· 161

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LIST OF FIGURES

Figure 2.1 Conceptual Flow of Southern California Planning Models ·························· 19
Figure 3.1 Framework of Urban Computable General Equilibrium Model ·················· 24
Figure 4.1 Zone System and Network Configuration of the Linear-City ······················ 78
Figure 4.2 Land Use Pattern in Baseline of Linear-city ················································ 85
Figure 4.3 Household Allocation Pattern in Baseline of Linear-city ···························· 89
Figure 4.4 Firm Allocation Pattern in Baseline of Linear-city ······································ 94
Figure 4.5 Intermediate Transactions in Baseline of the Linear-city ···························· 96
Figure 4.6 Transportation System in Baseline of Linear-city ······································ 101
Figure 4.7 Relative Office Rent and Wage to Commodity Price by Varying
Elasticity of Substitution,
3
θ ······································································ 107
Figure 4.8 Industrial Allocation Pattern by Varying Elasticity of Substitution,
3
θ ···· 108
Figure 4.9 Price and Output Value of µ=10 case and Baseline by Iteration ················ 113
Figure 4.10 Intermediate Transactions by Distance Decay Coefficients ······················· 115
Figure 4.11 Zonal Production Compositions by Distance Decays ·································· 117
Figure 4.12 Traversing Travel Time by Number of Households ··································· 121
Figure 4.13 Household Allocation Patterns in 200K and 240K Cases ··························· 122
Figure 4.14 Output and Income Distribution in 240K Case ············································ 123
Figure 4.15 Average Travel Time by Network Configurations ···································· 127
Figure 4.16 Output Value Produced by Network Configurations ································· 128
Figure 4.17 Convergence of the Model ········································································· 131

viii
Figure 5.1 Analysis Zone System for SCAG Applications ·········································· 146
Figure 5.2 Ratio of Developed land in SCAG Application ·········································· 148
Figure 5.3 Network Data Created for SCAG Application ··········································· 149
Figure 5.4 Distribution of the Scale Factor for Initial Output Value ··························· 152
Figure 5.5 Household Allocation Pattern in Baseline of SCAG ·································· 158
Figure 5.6 Industrial Production Allocation Pattern in Baseline of SCAG ·················· 158
Figure 5.7 Intermediate Transactions in Baseline of SCAG Application ····················· 159
Figure 5.8 Output Changes Due to Housing Development Option ······························ 162
Figure 5.9 Output Changes Due to Downtown Development Option ··························· 163
Figure 5.10 Output Changes Due to ‘No Land Use’ Option ·········································· 164
Figure 5.11 Output Changes Due to Unexpected Event in a Coastal Zone ···················· 165
Figure 5.12 Output Changes Due to Puente-Hill Earthquake Scenario ························· 166
Figure 5.13 Output Changes Due to 5% Free-Flow Speed Increase Option ·················· 167
Figure 5.14 Output Changes Due to 5% Capacity Increase Option ······························· 168
Figure 5.15 Output Changes Due to I-710 Completion Option ····································· 170

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ABSTRACT

The objective of this study is to establish an alternative modeling scheme for
comprehensive urban transportation studies. The model developed in this study is based on
Anas-Kim’s computable general equilibrium model and expanded in applicability to address
real world problems existent in complex urban systems.
The early conceptualization of a comprehensive general equilibrium (Anas, 1980)
has turned into an operational model by Anas and Kim (1990, 1996), Anas and Xu (2000)
and has been applied to small theoretical examples. Despite the sound theoretical
background on which their model was developed, the equilibrium price formulation has an
undesirable property of depending on number of factors, i.e., combination of input
commodities and input locations in multi-regional context. A numerical example proves the
model may not be computable, unless input data is strictly manipulated.
This study updates three aspects of the Anas-Kim’s comprehensive general
equilibrium model. First, the model has been expanded to accommodate for multiple
households and industries. This allows for diverse preference and production technologies.
Second, developers and local government are included as additional economic agents. By
including developers, land use pattern is explicitly modeled. Finally, a CES function replaces
the production function in the model, and a stable equilibrium price formulation is derived.
The CES function used is constant return to scale, yet the unity sum of input share
coefficients is held without any exponent on the coefficients.

x
To fulfill the object of study, the proposed model is applied to two distinctive data sets. A
small data with 11 zones is created, and applied to evaluate characteristics of proposed
model for the following selected parameters: elasticity of substitution, distance decay for
intermediate goods movement, growing population, and spatial distribution of network
capacity. The proposed model also demonstrates its applicability to real-world planning
options, through a Southern California dataset. Impacts from changes in land use policies
are analyzed with respect to regional production, household utility, and transportation system
performance. In testing transportation system improvement options, the model shows
capacity is of greater importance than free flow speed to improve regional mobility.

1
1. INTRODUCTION

1.1. BACKGROUND

Many tools have been developed to analyze urban transportation systems, but most
have gained only partial success. Although ‘abstraction’ by simplification is the only proper
way to model real-world subjects, lack of comprehensiveness in describing systems often
results in unacceptable ‘ceteris paribus’ assumption. And urban transportation study is not
an exception, especially when the models are applied to policy analysis.
The partial success is related to mathematical description of selfish economic agent
behavior. Optimization models of profit and utility are some examples of such
accomplishment. In these models, individual agents are purely rational, and choose the best
alternative to maximize interest – such as the choice of shortest travel time paths against
given constraints like restricted resources. Usually the constraints are not the focus of study,
so the models keep the constraints unchanged throughout the applications. However, as an
urban system evolves into a system with diverse character, invariability assumption is not
valid in many cases.
As the system becomes more diverse, policy designers need to consider even more
sophisticated ways to control various issues in the system. Direct effect from an
implemented policy cascades to all corners of the system with different intensities, and
creates new environments. As the agents adapt themselves to the new environment under
the policy, indirect effects are generated and have a ripple effect across the system. In many

2
cases, the secondary impacts from an implemented policy are greater than the primary or
direct impact.
The four-step transportation planning model system is an example that was
developed in 1920s based on limited interpretation of the relation between urban systems
and travel demand. Transportation activity has been considered as a repetitive daily
phenomenon with negligible variations. Consequently, travel demand was modeled by snap-
shot of socio-economic factors since the first generation models implemented to
metropolitan areas (Wiener, 1997, pp.12~16).
As a specific example, observed travel demand is regressed against surveyed
parameters such as population and number of jobs. The regressed relationship is applied to
forecasted parameters to estimate future travel demand. Using regression to estimate trip
generation means that there is a causality between transportation system and urban activity
system, and side of equation subordinates the other. This violates the second equilibrium
state that Manheim classified (1979, p.29, 326). According to the classification, there are
three types of equilibrium states in urban activity systems, which are 1) equilibrium in short
term transportation market between travel demand and network capacity; 2) equilibrium in
activity system under given transportation system, and exogenous factors; and 3) equilibrium
of transportation operators, seeking maximized utility from infrastructure. The first and
second equilibrium conditions state that, activity system evolves under a given transportation
system, as the transportation system supports the activity system. In other words, capacity of
transportation system is a function of socio-economic environments. Socio-economic
factors also rely on the performance of transportation system, and the system performances

3
is affected by each other. By assuming ‘ceteris paribus’, researchers discard the evolving
mechanism in transportation and activity system.
The conventional four-step process has been improved in various ways, and the
following are some seminal researches. Introduction of combined modeling techniques,
such as the works by Evans (1976), Boyce (1981), and, LeBlanc et al. (1983) enhanced
internal consistency between component models of the four-step procedure. Beckman et al.
(1956,) strengthened the reality in modeling the route choice behavior of a given travel
demand, while Frank-Wolfe (1956) increased the applicability of network model with an
efficient solving algorithm for the user equilibrium assignment framework. Wilson’s work
(1970) demonstrated that social science converges to natural science by deriving gravity
formula from entropy maximization concept in distribution of markets. McFadden (1976,
pp.47~99) opened a way to tractable model of human behavior under heterogeneity in
choices. Even with these progresses in transportation models over few decades, the
paradigm has not changed – transportation activity is a separable activity that can be
modeled independently from other urban system models.
Including Kitamura, many researchers studied about activity based travel demand
models since 1970s as an alternative approach of trip generation model in the four-step
procedure (Golob 1997). Detailed records of daily movement allow the researchers to
identify activity chains, and help them to create a reliable travel demand model (Kitamura
1984, 1988). However, even the microscopic record on activities only provides finely
disaggregated independent variables for demand models that might be calibrated with the
way that was used in the traditional method. Even though it is “activity based”, the model
does not present surrounding environments that the travelers interact with. Socio-economic

4
environments are given to them unchangeable, and it does not explain the trip generation
mechanism.
Urban transportation systems need models that preserve both of actors and activity
systems, and spatial computable general equilibrium (CGE) analysis can be an alternative
because the model structure fit the needs. CGE models are based on explicit description of
agents , such as households, industries, and governments, and equilibrium states in the
economic systems (Dixon et al. 1996, pp. 3~84).
Many spatial versions of CGE models describe in detail economic agents and
various activity systems, but fail to appropriately address the transportation network which is
an import aspect of an urban system. Br őcker (1998), and Buckley (1992) and various other
economists tried to incorporate multi-zonal figure into conventional CGE analysis so that the
transportation sector is modeled explicitly. However, as Tavasszy and Thissen (2002)
summarized, inclusion of transportation analysis module in a CGE model might have several
‘pitfalls’. In their models, transportation systems are presented by an industrial sector that
produces delivery service and other industries pay travel cost to the transportation sector.
Travel cost of commodity is modeled by discounting the original value of commodity, as if
drifting ‘iceberg’ is melting over the trajectory, which is an unrealistic representation. A
transportation network model should endogenizes congestion in scarce network capacity and
as such the price should increase with delivery distance, instead of reducing the value of
commodity.
Anas (1984) described highly comprehensive urban system that includes
transportation network in a CGE model by tying five prominent modeling disciplines
together – Mills’ general equilibrium of urban economics, Wilson’s entropy based

5
allocation, Lowry’s urban interaction, Beckmann’s network equilibrium, and McFadden’s
discrete choice model. Using the exogenous policy variables such as population, level of
export, zonal employment, production technology, and land use pattern, his model estimates
endogenous economic variables – the zonal price vectors of land rent, office rent, housing
rent, intermediate commodity prices, and wage. The model also estimates travel demand
from households and industries that produce activities, such as, working trips, shopping trip,
and intermediate goods deliveries, and compete for the given network capacity, while the
endogenous travel time and cost are used as key factors for location choices. Although Anas
emphasized tractability and detailed the desirable extensions to enhance reality in the model,
as well a thorough review on the existence of equilibrium condition, it was developed as a
theoretical model, only to demonstrate the possibility of ‘elimination of ad hoc links’
between paradigms. This model is yet to be applied to any practical analysis.
Based on Anas’ early effort, Kim (1990) developed a similar CGE model to describe
highly autonomous urban systems, and applied to explain urban structure under
heterogeneous choices. The model presented interactions between idiosyncratic households
and rational firms over transportation network and various markets. Households sell the
endowment to earn income, and spend the earnings on various commodities including
location-specific housing to maximize their utility. Firms choose the location of business,
level of production, and input requirements for production to maximize profits. Both
economic agents meet at labor market, land market, commodity market, and transportation
network. The interactions determine the wage, rent, commodity price, level of production,
as well as network congestion. Estimated equilibrium price vectors are fed back into
behavior models to estimate responses of the economic agents would make. Kim provided

6
numerical analysis using a simple linear urban system, and explained the creation
mechanism of non-monocentric urban structure (Kim 1990, Anas and Kim 1996), and the
similar example was applied by Anas and Xu (1999).
Regarding transportation analysis, his model is equivalent to combined model of trip
production, distribution, and assignment models in terms of the conventional four-step
modeling system. As a trip production step, home-based working trips and shopping trips
are estimated as byproducts of modeling household activities. Freight movement is also
estimated from delivery of the intermediate inputs. As residential locations and firms’
business locations are endogenously determined, the working places, shopping places, and
destination of intermediate goods movement are also spatially defined. So the estimated
travel demand is distributed spatially, and result in O-D matrices. Network analysis model
loads the estimated travel demand onto network, and calculates zone-to-zone congested
travel time. In short, Kim’s CGE model is a highly comprehensive urban system model that
includes transportation analysis system in it, along with desirable interaction models, such as
land use, industrial activity, household activity, and market price.
However, three aspects of Kim’s model has to be expanded or revised to enhance
applicability to real-world problems. First, various economic agents, especially households,
should be explicitly included. In his CGE model, heterogeneity of households is described
by a stochastic variable in the utility function. The explicit differentiation of households by
using separate sets of preference vectors would helpful to model the wide range of
household characteristics in residential allocation, income distribution and travel mode
choice behavior.

7
Second, including more household types and additional agents will enhance the
applicability of the CGE model. Although households and industries are the most important
agents, decisions by developers and local governments are also significant in shaping the
urban systems. Incorporating developers would make the unambiguous presentation of land
use behavior in a model. Developers turn undeveloped land into spaces for living or
business, and the land market in Kim’s CGE model is divided into markets of raw land,
residential space, and office space. Land use control can be modeled by supply of limited
amount of land for specific usage. Distinct equilibrium rents due to discriminated land
supply will differentiate the level of activities.
In addition, inclusion of local government in a model as an agent who produces
‘worker amenity’ as a tradable good would improve our understanding of dynamics in urban
systems. Researchers, such as Heikkila et al. (1989), and Michaels and Smith (1990),
demonstrated that various amenity measures differentiate intra-urban residential property
value. Also, Crane et al (1997) showed that accessibility to public service is a meaningful
factor in residential allocation pattern. However, as Anas reviewed (1982, p.264), behavior
of local governments has been rarely studied for regional models. Considering local
governments who levy sales taxes on firms and produce amenity would allow a more
realistic residential allocation, without increasing model complexity.
Third, Kim’s equilibrium price is highly sensitive to the size of problem, i.e., the
number of zones as well as the number of industrial sectors considered in application, as
reviewed in Section 4.6. Governed by distance decay input function, firms in the model are
allowed to interact with any other firms for intermediate inputs independent of location. The
spatial equilibrium price equation derived from the Cobb-Douglass production function

8
practically prohibits the model from working in the way it is supposed to when the
application becomes bigger than the example that he used for his research. To be applied to
undetermined systems, the model needs a more general production function to drive a stable
commodity price equation.

1.2. OBJECTIVES OF THE STUDY

The objective of this study is to establish an alternative modeling scheme for
comprehensive urban transportation studies. The new model developed in this study is
based on Kim’s computable general equilibrium model, expanded in applicability to address
real world problems existent in complex urban systems. This study demonstrates the
versatility of CGE model in urban environments with emphasis on policy analysis for
transportation and land use control.
The objective of this study will be fulfilled by performing the following tasks:
• Revise Kim’s CGE model (1990) into an applicable urban computable general
equilibrium (UCGE) model for arbitrary urban systems. The urban CGE model will
describe interactions between households, firms, developers, and local governments over
the labor market, commodity market, land market, rent market and transportation
network. Detailed solution algorithm will be provided along with a brief discussion on
existence and convergence of equilibrium prices. To demonstrate applicability of the
model, two sets of synthesized urban systems will be used.
• Review the characteristics of suggesting urban CGE model in various conditions. The
model will be applied to a small synthesized example with perturbed input parameters,
as well as assumed baseline condition. From the baseline analysis, the model will be

9
demonstrated for wealth of products that can be modeled by listing the individual
outputs from one run. Responses to the perturbed inputs will be reviewed to express the
capability of the proposed model to be used as a policy analysis tool.
• Apply suggested model to an example of practical size. The application will analyze
transportation planning alternatives such as increment of highway network capacity and
land use control in specific sub-regions. Model results will be examined focusing on
variations of spatial distribution.

1.3. ORGANIZATION OF THE THESIS

Chapter 2 reviews literature on urban models that focus on a combination of
transportation and other activity models. Southworth (1995) classifies these combined
models into 1) Lowry and DRAM/EMPAL; 2) spatial modeling using Input-Output
framework; and 3) integrated models based on urban economic theories. In addition, CGE
model is reviewed with special attention to Kim’s model (1990).
Chapter 3 details the proposed urban CGE model based on Kim’s CGE model. The
model is described considering 1) households’ utility maximizing behavior; 2) firms’ profit
maximizing behavior; 3) developers’ profit maximizing behavior; 4) local governments’
production maximizing behavior under balanced budget; 5) equilibrium conditions on land
market, rent market, labor market, commodity market, and transportation network. Solution
algorithm and discussion on existence of equilibrium prices are also provided in Chapter 3.
Production functions for industries, developers, and the local governments are based on a
variation of CES (Constant Elasticity of Substitution) production function in the proposed

10
urban CGE model. This form of production function is different from Kim’s model. The
effect of replacing the form of production function is reviewed in Chapter 4.
Chapter 4 demonstrates the rich characteristics of the proposed urban CGE model
using a small synthesized urban system. The sample system consists of 11 zones, 3 types of
households, 3 types of industries, developers and local governments in each zone. The
transportation network consists of 26 directional links. The model is applied to perturbed
conditions of regional population, agglomeration and distance decay, elasticity of
substitution of inputs, dispersion of households, and travel costs. As a type of perturbations,
two different production functions – Cobb-Douglass against modified CES – are compared
in terms of stability of driven equilibrium price equations.
In chapter 5, a separate set of synthesized data is used with the proposed model for
practical issues. Data was created for the Southern California Association of Governments
(SCAG) planning region based on public domain data. The system consists of 266 zones, 3
types of households, and 3 types of industries. The transportation network consists of
15,896 directional links, and 5,146 nodes. Using this data, three types of policy options are
tested – i) effects of various land use policies on regional economy and transportation
system, ii) comprehensive impacts from unexpected events to regional economy, and iii)
effects of enhancing of network capacity, speed, and connectivity. Chapter 6 summarizes
application of the proposed urban CGE model to hypothetical planning alternatives, and
provides future research directions.

11
2. OVERVIEW ON THE SPATIAL INTERACTION MODELS

2.1. INTRODUCTION

This chapter reviews spatial interaction models focusing on operational integrated
transportation / land use models. Theoretically, an integrated transportation / land use model
analyzes activity allocation patterns with travel demand and provides travel time estimation
capabilities. A review of such models allows establishing a base condition for the modeling
efforts undertaken in this research. Southworth (1995) provides a comprehensive review on
these integrated transportation/landuse models by classifying them into: 1) Lowry based
Putman’s DRAM/EMPAL (1983, 1991); 2) models rooted on urban economic theories; and
3)models combined with Input-Output model. This chapter reviews the models based on
Southworth’s classification and discusses CGE model as an alternative framework for spatial
interaction models.

2.2. LOWRY MODEL AND PUTMAN’S DRAM/EMPAL

Lowry’s “metropolis” model maybe the first operation integrated model.
Incorporating a limited number of equations and constraints, his model distributes
population, basic sector employment, service sector, and land use. Given employment
requirements and ratio to population, which is the basis of demand for service, the model
allocates basic sector employment over exogenous set of land use zones. Then, the demand

12
from households is equilibrated to service supply at each zone by means of allocation of
service sector at given travel distance between zones. In essence, the approach consists of a
linkage between the economic base model that defines the basic activity against non-basic
activities in a region economy, and a spatial allocation model which considers travel
impedance over space.
Putman is one of the most well known successors of the Lowry model. He
developed an integrated urban model named Transportation Land-Use Package (ITLUP), by
integrating his Disaggregated Residential Allocation Model (DRAM), the Employment
Allocation Model (EMPAL), and some of the Urban Transportation Planning System
(UTPS) components, such as mode choice module and traffic assignment module (Putman,
1983, 1991). Based on zonal accessibility, total population, and total employment, EMPAL
allocates employments over zones. Then, DRAM allocates households and related land
consuming activities.
As a byproduct, DRAM also generates travel demand matrices. Allocated activities
are translated into vehicle trip production by DRAM, and classified as home-to-work, home-
to-shop, and work-to-other trip matrices. Integrated multinomial logit model then splits the
origin-destination requirement (O/D) for travel modes of private and public vehicles.
Capacity restraint traffic assignment model loads the vehicle O/Ds to transportation network
and updates travel cost matrix. Based on marginal changes of travel cost, new accessibility
measure is calculated and is fed back to EMPAL and DRAM to update employment and
residential allocation.
Southworth (1995) observed that Putman’s integrated model is operational because
it is based on data that is generally available. For this reason, Putman’s model is renowned

13
as the most popular urban integrated model. However, no market clearing mechanism or
behavioral modeling in activity allocation are two critical weaknesses of the model as a
comprehensive policy analysis tool. Competition over land and service induce changes in
land and service prices, and, the altered price affects allocation decision. But, Putnam’s
model explains all the allocation and distribution of activities by only using the endogenous
“accessibility.” Despite including many variables in calculation of accessibility, such as
latent development potentials and network access cost, lack of behavioral consideration in
his model limits credibility of the results.

2.3. INTEGRATED MODELS BASED ON URBAN ECONOMIC THEORIES

Models based on this approach attempt to endogenize the land market clearing
mechanism by applying neoclassical economic theory. Initiated by Wingo (1961) and
Alonso (1964), households choose their residential location in such a way that their utility is
maximized, under the constraint of a “bid-rent” function. The “bid-rent” function is an
expression of a trade-off between the housing price and travel cost to the urban center, and
each location is rented to the highest bidder. This framework was addressed by Mills (1979)
with linear programming, Anas (1984) with nonlinear, entropy/utility-maximizing and
network-based programming forms.
In a similar effort, Kim (1989) developed an integrated urban model for Chicago by
combining Mill’s general urban system equilibrium, Wilson’s entropy maximization in
spatial interaction, Boyce’s combined transportation-facility location models, and
Beckman’s user equilibrium network assignment model. He named the model as “3-
dimensional” urban land use model for his introduction of “density related production

14
technology”. The model expresses a strong bond between interregional input-output, land
use and transportation within a single mathematical programming. The objective function of
the model is a joint minimization to the Wardrop’s user equilibrium principle in a network
system, and the total exporting costs out of the region, as well as the total land rental costs.
As constraints, he included conservation rule of path traffic flow, predefined export
requirement, input-output relationship between production activity, commodity market
equilibrium condition, and entropy maximization principle. Substitution mechanism of a
production side was expressed by a regional input-output model, with differentiated
technologies for land use intensity in production.
Even though he demonstrated the possibility of integration of urban economic
theories into multi-regional context, Kim’s model excluded households. To evaluate
network system cost properly, the model should have to use a full O/D set that consists of
personal trip and freight movement. But in his model, it is not clear that if travel demand
from households is a given or ignored. In either case, the estimated travel impedance of time
and cost are not acceptable if household behavior is excluded.

2.4. SPATIAL MODELS BASED ON INPUT-OUTPUT FRAMEWORK

Since Isard (1951) suggested the “ideal” interregional input-output formulation,
many researchers, such as Rieffler and Tiebout (1970), Moses (1955), and Leontief, and
Strout (1963) focused on spatial trade models. Isard’s model is an extended version of
Leontief’s input-output model to inter-region as well as inter-industry (Battern et al. 1985).
The original Leontief’s input-output model can be expressed as equation (2-1);

15
i
j
j j i i
y x a x + ⋅ =
∑
····················································································· (2-1)

where
i
x is total output of industry i,
ij
a is input to the production of industry j from
industry i, and
i
y is the final demand for the product in the region as a whole. By
disaggregating the economy of a region over space as well as industrial sectors, the equation
(2-1) is expended multi-regional counterpart;

∑∑ ∑
+ ⋅ =
qj q
i
pq
j
q
ij
pq
i
p
y x a x ·········································································· (2-2)

where
i
p
x is output from industry i produced in region p,
ij
pq
a is input from
industry i in region p to industry j in region q, and
i
pq
y is the final demand in region
q that is satisfied by the production of industry i in region p (Battern, et al, 1986).
Riefler named Isard’s model as “ideal” because Isard’s model perfectly
explains the trade flow not only between industries but also between geographic
origin and destination (Riefler, 1973). To represent the trade by industrial sectors
and geographic origin and destination, a great amount of information is required to
establish the technical coefficient matrix,
ij
pq
a . Due to this heavy data dependency,
Isard also mentioned that the geographical unit of the model could be coarse as much
as states if any trade pattern data is available.
To circumvent the data problem of the “ideal” interregional input-output
formulation, the most of subsequent models simplify the equation (2-2). For
example, Riefler and Tiebout (1970) applied several regional input coefficient matrices,

16
instead one big multi-regional matrix. In contrast, Chenery and Moses (1953), Leontief and
Strout (1963) suggested simplified version of interregional models by employing the theory
of trade pool or demand by production region (Hartwick, 1971; Miller, et al, 1985, pp 45-97).
The concept was introduced by Chenery and Moses (Bröcker, 1998). All commodities
produced by an industry from various regions are transported to a pool in a region. Once
goods are merged into a pool, the pooled goods are delivered for use of intermediate or final
demand in the same region. With this assumption, the location of the goods production or
the industry for the input does not need to be specified.
In spite of the wide popularity due to its simplicity, the input-output model does not
represent the price effect or more precisely, the elasticity in substitution, for expensive
products. Predefined and rigid technical coefficients determine the amount of transaction
between industries regardless of scarcity of commodity. For a multi-regional version,
intermediate transaction over space should be formulated in terms of delivery cost and
appropriate elasticity. Some operational models overcome these problems by integrating
with transportation modules. As examples, reviews on MEPLAN, and SCPM are provided.

2.4.1. MEPLAN

Among the integrated models that include input-output modeling, MEPLAN is one
of the elaborate models (Hunt and Simmonds, 1993). In this model, land market,
commodity market and transportation interact by means of price and transport cost.
Different from the traditional input-output framework, the factor price is endogenously
determined by means of either of weighted average of the production cost plus shipping, or
by an iterative process that establishes the market price, which results from equilibration

17
between supply and demand for factor in each zone. One notable feature is that MEPLAN
allows flexible input-output relationship by applying factor price sensitive coefficient, which
is similar to elasticity of substitution. MEPLAN estimates transportation demand from
spatial input-output and land use sub-models. A mutinomial logit model splits the demand
into different modes and Dial’s (1971) probabilistic, multi-path assignment model allots the
demand on to the network. As with other operational models, MEPLAN feeds the updated
accessibility (travel time) back to activity-location model and market pricing variables.
It is not clear if the equilibrium price estimated in MEPLAN is determined by any
market clearing mechanism. The developers describe the equilibrium procedure as a “tug-
of-war” between two pulling forces in opposite direction: input factor price from other zones
and output price to be traded to other zones (Hunt and Simmonds, 1993). This explanation
may have some conceptual difference from the concept of market clearing process in which
price is determined from a tug-of-war between demand and supply. Since the model does
not have explicit production mechanism other than the traditional input-output model, the
equilibrated price might be misinterpreted.

2.4.2. SCPM

By incorporating Garin-Lowry style allocation model with regional input-output
model, researchers created a series of regional economic impact analysis models, named
Southern California Planning Model (SCPM) (Richardson and Gordon,1989; Jun, 1999; Cho
et. al., 2000). As standard application of the input-output model, given marginal changes in
final demand (direct impacts) will be amplified as the direct impacts are going through the

18
inter-industrial linkage represented by regional input-output model (indirect impacts), and
also will cause household consumption change (induced impacts) in SCPM.
Estimated impacts are allocated over space according to regional activity system in
SCPM, and the zonal employment, the journey- to-work, and journey-to-shop matrices are
used as pseudo variables that might represent the activity system. In earlier version of
SCPM, the estimated indirect impacts are allocated in proportion of employment. Allocation
of induced impacts involves three steps. First, the aspatial induced impacts are allocated
over the place of work in proportion of zonal employment. Second, those are transferred to
the place of residences from the work place by using (transposed) the journey-to-work
matrix. Last, the induced impacts are further relocated from the place of residence to the
place of consumption by means of journey-to-shop matrix. Figure 2.1(a) depicts this
allocation process.
The subsequent version of SCPM integrates transportation models, especially, trip
distribution and assignment combined model, into the input-output model. This integration
endogenizes journey-to-work and journey-to-shop matrices, as well as intermediate freight
trips, and replaces the inflexible journey matrices and employment data. The allocation of
impacts over space becomes more realistic than the earlier one, in response to the distance
(or travel cost) over transportation network from where the direct impacts are imposed. The
improved allocation model is depicted in the dashed box in Figure 2.1 (b).

19

(a) SCPM 1 (b) SCPM2

Figure 2.1 Conceptual Flow of Southern California Planning Models
Direct Impact
I/O model
w/o HH
I/O model
w HH
Total Indirect
Impact
Total Induced
Impact
Employment
by place of work
by industry
Allocated
Indirect impact
Induced impact
by place of work
Induced impact by
place of resident
Induced impact by
place of consumption
Transpose of
Home-to-work trip
Home-to-shop trip
Transportation
Network Model
Travel
Cost
Freight
Movement
Passenger
trip demand
Impact
allocation
Activity Allocation Model
Travel
Demand
Direct Impact
I/O model
w/o HH
I/O model
w HH
Total Indirect
Impact
Total Induced
Impact

20
In spite of several application to practical planning problems, such as evaluation of
regional growth plan (Heikkila, et.al. 1991), an earthquake impact analysis (Cho et.al. 2000),
a tsunami impact analysis (Borrero, et. al. 2003), SCPM has two critical problems. First, as
Figure 2.1 shows, there is no feed-back from allocation to the input-output model. It means
the total impact is only determined by exogenous final demand change, and no secondary
effect is involved. Second, even though the model has been applied to natural hazards, as
long as the impact estimation is performed by the inflexible input-output model, it is not able
to incorporate the effects from rapid price shift due to demand surge.

2.5. CGE AS AN ALTERNATIVE MODEL

According to Dixon et al. (1996), CGE model includes more than one economic
actor whose behaviors are expressed by optimization formulations, and the level of the
activities is influenced by input prices. Typical actors included in such models are
households as utility maximizers and forms profit maximizers or cost minimizers. The
behavior models describe how the actors make the decisions on demand and supply in
responding to the factor price.
This framework is distinguished from the input-output models for lack of sufficient
behavioral specifications of the economic actors and role of price in the input-output model.
However, a set of input-output accounts is used for calibration purpose. The social account
matrix provides information about commodity flow between industries, households,
governments, importers, and exporters. The inflexible input-output relationship is
supplemented by numerical estimates of various elasticity parameters. And due to the

21
efforts that make the CGE flexible in modeling in industrial linkage, CGE is a natural
candidate that might overcome the drawbacks of multiregional input-output analysis.
Besides the anticipated usefulness, CGE models are known to be data-hungry. Even
Bröcker’s (1998) “operational” spatial CGE model shows that a CGE model has extensive
data requirement. As a typical CGE model, his model has three economic actors, which are
households, firms and transporter. Transporter takes responsibility of delivery of
commodities for intermediate and final demand. For simplicity, he employed several
assumptions such as trade pool, simple transport rate, and fixed travel cost in modeling the
transporters’ behavior. In spite of his effort to make the model simple, the statement on
operability was conditional. If there is data, his model could be calibrated and, in turn,
could be operational. For this reason, Oosterhaven (1984) exclaimed that CGE models are
not operational at all because it is awfully costly.
As reviewed above, CGE models require data to represent industrial linkage as a
function of price, as well as the actors’ behaviors in tangible formulations. Examining the
works by Anas and Kim (1996), Kim (1990), and Anas and Xu (1999), the requirement can
be fulfilled by various considerations for multi-regional context. Applying a gravity model
to disaggregate the technical coefficients is an example. In their hypothetical computable
general equilibrium models, a typical Cobb-Douglass production function needs spatial input
coefficients by sectors,
ij
rq
α , which determines the relative amount of intermediate input
purchased from industry j at a zone q, to industry i at zone r. Instead of assuming that the
coefficients will available, they started with an aspatial version,
ij
α , which comes from the
social accounting matrix, or region input-output tables as the technical coefficient matrix.
And then, a simple decay function is assumed, and disaggregate it spatially according to

22
travel time. This effort reduces required coefficients from (I×R)
2
to I×R+n, where I is
number industries, R is number of zone considered respectively, and n is number of
additional coefficients needed for decay function.
Application of gravity model to explain the interaction between firms in
metropolitan context has plenty of empirical basis. Richardson anticipated that gravity model
could be a good tool to summarize the effect of agglomeration and dispersion, when the
coefficients are available (1979, p 159). But as Wilson (1970) developed a general gravity
model from the entropy maximizing principles, he also demonstrated an iterative method to
calibrate a negative exponential distance decay function in studies of commodity flow. Also,
Cho et. al. (2000) applied similar technique for SCPM2 to calibrated distance decay
coefficients for selected intermediate commodity flows (mining, wholesale, durable and non-
durable manufacturing) in Los Angeles area. And similar efforts for other data requirements
can be found from various disciplines to make the CGE model operational.

23
3. FORMULATION OF URBAN CGE MODEL

3.1. OVERVIEW OF THE MODEL

This chapter details the architecture of the urban CGE model. By simulating
behaviors of economic agents, the urban CGE model describes key activities taking place in
a region. A closed region is divided into a finite number of zones. A transportation network
system connects these spatially disaggregated zones. Land is allotted to activities which
include residences, businesses, transportation facilities and public services provided by local
governments.
Four economic agents - households, firms, developers and local governments,
interact by means of production, consumption and transport in a closed economic system.
Figure 3.1 depicts the general framework of the urban CGE model.
Households earn their income by selling endowments to firms, developers, and local
governments located throughout a region. Additional household income may accrue from
renting raw land to developers. As price-takers, households choose locations of residence
and employment simultaneously. They spend the earnings on housing, leisure, trips and a
set of composite commodities, so that their utility is maximized.
Seeking maximum profit, firms and developers decide on their business location and
level of production at given input prices. Input prices include wages, rents for office space,
the prices of intermediate inputs provided by other firms and associated transport costs.
Optimal output prices are also determined by input prices and the production technology.

24

Figure 3.1 Framework of Urban Computable General Equilibrium Model

Two types of developers namely, housing developers and office developers, use
distinct technologies to develop floor space for housing and office. Local governments levy
a given level of sales tax on firms and developers operating within their jurisdiction. In order
to produce a worker amenity, local governments purchase labor, raw land, goods and
services with the collected tax and any other subsidy from outside the region. Local
government spending seeks to maximize the produced amenity and ensure a balanced
government budget.
The economic agents compete in the market for land, labor, developed floor space,
commodities, and transportation, realizing the market clearing prices in each zone. When no
restrictions are imposed on land use, developers and local governments compete over raw
Market
Transportation Network
Land / Space
Amenity
Labor
Commodity
Developers
Firms
Local Governments
Households
Tax
Economic agent
Tradable goods
Flow

25
land in each zone. However, when land use is controlled, for example, the quantity of land
available for specific use such as developing houses and offices is predefined, the land
market is divided into several independent markets according to these land uses. Thus, in a
controlled situation, land will be used according to different rents, depending on its use. A
labor market is established in each zone and the labor supply comes from all across the
region. Facilities operating in the zone such as firms, developers, and local governments
create a demand for labor. Goods and services are traded through markets established in the
zone where they are produced, and transported for consumption as final demand and
intermediate inputs for industries located at different sites. Working, shopping, and goods
movements generate travel demand. Travel demands compete over congestible, mode-
specific transportation networks.
The model describes a highly autonomous society, but uses only a limited number of
inputs. The model requires five types of input data: 1) size and number of zones in study
area; 2) total number of households distinguished by preferences; 3) household consumption
preferences; 4) transportation network configuration; and 5) parameterized production
technologies. The model computes endogenous prices and quantities simultaneously for all
goods and services in each geographic zone.
The following sections detail the urban CGE model. Behaviors of households,
firms, developers, and local governments are described in Sections 3.2 through 3.5. Section
3.6 describes equilibrium conditions in the markets of labor, land, housing, office,
commodity, and transportation modes. Section 3.7 summarizes the unknowns of the model
and includes a brief discussion of existence and uniqueness of the solution. This is followed
by a solution algorithm in section 3.8.

26
3.2. HOUSEHOLD MODEL

The basic household model with h-different preference types is described first. Then
the basic model is expanded to a multi-modal transportation system by incorporating various
travel modes for working and shopping trips.

3.2.1. Basic Household Model

The regional population includes distinct classes of households, each with a different
preference type, denoted by h. Price-taking households maximize a log-linear utility
function of commodity consumption (Y), housing floor consumption (H), household leisure
(N), and amenity given by residential place (G), subject to a budget constraint.

h
pq
G N H Y
U
, , ,
max

····· (3.1)
= ( ) ( ) ( ) ( )
∑∑
+ ⋅ + ⋅ + ⋅ + ⋅
i
h
pq p
h h
pq
h h
pq
h
k
ih
pqk
ih
pk
G N H Y ε φ δ β α ln ln ln ln

subject to
()
h
pq
h
pq pq
ik
pk
i ih
pqk
h
q
D N t T t Y E w +
⎥
⎦
⎤
⎢
⎣
⎡
− ⋅ − ⋅ ⋅ − ⋅
∑∑
ζ
······· (3.2)

= ( )
pq
u
p
h
pq
ik
pk i
i
k
ih
pqk
c T r H c p Y ⋅ + ⋅ + ⋅ + ⋅
∑∑
ζ ,

where c , and t represent travel cost, and travel time for two-way trips respectively, i.e.,
qp pq pq
c c c + = , and
qp pq pq
t t t + = .
The utility function is assumed to be homogeneous of degree one. Thus the sum of
utility coefficients is unity, i.e., ( ) 1
,
= + + +
∑
h h h
k i
ih
pk
φ δ β α . The given endowment is

27
allocated to travel time for shopping (
pk
i ih
pqk
t Y ⋅ ⋅ ζ ), travel time for working (
pq
t T ⋅ ), leisure
(
h
pq
N ), and labor, at the hourly wage of
h
q
w , as shown on the right hand side of constraint,
equation (3.2). The expenditure consists of commodity (
ih
pqk
Y ) at price
i
k
p and the travel
cost associated with shopping (
pk i
c ⋅ ζ ), housing (
h
pq
H ) at rent
u
p
r , and the cost of working
trips over the give time period (
pq
c T ⋅ ).
Households earn additional income by renting raw land. Because the region is
closed to the outside, all the land is owned by the households within the region. Place of
residence is not necessarily identical to the lot address that each household owns within the
region. Assuming that more disposable income means more land owned, the urban CGE
model distributes the total payment for raw land to households in proportion to labor income
less travel costs, as defined by
( )
() {}
∑∑∑
∑
⋅ + ⋅ − ⋅
⋅ + ⋅ − ⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ =
hp q
pq
h
q pq
h
q
pq
h
q pq
h
q
p
p p
h
pq
t w c T w E
t w c T w E
r A D . ··············· (3.3)

At given prices, each household chooses the optimal quantities for consumption
within its budget constraint. The quantities are solved for by applying the Lagrangian of the
systems of equations,

( ) ( ) ( ) ( )
h
pq p
h h
pq
h h
pq
h
ik
ih
pqk
ih
pk
G N H Y ε φ δ β α + ⋅ + ⋅ + ⋅ + ⋅ =
∑∑
ln ln ln ln L
– ()
⎢
⎣
⎡
⋅ ⋅ + ⋅ + ⋅ − + ⋅ ⋅
∑∑
ik
pk
i h
q pk
i i
k
ih
pqk
h
pq
h
q
t w c p Y D w E ζ ζ l (3.4)
– ()
⎥
⎦
⎤
⋅ − ⋅ − ⋅ + ⋅
h
q
h
pq
u
p
h
pq pq
h
q pq
w N r H t w c T

where l is Lagrangian coefficient for the budget constraint.

28
The first-order conditions of the Lagrangian are :
( ) 0
1 L
= ⋅ ⋅ + ⋅ + ⋅ + ⋅ =
∂
∂
pk
i h
q pk
i i
k
ih
pqk
ih
pk
ih
pqk
t w c p
Y Y
ζ ζ α l , ·········· (3.5a)
0
1 L
= ⋅ + ⋅ =
∂
∂
u
p
h
pq
h
h
pq
r
H H
l β , ·········· (3.5b)
0
1 L
= ⋅ + ⋅ =
∂
∂
h
q
h
pq
h
h
pq
w
N N
l δ , ·········· (3.5c)
( )
∑∑
⋅ ⋅ + ⋅ + ⋅ − + ⋅ =
∂
∂
ik
pk
i h
q pk
i i
k
ih
pqk
h
pq
h
q
t w c p Y D w E ζ ζ
l
L
········· (3.5d)
– ( ) 0 = ⋅ − ⋅ − ⋅ + ⋅
h
q
h
pq
u
p
h
pq pq
h
q pq
w N r H t w c T .

The Lagrangian coefficient is solved for by rearranging equations (3.5a), (3.5b), and
(3.5c),
()
pk
i h
q pk
i i
k
ih
pk
ih
pqk
t w c p
Y
⋅ ⋅ + ⋅ + ⋅
⋅ − =
ζ ζ
α
l
1
,
u
p
h h
pq
r
H
⋅
⋅ − =
l
1
β ,
h
q
h h
pq
w
N
⋅
⋅ − =
l
1
δ .

and substituting them into equation (3.5d),
()
pq
h
q pq
h
pq
h
q
ik
h h ih
pk
t w c T D w E ⋅ + ⋅ − + ⋅
+ +
− =
∑∑
δ β α
l
.

This permits the optimal consumption quantities to be expressed in terms of consumption
preference, and associated prices.
( )
pk
i h
q pk
i i
k
pq
h
q pq
h
pq
h
q
h
ih
pk
ih
pqk
t w c p
t w c T D w E
Y
⋅ ⋅ + ⋅ +
⋅ + ⋅ − + ⋅
⋅
−
=
ζ ζ φ
α
1
, ······································(3.6a)

29
( )
u
p
pq
h
q pq
h
pq
h
q
h
h
h
pq
r
t w c T D w E
H
⋅ + ⋅ − + ⋅
⋅
−
=
φ
β
1
, ····································(3.6b)

( )
h
q
pq
h
q pq
h
pq
h
q
h
h
h
pq
w
t w c T D w E
N
⋅ + ⋅ − + ⋅
⋅
−
=
φ
δ
1
. ·····································(3.6c)

Each household maximizes utility by consuming these optimal quantities, which are
defined by equations (3.6). As these equations show, demand is proportional to total
disposable income (the sum of labor income and earnings from land rent, minus travel cost,
( )
pq
h
q pq
h
pq
h
q
t w c T D w E ⋅ + ⋅ − + ⋅ ), and is inversely proportional to price. The price term for
each commodity demand consists of three elements. These are the price of the commodity at
the shopping center, the travel cost of shopping, and the time cost for shopping. Kim (1990,
p.97) defined this as effective price. Wage is implicitly defined as the value of time, and is
used to convert time cost to the money cost of travel for shopping, and the share of
endowment allocated to leisure.
Note that in equations (3.6) that household demand is scaled by utility coefficients,
excluding the coefficient for amenity term, φ
h
. In general, factors in the utility function
have associated prices, and thus, the factors are included in the budget constraint. The
model also estimates optimal demand as the ratio of disposable income to effective price,
scaled by utility coefficients for terms with prices. The urban CGE model, however,
assumes that worker amenity is given freely based on place of residence. Because it is a
consumer good in the model, households evaluate the worker amenity as a factor that
contributes to their respective utilities. In other words, households do not determine the
optimal consumption of amenity consumption, and the Lagrangian does not include the

30
amenity coefficient, but it is included in the household’s indirect utility function as a
location-specific constant.
Given the fixed time endowment, the household assigns time to leisure, travel (for
working and shopping), and labor supplied to the market. The optimal quantity of leisure at
the given hourly wage rate is given by equation (3.6c). Work trip time is calculated based on
the total number of working days (T), and two-way travel time between place of residence
and the job (
pq
t ). Since network congestion varies depending on time of day, the travel
times for working and return trips are estimated by using a representative work day
congestion level. For a given time period, such as a month, the time allocated to shopping
trips, is calculated based on three factors: 1) two-way travel time between shopping centers
and the residential zone,
pk
t ; 2) the quantity of purchased commodity,
ih
pqk
Y , and ; 3) the
frequency of shopping trips per quantity of the commodity,
i
ζ . For simplicity, the
frequency is exogenously given for each type of commodity, and assumed to be invariant.
The remaining endowment will be offered to the market as labor supply. The amount of
labor supplied to the labor market in zone q by a household h, living in zone p,
h
q
L is
calculated in equation (3.7) ;

h
q
L =
h
pq pq
ik
pk
i ih
pqk
N t T t Y E − ⋅ − ⋅ ⋅ −
∑∑
ζ ·································· (3.7)

=
( )
pq
ik
pk
i
pk
i h
q pk
i i
k
pq
h
q pq
h
pq
h
q
h
ih
pk
t T t
t w c p
t w c T D w E
E ⋅ − ⋅ ⋅
⋅ ⋅ + ⋅ +
⋅ + ⋅ − + ⋅
⋅
−
−
∑∑
ζ
ζ ζ φ
α
1

–
( )
h
q
pq
h
q pq
h
pq
h
q
h
h
w
t w c T D w E ⋅ + ⋅ − + ⋅
⋅
− φ
δ
1
.

31
Because of industrial and spatial aggregation, a household may shop for a given
commodity in multiple places. The commodities produced by an industry might be
distinguished by the production location. Because region-wide industrial activity is modeled
in terms of only a limited number of sectors, there might be distinctions between firms
classified into the same industry, but which operates in different zones. Households can visit
many places to purchase the set of distinct commodities that have been aggregated into a
single industrial output. In addition to the aggregation of industries, choice of shopping
locations vary due to the different distances to the nearest shopping center. The model
assumes that all the zonal activities are concentrated at the zone centroid. Households in a
zone are assumed to be located at one geographic point, and any distance measure from/to
the centroid is identical for all households. However, in reality, households are located
throughout a region, and the distances between shopping centers and residential buildings
vary.
Choice of shopping location is modeled based on quantity produced and the
effective price. Households would prefer shopping at the center that sells a greater variety of
products. A greater quantity of production in a given zone also implies more variety in the
product from the zone. In the model, preferences for the choice of shopping center are
described by the coefficient for commodity consumption in the utility function, i.e., the
aspatial utility coefficients for commodities are disaggregated spatially based on the variety
(and quantity) of products and effective prices (the sum of the commodity price and cost of
the shopping trip) associated with each zone. The aspatial coefficient for household h
consuming commodity i is given by
ih
p
ih
α α = . Equation (3.8) aggregates this utility
coefficient over the shopping center k , i.e.,
ih
pk
α , so that
ih
p
k
ih
pk
α α =
∑
.

32
ih
pk
α =
() []
() []
∑
⋅ + ⋅ ⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + ⋅ ⋅ ⎟
⎠
⎞
⎜
⎝
⎛
⋅
z
pk
i i
k
ih
i
i
z
pk
i i
k
ih
i
i
k
ih
c p
X
X
c p
X
X
ih
ih
ξ µ
ξ µ
α
η
η
exp
exp
(
(
························· (3.8)

where
i
k
X : Commodity i produced in zone k, { }
i
z
z
i
X X min =
(
,
ih
η : Coefficient for the quantity of output of household h’s preference for commodity i,
ih
µ
: Coefficient for the effective price of household h’s preference for commodity i.

In equation (3.8), the units for the variety of commodity, and effective shopping
price are not comparable. As such, production is scaled by the minimal zonal output, i.e., it
is assumed that the relative quantity,
i i
k
X X
(
/ , where { }
i
z z
i
X X min =
(
, denotes the variety
of zonal production.
Following the Anas and Kim CGE model, the utility function has a random term,
denoted by
h
pq
ε . The distribution is assumed to 1) be independent of any of the variables, 2)
be identical over the choice set (p, q), i.e. the places of residence and work, and 3) follows
the Weibull distribution (Anas, 1982). Then the utility function can be rewritten as equation
(3.9),
h
pq
h
pq
h
pq
V U ε + = . ························································································ (3.9)

where
h
pq
V is the indirect utility function, which is observable from data. By replacing the
quantity terms in the utility function in equation (3.1) by equation (3.6), the indirect utility
can be presented by just prices, i.e., commodity prices, rent wage, and travel time and cost,

33
h
pq
V =
()() ( )
⎥
⎦
⎤
⎢
⎣
⎡
− − ⋅ ⋅ − ⋅ − + ⋅ ⋅ −
h
pq
h
q pq
h
pq
h
q
h
t w T c T D w E φ φ 1 ln ln 1 ··(3.10)
( )
∑∑
⋅ ⋅ + ⋅ + ⋅ −
ik
pk
i h
q pk
i i
r
ih
pk
t w c p ξ ξ α ln
( ) ( ) ( )
p
h h
q
h u
p
h
G w r ln ln ln ⋅ + ⋅ − ⋅ − φ δ β .

The spatial distribution of households h who live in zones p, and work at zones q
follows the logit model given in equation (3.11);
∑∑
⋅
⋅
⋅ Θ = Ω
pq
h
pq
h
h
pq
h
h h
pq
V
V
) exp(
) exp(
ψ
ψ
. ································································ (3.11)

where
h
ψ is the dispersion factor of population, and
h
Θ is the given total number of
households of type h.

3.2.2. Extended Household Model for Multi-Modal Transportation System

While the basic household model does not specify travel modes for trips, numerous
modes might be available. For example, transportation modes might be categorized into
public transit and private automobiles according to whether the mode has predefined route
and/or schedule. For modeling purposes, transportation modes are represented by associated
travel cost and time. When more than one travel mode is available in the transportation
system, households face choices. As utility maximizers, households select the best
alternative mode.
Since travel cost and time will be differentiated by mode, the indirect utility will
vary with mode choices. Changes in utility, in turn, prompt changes in household location

34
through the indirect utility function. As shown by equation (3.10), the indirect utility
function includes travel time and cost for working and shopping trips (
pq
t ,
pk
t ,
pq
c , and
pk
c ,
respectively). The mode choices for the trips are assumed to be independent of each other.
Hence, if M modes are available between residential (p), working (q) and shopping (k)
places, households will select one set of travel modes for working (p-q) and shopping (p-k)
trips from M×M combinatorial alternatives.
By incorporating travel modes for working and shopping denoted by m and n
respectively, the indirect utility function (3.10) is updated to equation (3.12) ;
hmn
pq
V =
()() ( )
⎥
⎦
⎤
⎢
⎣
⎡
− − ⋅ ⋅ − ⋅ − + ⋅ ⋅ −
h m
pq
h
q
m
pq
hm
pq
h
q
h
t w T c T D w E φ φ 1 ln ln 1 ··(3.12)
( )
∑∑
⋅ ⋅ + ⋅ + ⋅ −
ik
n
pk
i h
q
n
pk
i i
r
ih
pk
t w c p ξ ξ α ln
( ) ( ) ( )
p
h h
q
h u
p
h
G w r ln ln ln ⋅ + ⋅ − ⋅ − φ δ β

where the income from land rent is given as follows;
( )
() {}
∑∑∑∑
∑
⋅ + ⋅ − ⋅
⋅ + ⋅ − ⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ =
hp q m
m
pq
h
q
m
pq
h
q
m
pq
h
q
m
pq
h
q
p
p p
hm
pq
t w c T w E
t w c T w E
r A D . ·······(3.13)

Since indirect utility is disaggregated by travel modes m and n, households will be allocated
to zones according to equation (3.14).
∑∑∑∑
⋅
⋅
⋅ Θ = Ω
pq m n
hmn
pq
h
hmn
pq
h
h hmn
pq
V
V
) exp(
) exp(
ψ
ψ
. ················································· (3.14)

The optimal consumption choices depicted by equation (3.6) also extend to the
various travel modes for work and shopping trips, as in equation (3.15).

35
( )
n
pk
i h
q
n
pk
i i
k
m
pq
h
q
m
pq
hm
pq
h
q
h
ih
pk ihmn
pqk
t w c p
t w c T D w E
Y
⋅ ⋅ + ⋅ +
⋅ + ⋅ − + ⋅
⋅
−
=
ζ ζ φ
α
1
, ······································(3.15a)
( )
u
p
m
pq
h
q
m
pq
hm
pq
h
q
h
h
hm
pq
r
t w c T D w E
H
⋅ + ⋅ − + ⋅
⋅
−
=
φ
β
1
, ······································(3.15b)
( )
h
q
m
pq
h
q
m
pq
hm
pq
h
q
h
h
hm
pq
w
t w c T D w E
N
⋅ + ⋅ − + ⋅
⋅
−
=
φ
δ
1
. ······································(3.15c)

3.3. FIRM MODEL

Firms are identified according to product and location of operations. The urban
CGE model describes the collective behavior of firms rather than tracing individual
businesses. Therefore, for all firms operating in a given zone and producing a specific kind
of product are assumed to use identical production technologies. As price takers, firms
maximize profit by determining the optimal quantity inputs, and the optimal level of
production in response to given prices. Prices include wages (
h
q
w ), rent of developed floor
space for industrial use (
v
q
r ), free-on-board price of intermediate inputs (
i
r
p ), and transport
cost (
rq
c ), including delivery frequency
i
ξ of inputs produced by other firms. The
production technology is represented by a variation of the constant elasticity of substitution
(CES) production function. Equation (3.16) describes the profit maximizing behavior of a
firm that produces commodity j at zone q;
j
q
F L Z X
ϖ
, , ,
max ······················ (3.16)

= ( ) ( )
j
q
v
q
h
hj
q
h
q
ir
rq
i i
r
ij
rq
j
q
j
q
j
q
F r L w c p Z X p ⋅ − ⋅ − ⋅ + ⋅ − − ⋅
∑ ∑ ∑
ξ τ 1
subject to

36
j
q
X ··························· (3.17)
= () () () () () ()
1
1 1 1 1 1 1
−
− − −
⎥
⎦
⎤
⎢
⎣
⎡
⋅ + ⋅ + ⋅
∑ ∑∑
j
j
j
j
j j
j
j j
j
j j
q
j
h
hj
q
hj
ir
ij
rq
ij
rq
F L Z
θ
θ
θ
θ
θ θ
θ
θ θ
θ
θ
σ ρ ϕ ,

where
j
θ >0, and ≠
j
θ 1 is the exogenous elasticity of substitution, with constant returns to
scale. In the standard CES function, the unity is imposed on the sum of input share
coefficients with the elasticity of substitution,
j
θ , () () () 1
1 1
,
1
= + +
∑ ∑
j j j j
h
hj
r i
ij
rq
θ θ θ
σ ρ ϕ .
Instead, the proposed model assumes 1
,
= + +
∑ ∑
j
h
hj
r i
ij
rq
σ ρ ϕ . This modification
leads to a stable output price calculation. The prices estimated based on the standard CES
function are sensitive to the number of zones and number of sectors, which makes it difficult
to compute prices in practical applications. Section 4.6 provides a review of the
characteristics of output prices derived from original CES function in multi-regional context,
and the prices generated by the proposed variation.
Firms spend their revenues to pay taxes and purchase inputs. Firms pay taxes to
local government from selling the product in quantity
j
q
X at the price
j
q
p . The local
government of zone q determines the tax rate
j
q
τ for each industry j that is operating within
its jurisdiction. The production technology requires labor inputs from all or some of the
household types, H ∈ h , located at any place of residence in the model area (
hj
q
L ). Different
household types might represent various kinds of labor expertise, and firms will want to
employ various quantities of skilled labor compensated at differentiated wages,
h
q
w . Firms
also require intermediate goods as inputs (
ij
rq
Z ) from industries, I ∈ i , including

37
itself, I ∈ j . In addition to labor, and intermediate inputs, production activity takes place in
the locations developed for industrial use, and the firm demands an associated quantity of
floor space (
j
q
F ).
Based on the given production technology by equation (3.17), the firm finds the
optimal quantities of input and the output at given prices. The Lagrangian of the systems of
equations (3.16) and (3.17) is used to solve for the optimal quantities;
L = ( )
j
q
j
q
j
q
X p τ − ⋅ 1
– ( )
∑∑
⋅ + ⋅
ir
rq
i i
r
ij
rq
c p Z ξ
–
j
q
v
q
h
hj
q
h
q
F r L w ⋅ − ⋅
∑
····················· (3.18)
–
j
q
X ⋅ l
+ () () () () () ()
1
1 1 1 1 1 1
−
− − −
⎥
⎦
⎤
⎢
⎣
⎡
⋅ + ⋅ + ⋅ ⋅
∑ ∑∑
j
j
j
j
j j
j
j j
j
j j
q
j
h
hj
q
hj
ir
ij
rq
ij
rq
F L Z
θ
θ
θ
θ
θ θ
θ
θ θ
θ
θ
σ ρ ϕ l ,

where l is the Lagrangian coefficient to the production technology. The first-order
conditions of the Lagrangian are the following;
ij
rq
Z ∂
∂ L
= – ( )
rq
i i
r
c p ⋅ + ξ
+ () ()
j j ij
rq
ij
rq
Z
θ θ
ϕ
1 1
−
⋅ ⋅ l ×
() ()
⎢
⎣
⎡
⋅
∑∑
−
ir
ij
rq
ij
rq
j
j
j
Z
θ
θ
θ
ϕ
1 1
() ()
∑
−
⋅ +
h
hj
q
hj j
j
j
L
θ
θ
θ
ρ
1 1
() ()
1
1
1 1
−
−
⎥
⎦
⎤
⋅ +
j
j
j
j j
q
j
F
θ
θ
θ
θ
σ ,
()() () ()
j j j ij
rq
ij
rq
j
q rq
i i
r
Z X c p
θ θ θ
ϕ ξ
1 1 1
−
⋅ ⋅ ⋅ + ⋅ + − = l ,

38
() () ()
j j j hj
q
hj j
q
h
q hj
q
L X w
L
θ θ θ
ρ
1 1 1
L −
⋅ ⋅ ⋅ + − =
∂
∂
l ,

() () ()
j j j j
q
j j
q q
j
q
F X R
F
θ θ θ
σ
1 1 1
L −
⋅ ⋅ ⋅ + − =
∂
∂
l .

The optimal input quantities are derived from these first-order conditions.
j
rq
i i
r
j
q
ij
rq
ij
rq
c p
X Z
θ
ξ
ϕ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
⋅ ⋅ =
l
, ··························································· (3.19a)
j
h
q
j
q
hj ij
q
w
X L
θ
ρ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ ⋅ =
l
, ····························································· (3.19b)
j
h
q
j
q
j j
q
r
X F
θ
σ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ ⋅ =
l
. ····························································· (3.19c)

By inserting equations (3.19) into equation (3.17), the Lagrangian multiplier l is derived as
equation (3.20):
() () ()
j
j j j
v
q
j
h
h
q
hj
ir
rq
i i
r
ij
rq
r w c p
θ
θ θ θ
σ ρ
ξ
ϕ
−
− − −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
=
∑ ∑∑
1
1
1 1 1
l . ······················ (3.20)

Based on equations (3.19) and (3.20), the firm’s profit, equation (3.16), can be simplified as
follows;
j
q
ϖ = ( )
j
q
j
q
j
q
X p τ − ⋅ 1
– ()
()
()
∑∑
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⋅ +
⋅ ⋅ ⋅ ⋅ +
ir
rq
i i
r
j
q
ij
rq rq
i i
r j
j
c p
X c p
θ
θ
ξ
ϕ ξ
l

–
()
()
∑
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⋅ ⋅ ⋅
h
h
q
j
q
hj h
q j
j
w
X w
θ
θ
ρ
l
–
()
() ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⋅ ⋅ ⋅
j
j
v
q
j
q
j v
q
r
X r
θ
θ
σ
l
.

39
Therefore
j
q
ϖ = ( )
j
q
j
q
j
q
X p τ − ⋅ 1
– ()
() () ()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
⋅ ⋅
− − −
∑ ∑∑
1 1 1
j j j
j
v
q
j
h
h
q
hj
ir
rq
i i
r
ij
rq j
q
r w c p
X
θ θ θ
θ
σ ρ
ξ
ϕ
l ,
= ( )
j
q
j
q
j
q
X p τ − ⋅ 1 – l ⋅
j
q
X . ················································ (3.21)

In a market with perfect competition and no barriers to entry and exit, the long-run
profit of any firm (i.e.,
j
q
X > 0) will be zero. Therefore, the expression for the output price
is simplified as shown in equation (3.22) as a general (or, power) mean of input prices
weighted by input share coefficients.
j
q
p =
j
q
τ − 1
l
, ··································································· (3.22)
=
() () ()
j
j j j
i
v
q
j
h
h
q
hj
r
rq
i i
r
ij
rq
j
q
r w c p
θ
θ θ θ
σ ρ
ξ
ϕ
τ
−
− − −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
⋅
−
∑∑ ∑
1
1
1 1 1
1
1
.

According to the equation, the output price depends on the tax rate. In addition, for
any imperfect elasticity of substitution, ≠
j
θ 1, the output price increases as the input prices
of intermediate goods, wages and office rent rise. Also, note that the output price does not
include any quantity term for inputs, and thus its homogeneity is degree zero, i.e., the output
price is not affected by any of input and output quantities in the production process.
Equations (3.23) give the optimal input quantities. These equations are derived by
replacing the Lagrangian coefficient in equations (3.19) with output price from equation
(3.22). The optimal input quantities are formulated as functions of input and output prices,
and the output quantities.

40
( )
j
rq
i i
r
j
q
j
q j
q
ij
rq
ij
rq
c p
p
X Z
θ
ξ
τ
ϕ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
− ⋅
⋅ ⋅ =
1
, ·································· (3.23a)
( )
j
h
q
j
q
j
q j
q
hj ij
q
w
p
X L
θ
τ
ρ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ − ⋅
⋅ ⋅ =
1
, ·································· (3.23b)
( )
j
v
q
j
q
j
q j
q
j j
q
r
p
X F
θ
τ
σ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ − ⋅
⋅ ⋅ =
1
. ·································· (3.23c)

Firms purchase intermediate goods from various locations because of industrial and
spatial aggregation. In essence, this is similar as to reasons why households shop at a variety
of shopping centers. . The variety of input locations is modeled by spatial input share
coefficient,
ij
rq
ϕ . The exogenous share coefficient for intermediate input,
∑
= =
r
ij
rq
ij
q
ij
ϕ ϕ ϕ is distributed over zones, from which input is purchased, as a function
of the exponent of zonal effective price,
rq
i i
k
c p ⋅ + ξ , as shown in the following equation:
()
() []
() []
j
j
k
kq
i i
k
ij
rq
i i
r
ij
ij ij
rq
c p
c p
θ
θ
ξ µ
ξ µ
ϕ ϕ
1
1
exp
exp
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⋅ + ⋅
⋅ + ⋅
⋅ =
∑
, ································· (3.24)

where
ij
µ represents the decay coefficient of the input share relative to the effective input
price. The decay coefficient describes how the firm is sensitive to the effective price of
intermediate inputs when selecting a supplier. In addition, the exogenous delivery
frequency,
i
ξ of intermediate input translates the zone-to-zone travel cost,
rq
c , into the unit
transport cost of the intermediate trade.

41
3.4. DEVELOPER MODEL

In the Urban CGE model, the sites for residential and industrial uses are tangible
goods to be produced. Households consume housing floor space (
h
pq
H ), according to
equation (3.2), whereas firms demand floor space for their production activities (
j
q
F )
according to equation (3.16). Developable land is supplied by developers, who have
characteristics similar to firms. Like the firms, the developers are profit maximizers. For
given levels of wages (
h
k
w ), intermediate input prices (
rk
i
r
c p + ), and rents for raw land
(
k
r ), developers determine the level of inputs and production. In this case, the quantity of
developed land is the produced goods.
The developer’s production technology follows the modified CES production
function, same as the production functions for firms, with constant returns to scale.
However, the technology for developing residential structures need not be same as that for
industrial spaces. Residential developers are distinguished from industrial developers in the
model due to this difference in technology. Superscripts u, and v denote residential
developers, and industrial land developers, respectively. The residential developers who are
operating in zone p to produce residential space H
p
are described as following:
u
p
Q L Z H
ϖ
, , ,
max ·············· (3.25)
= ( ) ( )
u
p p
h
hu
p
h
p
ir
rp
i i
r
iu
rp
u
p p
u
p
Q r L w c p Z H r ⋅ − ⋅ − ⋅ + ⋅ − − ⋅ ⋅
∑ ∑ ∑
ξ τ 1

subject to
p
H ··············· (3.26)
= () () () () () ()
1
1 1 1 1 1 1
−
− − −
⎭
⎬
⎫
⎩
⎨
⎧
⋅ + ⋅ + ⋅
∑ ∑∑
u
u
u
u
u u
u
u u
u
u u
p
u
h
hu
p
hu
ir
iu
rp
iu
rp
Q L Z
θ
θ
θ
θ
θ θ
θ
θ θ
θ
θ
σ ρ ϕ .

42

The industrial developers in zone q produce F
q
based on the systems of equations:
v
q
Q L Z F
ϖ
, , ,
max ·············· (3.27)
= ( ) ( )
v
q q
h
hv
q
h
q
ir
rq
i i
r
iv
rq
v
q q
v
q
Q r L w c p Z F r ⋅ − ⋅ − ⋅ + ⋅ − − ⋅ ⋅
∑ ∑ ∑
ξ τ 1

subject to
q
F ················ (3.28)
= () () () () () ()
1
1 1 1 1 1 1
−
− − −
⎭
⎬
⎫
⎩
⎨
⎧
⋅ + ⋅ + ⋅
∑ ∑∑
v
v
v
v
v v
v
v v
v
v v
q
v
h
hv
q
hv
ir
iv
rq
iv
rq
Q L Z
θ
θ
θ
θ
θ θ
θ
θ θ
θ
θ
σ ρ ϕ

These systems are similar to equations (3.16) and (3.17). In equations (3.25) to
(3.28), θ is the elasticity of substitution,
u
r ,
v
r , and r are housing rent, industrial rent, and
rent for undeveloped land respectively, and Z, L, and Q represent intermediate goods, labor,
and undeveloped land respectively.
The two distinct developers co-exist in each zone and compete over undeveloped
land, along with the local government. See section 3.5 for model of local governments, and
3.6 for equilibrium conditions for land market. Although the zones where the residential and
industrial developers are operating are denoted separately by ∈ p R, and ∈ q R in the
systems of equations, this is merely to associate the developed spaces with households’
residential location and firms’ operating locations.
The model does not specifically restrict the use of undeveloped land. Rather than
using the pre-specified land use plan as input data, zonal land use is determined as a
consequence of developers’ production decision in light of the available production
technologies. Developers decide the amount of raw land needed to maximize profit, based
on the demand for developed spaces, and given input prices. The optimal quantity of

43
undeveloped land used as an input is determined by solving the first-order Lagrangian.
Equations (3.29) are the optimal demands from residential developers in zone p, and
industrial developers in zone q:
( )
u
p
u
p
u
p
p
u u
p
r
r
H Q
θ
τ
σ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ − ⋅
⋅ ⋅ =
1
, ··································································(3.29a)
( )
v
q
v
q
v
q
q
v v
q
r
r
F Q
θ
τ
σ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ − ⋅
⋅ ⋅ =
1
. ··································································(3.29b)

In addition to raw land, developers’ production processes also require labor with
various levels of expertise, and intermediate goods. Equations (3.30) and (3.31) are the
optimal quantities derived from the first-order conditions of the Lagrangian. The market
clearing mechanism is derived in section 3.6 based on these optimal demands.
( )
u
rp
i i
r
u
p
u
p
p
iu
rp
iu
rp
c p
r
H Z
θ
ξ
τ
ϕ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
− ⋅
⋅ ⋅ =
1
, ································································(3.30a)
( )
v
rq
i i
r
v
q
v
q
q
iv
rq
iv
rq
c p
r
F Z
θ
ξ
τ
ϕ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
− ⋅
⋅ ⋅ =
1
, ································································(3.30b)
( )
u
h
p
u
p
u
p
p
hu iu
p
w
r
H L
θ
τ
ρ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ − ⋅
⋅ ⋅ =
1
, ································································(3.31a)
( )
v
h
q
v
q
v
q
q
hv iv
q
w
r
F L
θ
τ
ρ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ − ⋅
⋅ ⋅ =
1
. ································································(3.31b)

Developers purchase mobile commodities from various locations depending on the
prices of the commodity and transportation costs. Due to spatial and sectoral aggregation in

44
the model, the input shares for commodity,
u
p
φ and
v
q
φ are spatially disaggregated to
iu
rp
φ
and
iv
rq
φ according to the equation (3.24).
In a perfectly competitive market with no barriers to entry and exit, the operating
developers make zero profit in the long run. The price of the developers’ products, i.e., the
rent for developed space, is derived based on this condition. The long run equilibrium rent
for residential floor space and industrial floor space follow equations (3.32), which show
homogeneity of degree zero.
() ()
()
u
u u u
p
u
h
h
p
hu
ir
rp
i i
r
iu
rp
u
p
u
p
r
w c p
r
θ
θ θ θ
σ ρ
ξ
ϕ
τ
−
− − −
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
+ +
⋅ +
⋅
−
=
∑ ∑∑
1
1
1 1 1
1
1
··········(3.32a)
() ()
()
v
v v v
q
v
h
h
q
hv
ir
rq
i i
r
iv
rq
v
q
v
q
r
w c p
r
θ
θ θ θ
σ ρ
ξ
ϕ
τ
−
− − −
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
+ +
⋅ +
⋅
−
=
∑ ∑∑
1
1
1 1 1
1
1
··········(3.32b)

3.5. LOCAL GOVERNMENT MODEL

The opportunity for mathematical representation for the local government’s
economic behavior is limited because it is unclear if the objectives of local government can
be described in terms of optimization concepts. Furthermore, if the output of local
government is public goods, then the production of zero-price goods is not economical.
However, it has been suggested that the local government is also an economic agent because
it demands inputs (Deacon 1978), and produces public services to enhance local amenity
(Crane, Daniere et al. 1997).

45
In the urban CGE model, the local government maximizes the production of
amenity,
p
G in zone p, by consuming intermediate commodities (
ig
rp
Z ), labor (
hg
p
L ), and
undeveloped land (
g
p
Q ). The total production for local government is divisible; it is shared
by the households who reside in its jurisdiction, and used to enhance household utility.
The local government operates under a balanced budget rule. Revenue, denoted by
Γ
g
, consists of taxes that are extracted from firms and developers in the local jurisdiction,
along with a subsidy from outside the jurisdiction. Exogenous sale tax rate is imposed on
each individual industry, the total tax revenue is endogenous as industrial output (and sale) is
determined by the model. Revenue is spent on purchasing inputs at given commodity prices
(
i
r
p ), including transport cost (
rp
i
c ⋅ ξ ), wages (
h
p
w ), and land rents (r
p
).
A modified CES production function with constant returns to scale is assumed for
the local government production function, subject to the budget constraint. This is shown in
the following systems of equations;

g p
Q L Z
G
∈
, ,
max = ···························(3.33a)
() () () () () ()
∑∑ ∑∑
∈
−
− − −
⎭
⎬
⎫
⎩
⎨
⎧
⋅ + ⋅ + ⋅ ⋅ Κ
g p
g
p
g
h
hg
p
hg ig
rp
ir
ig
rp
g
g
g
g
g
g g
g
g g
g
g
Q L Z
1
1 1 1 1 1 1
θ
θ
θ
θ
θ θ
θ
θ θ
θ
θ
σ ρ ϕ

subject to
Γ
g
=
∑∑
∈
⎟
⎠
⎞
⎜
⎝
⎛
⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅
g pi
v
p p
v
p
u
p p
u
p
i
p
i
p
i
p
F r H r X p τ τ τ ···················(3.33b)
= ()
∑∑ ∑∑
∈ ⎭
⎬
⎫
⎩
⎨
⎧
⋅ + ⋅ + ⋅ + ⋅
g p
p
g
p
h
h
p
hg
p
ir
rp
i i
r
ig
rp
r Q w L c p Z ξ

where
∑∑∑
∈
Ω
= Κ
g pq h
h
pq
g
1

46
The system of equations accounts for the number of households in the zones p, while
the external subsidy is omitted. Κ
g
denotes the inverse of total households in the
jurisdiction.
h
pq
Ω , from equation (3.14), represents the number of h-type households who
live in zone p and work at zone q. The produced amenity is divided evenly among all
households in a zone in which no household is discriminated against. Since the proposed
model focuses on a closed, autonomous system, an external subsidy term is omitted from the
budget constraint. So the sales tax is the sole source of government revenue.
The local government seeks optimal quantities of inputs to maximize production of
the worker amenity at the given input prices. The Lagrangian for the systems of equations is
used to solve for the optimal quantities.

L =
() () () ()
∑∑ ∑∑
⎩
⎨
⎧
⋅ + ⋅ ⋅ Κ
− −
ph
hg
p
hg ig
rp
ir
ig
rp
g g
g
g g
g
g
L Z
θ
θ
θ θ
θ
θ
ρ ϕ
1 1 1 1

+ () ()
1
1 1
−
−
⎭
⎬
⎫
⋅
g
g
g
g
g g
p
g
Q
θ
θ
θ
θ
θ
σ
()
∑∑ ∑∑
∈ ⎩
⎨
⎧
⋅ + ⋅ + ⋅ + ⋅ ⋅ −
g p
p
g
p
h
hg
p
h
p
ir
rp
i i
r
ig
rp
r Q L w c p Z ξ l
–
⎭
⎬
⎫
⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅
∑
v
p
v
p
v
p
u
p
u
p
u
p
i
i
p
i
p
i
p
H r H r X p τ τ τ , ················(3.34)

where l is the Lagrangian coefficient to the budget constraint. The optimal quantities that
lead to maximum amenity are derived from the Lagrangian’s first-order conditions for input
quantities. The first-order conditions are as follow.

47
ig
rp
Z ∂
∂L
= () () () ()
∑∑ ∑∑
⎩
⎨
⎧
⋅ + ⋅ ⋅ Κ
− −
ph
hg
p
hg
ir
ig
rp
ig
rp
g g
g
g g
g
g
L Z
θ
θ
θ θ
θ
θ
ρ ϕ
1 1 1 1

() () () ()
g g
g
g
g
g ig
rp
ig
rp
g
p
g
Z Q
θ θ
θ
θ
θ
θ
ϕ σ
1 1
1
1
1 1
−
−
−
⋅ ⋅
⎭
⎬
⎫
⋅ +
( )
rp
i i
p
c p ⋅ + ⋅ − ξ l

= () () () () ()
rp
i i
p
ig
rp
ig
rp p
g
c p Z G
g g
g
g
g
⋅ + ⋅ − ⋅ ⋅ ⋅ Κ
−
−
ξ ϕ
θ θ
θ
θ
θ
l
1 1
1
1
=0 , ············(3.35a)

hg
p
L ∂
∂L
= () () () ()
h
p
ig
rp
ig
rp p
g
w Z G
g g
g
g
g
⋅ − ⋅ ⋅ ⋅ Κ
−
−
l
θ θ
θ
θ
θ
ϕ
1 1
1
1
= 0 , ············(3.35b)

g
p
Q ∂
∂L
= () () () ()
p
ig
rp
ig
rp p
g
r Z G
g g
g
g
g
⋅ − ⋅ ⋅ ⋅ Κ
−
−
l
θ θ
θ
θ
θ
ϕ
1 1
1
1
= 0 . ·············(3.35c)

If these first-order conditions are substituted into the budget constraint, the Lagrangian
coefficient is expressed as follows:
()
g
θ
l
=
()
() ()
()
∑∑∑ ∑
∈
− − −
−
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
+ +
⋅ +
⋅
Γ
⋅ Κ
g pi r
p
g
h
h
p
hg
rp
i i
p
ig
rp
g
g g
g g g
g
r
w c p
G
1 1 1
1
θ θ θ
θ
σ ρ
ξ
ϕ

=
()
g
g
g g
G
g
Λ ⋅
Γ
⋅ Κ
−1 θ
, ·············································(3.36)

where Λ
g
=
() ()
()
∑∑∑ ∑
∈
− − −
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
+ +
⋅ +
g pi r
p
g
h
h
p
hg
rp
i i
p
ig
rp
g g g
r
w c p
1 1 1 θ θ θ
σ ρ
ξ
ϕ

Note that Λ
g
denotes a type of aggregated value of input prices, with input share coefficients
as weights. The ratio of the total revenue ( Γ
g
) to aggregate input price ( Λ
g
) gives the
aggregate amenity produced in quantity. Equations (3.37) show that, the optimal input is

48
derived from the aggregated quantity and input prices. Market equilibrium will be modeled
based on these optimal quantities.
()
g
rp
i i
p
ig
rp
g
g
ig
rp
c p
Z
θ
ξ
ϕ
⋅ +
⋅
Λ
Γ
= , ···························································(3.37a)
()
g
h
p
hg
g
g
hg
p
w
L
θ
ρ
⋅
Λ
Γ
= , ···························································(3.37b)
()
g
p
g
g
g
g
p
r
Q
θ
σ
⋅
Λ
Γ
= . ···························································(3.37c)

Equation (3.38) shows the quantity of amenity used by individual households.
Production is proportional to revenue ( Γ
g
), while it is inversely proportional to the number of
households ( Κ
g
=
∑
Ω
h q p
h
pq
, ,
/ 1 ) in the jurisdiction, and the aggregated input price ( Λ
g
).

g p
G
∈
································(3.38)
=
() ()
()
1
1 1 1
−
∈
− − −
∑∑∑ ∑
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
+ +
⋅ +
⋅
Λ
Γ
⋅ Κ
g
g
g g g
gpi r
p
g
h
h
p
hg
rp
i i
p
ig
rp
g
g
g
r
w c p
θ
θ
θ θ θ
σ ρ
ξ
ϕ

= ()
1
1
−
Λ ⋅ Γ ⋅ Κ
g
g g g
θ

Unlike firms’ input share coefficient for intermediate goods, the coefficients for
local government may not be distributed over the zones. Due to spatial and sectoral
aggregation, firms are assumed to purchase intermediate goods from various zones, and the
input share coefficients. As equation (3.24) shows,
ij
rq
ϕ , are used to model the spatial
distribution. However, only one local government administers in each zone, and the variety
of intermediate inputs to local government might not be as various as it is for firms’

49
intermediate input requirements. For this reason, local government is assumed to purchase
from zones where the effective price is cheapest. Thus the input share coefficient follows
equation (3.39)
( )
⎪
⎩
⎪
⎨
⎧ ⋅ + =
=
otherwise , 0
min arg if ,
kp
i i
k
k
ig
ig
rp
c p r ξ ϕ
ϕ ·······················································(3.39)

3.6. EQUILIBRIUM CONDITIONS

The previous section outlined the partial equilibrium conditions for each economic
agent’s behavior. Each economic agent pursues the optimal input and output quantities
derived by their own interests. However, the optimal quantities in the individual models of
behavior were derived independently without considering the other agents’ behavior, and are
only valid under the following three assumptions: 1) the input price is given; 2) the supply of
input factors is unbounded, so there is no limitation in quantity at the given price; 3) the
demand for the output is also unbounded so that the entire product is sold. In a real market,
however, all three conditions might not be met.
This section describes equilibrium market conditions in each zone. Economic
agents interact with each other via markets. In a market, one agent’s demand is satisfied by
another’s supply so that the market has no excess demand. This is known as a static general
equilibrium state. To achieve a general equilibrium, the optimal quantities of demand and
supply from each economic agent are equated with respect to the markets of undeveloped
land, developed land, labor, commodities, and transportation services.

50
3.6.1. Equilibrium in the Market for Land

For given land rent, the demands of residential developers, industrial developers,
and the local government for undeveloped land follow equations (3.29a), (3.29b), and
(3.37c), respectively. In the equilibrium state, the available land for development in each
zone, denoted by A
p
, R ∈ p , which is exogenous to the urban CGE model, should be equal
to demand, so that no excess demand remains. Equation (3.40) depicts the equilibrium state
in the land market.

() ()
()
u
u
p
u
p
u
p
p
u
r
r
H
θ
θ
τ
σ
− ⋅
⋅
1
+
( ) ( )
()
v
v
p
v
p
v
p
p
v
r
r
F
θ
θ
τ
σ
− ⋅
⋅
1
+
()
0 = − ⋅
Λ
Γ
p
p
g
g
g
A
r
g
θ
σ
·····(3.40)

The market clearing rent for undeveloped land, r
p
is determined from this condition,
when
u
p
r ,
v
p
r , H
p
, F
p
, Γ
g
, and Λ
g
are known. However, rents in the model are
interdependent, as equations (3.32) show, rent for developed land,
u
p
r and
v
p
r includes the
rent for raw land r
p
, too. The calculation of equilibrium land rents may be done by one of
two following methods: i) solution of simultaneous equations, or ii) iterative methods. In
addition, even if all the endogenous variables (again,
u
p
r ,
v
p
r , H
p
, F
p
, Γ
g
, and Λ
g
) are known,
an exact analytic form for land rent is not readily derived from the equilibrium condition,
because the equation (3.40) is highly non-linear. The equation is associated with three
exponents to the unknown rent for land rent,
g v u
θ θ θ , , . In this case, the unknown rent
might be computed by applying numerical techniques, such as the golden section method.

51
The predefined land use plan may divide the land market in a zone into several
sections for the specified land use. In the urban CGE model, land use is one of the many
endogenous variables. However, if a plan is implemented, the amount of land for specific
purpose is fixed, and the total available land A
p
is split into fractions for each use. Individual
markets are then established for this use over fixed amount of land in each zone, and each
market is associated with its own equilibrium condition. In the proposed urban CGE model,
three types of land uses are considered, i.e., residential, industrial and municipal uses. As
shown in equations (3.41), land prices are differentiated according to uses, when the uses are
fixed.
() ()
()
0
1
= −
′
− ⋅
⋅
u
p
p
u
p
u
p
p
u
A
r
r
H
u
u
θ
θ
τ
σ , ································································(3.41a)
() ()
()
0
1
= −
′ ′
− ⋅
⋅
v
p
p
v
p
v
p
p
v
A
r
r
F
v
v
θ
θ
τ
σ , ·································································(3.41b)
()
0 = −
′ ′ ′
⋅
Λ
Γ
g
p
p
g
g
g
A
r
g
θ
σ
, ··································································(3.41c)

where
u
p
A ,
v
p
A , and
g
p
A are the quantities of land, given by the land use plan. The sum is
the total available land, i.e.,
g
p
v
p
u
p p
A A A A + + = . Also,
p p p
r r r ′ ′ ′ ≠ ′ ′ ≠ ′ denote the different
undeveloped land rents for each use.

3.6.2. Equilibrium in the Market for Housing Floor Space

Produced housing units should satisfy the housing demand from households in each
zone. Residential developers determine the optimal output to maximize their profit at the

52
given input prices. Also, households demand housing units to maximize their utility. On the
other hand, at the equilibrium state, the optimal quantity of housing produced, H
p
, in
equation (3.26) should meet this household demand, so that no excess demand exists in the
housing market. Demand from individual households is given by
h
pq
H in equation (3.6b).
Along with the number of households,
h
pq
Ω from equation (3.11), the equilibrium quantity of
housing supplied in each zone is as equation (3.42). When the wages, and travel time and
cost are given, this equation (3.42) and equation (3.26) for H
p
, determines the rent of
housing,
u
p
r .
H
p
=
∑ ∑
⋅ Ω
hq
h
pq
h
pq
H ·····························(3.42)
=
( )
∑∑
⋅ + ⋅ − + ⋅
⋅
−
⋅ Ω
hq
u
p
pq
h
q pq
h
pq
h
q
h
h
h
pq
r
t w c T D w E
φ
β
1
.

3.6.3. Equilibrium in the Market for Industrial Floor Space

Firms demand developed industrial or office space as summarized in equation
(3.23c) to maximize their profit. The supply from developers, F
q
, in equation (3.28) should
clear this demand in each zone, so that no excess demand exists in the market. Given F
q
,
j
q
X , and
j
q
p , equation (3.43) determines the rent for industrial space,
v
q
r .
F
q
=
∑
j
j
q
F =
( )
∑
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ − ⋅
⋅ ⋅
j
v
q
j
q
j
q j
q
j
j
r
p
X
θ
τ
σ
1
····························(3.43)

53
3.6.4. Equilibrium in the Market for Labor

Households allot the given time endowment E to travel to work (
pq
t T ⋅ ), travel to
shop (
pk
i ih
pqk
t Y ⋅ ⋅ ζ ), leisure (
h
pq
N ), and labor supply, as in equation (3.7). Simultaneously,
firms, residential developers, industrial developers, and the local government require labor
as an input, according to equations (3.23b), (3.31a), (3.31b), and (3.37b), respectively.
Equilibrium in the labor market implies that
∑
⋅ Ω
p
h
pq
h
pq
L =
hg
q
hv
q
hu
q
j
hj
q
L L L L + + +
∑
. ························································(3.44)

By substituting the associated supply and demand equations from the models of the various
individual agents, the equilibrium condition for the labor market is restated as
∑∑∑
⎭
⎬
⎫
⎩
⎨
⎧
− ⋅ − ⋅ ⋅ − ⋅ Ω
p
h
pq pq pk
ik
i ih
pqk
h
pq
N t T t Y E ζ = ····································(3.45)

() ( )
()
∑
− ⋅
⋅ ⋅
j
h
q
j
q
j
q j
q
hj
j
j j
w
p
X
θ
θ θ
τ
ρ
1

+
() ( )
()
u
u u
h
q
u
q
u
q
q
hm
w
r
H
θ
θ θ
τ
ρ
− ⋅
⋅ ⋅
1

+
() ( )
()
v
v v
h
q
v
q
v
q
q
hn
w
r
F
θ
θ θ
τ
ρ
− ⋅
⋅ ⋅
1

+
()
g
h
q
hg
g
g
w
θ
ρ
⋅
Λ
Γ
.

Just as in the market of undeveloped land, the labor market equilibrium condition
constitutes a wage term in which the elasticity of substitution plays a role as an arbitrary

54
power of the wage. As a result, it is difficult to derive an exact analytic form of market
clearing wage. A numerical solution technique is required.

3.6.5. Equilibrium in the Market for Commodities

The produced commodities are consumed by households, two types of developer,
the local government, and firms, including firms that have produced the commodity. The
household consumption of commodity i produced in zone r is given by equation (3.16a).
Equations (3.23a), (3.30a), (3.30b), and (3.37a) denote the commodity demand from various
firms, residential developers, industrial developers, and the local government, respectively.
The equilibrium condition in which no excess demand exists is
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+ + + + ⋅ Ω −
∑∑ ∑∑ ∑ ∑ ∑∑∑
∈ jq
ij
rq
gg p
ig
rp
q
iv
rq
p
iu
rp
hp q
ih
pqr
h
pq
i
r
Z Z Z Z Y X

= 0 . ····· (3.46)

According to equation (3.23a), the demand for intermediate input
ij
rq
Z from a firm is
proportional to the output
j
q
X . As a result, the equilibrium condition in the market for
commodity i (equation 3.46) can be restated as followings.
i
r
X =
i
r
FD +
( )
∑∑
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
− ⋅
⋅
j
j
q
q rq
i i
r
j
q
j
q ij
rq
X
c p
p
j
θ
ξ
τ
ϕ
1
·········································(3.47)
Therefore,

∑∑
⋅ −
jq
j
q
ij
rq
i
r
X a X =
i
r
FD ····························(3.48a)

where
i
r
FD =
∑ ∑ ∑ ∑ ∑∑ ∑
∈
+ + + ⋅ Ω
gg p
ig
rp
q
iv
rq
p
iu
rp
hp q
ih
pqr
h
pq
Z Z Z Y , and

55

ij
rq
a = ()
( )
j
j
rq
i i
r
j
q
j
q ij
rq
c p
p
θ
θ
ξ
τ
ϕ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
− ⋅
⋅
1
1

=
() []
() []
()
j
j
rq
i i
r
j
q
j
q
k
kq
i i
k
ij
rq
i i
r
ij
ij
c p
p
c p
c p
θ
θ
ξ
τ
ξ µ
ξ µ
ϕ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
− ⋅
⋅
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⋅ + ⋅
⋅ + ⋅
⋅
∑
1
exp
exp
1
·········(3.48b)

Although equation (3.48a) represents the equilibrium condition for a commodity i in
a zone r, the condition is also affected by equilibrium conditions for other commodities j
from other zones q. Consequently, the regional equilibrium condition can be stated as a
system of equations for all commodities from all zones. The matrix form of this system of
equation for I industrial sectors in R zones is

FD X A X = ⋅ − ·························································································(3.49)

where X is a (I ·R×1) column vector for output, A is a (I ·R × I ·R) matrix of endogenous
technical coefficients calculated via equation (3.48b) , and FD is another (I ·R×1) column
vector of final demands.
Equation (3.49) realizes the Isard’s ideal interregional input-output model (1953).
This well known input-output model is developed from the system-wide commodity market
equilibrium conditions. Many researchers have tried to simplify these dense interregional
relationship. For example, Chenery and Moses (1953) devised the concept of the trade pool,
and, Leontief and Strout (1963) further simplified the model of interregional trade (Bröcker
1998). However, the proposed urban CGE model estimates interregional technical
coefficients by distributing the given input share, φ, according to input and output prices, and
according to distance decay (µ) to capture the interaction between firms as in equation

56
(3.48b). In the model, the technical coefficients
ij
rq
a are non-zero when the distance decay is
reasonable, and, A is a very dense interregional technical coefficient matrix.
Final demand in the standard input-output model consists of household
consumption, government consumption, and exports. However, since the urban CGE model
describes a closed economy, no (net) exports are involved in final demand,
i
r
FD in equation
(3.48a). In addition, final demand includes intermediate demands from developers. Note
that it is possible to include additional final demand for export without changing the model
structure. In that, the price of exported goods is identical to that of internal consumption.

3.6.6. Equilibrium in the Transportation Network System

The transportation system supports the activity system by providing network
capacity so that the travel demand is accommodated with acceptable travel times and costs.
In the proposed model, the transportation network configuration, including capacity is an
exogenous input, while travel demand is completely endogenous. Three types of travel
demand are included in the model – home based working trips (
w
pq
f ), home based shopping
trips (
s
pk
f ), and freight movement between firms, developers and the local governments
(
t
rq
f ). Equations (3.50) specify the travel demands in total.

∑
Ω =
h
h
pq
w
pq
f ························································(3.50a)
∑∑∑
⋅ =
hq i
i ih
pqk
s
pk
Y f ζ ······················································(3.50b)
∑∑
⋅ =
ij
i ij
rq
t
rq
Z f ξ , { } g v u j , , ∪ ∈ ∀ I ····················································(3.50c)

57
Four additional factors are considered to load the travel demands of equations (3.50)
onto the network. First, trips may have different peak rates over time of day depending on
trip types. For example, work trips are concentrated within a narrow time window in the
morning, while shopping trips are distributed over a wide time span in the afternoon. Also
some truck movements take place at night for early delivery before markets open, while
small deliveries occur during daytime.
Second, the effect of congestion is explicit. A volume-delay function in the network
model estimates congestion based on a volume – capacity ratio, in which both the numerator
and denominator are expressed in terms of passenger-car unit. In general, home-based trips
are made by passenger cars, while deliveries are made in trucks, which contribute about
twice as much congestion as passenger cars do (HCM 1996). Third, the travel demand needs
to be increased in total to account for the return trips.
Finally, many activities can be achieved via just one chained trip. For example,
many deliveries to proximate destinations can be covered by one truck trip. Travel demands
estimated for from equations (3.50) are adjusted in the model for the four factors – peak rate,
passenger car equivalency, return trip, and combined trips – by using a simple combined
factor. Assuming exogenous coefficients γ
w
, γ
s
, and γ
t
are combined adjustment factors for
working trip, shopping trip, and freight movement respectively, then total travel demand, f
pq
,
that the network’s capacity must accommodate is

t
pq
t s
pq
s w
pq
w
pq
f f f f ⋅ + ⋅ + ⋅ = γ γ γ . ·····························································(3.51)

As stated above, the network configuration is exogenous. A network consists of
links and nodes. The nodes define the locations of intersections, and connect links to

58
transmit travel demand between nodes. The links are defined by two end nodes. Travel
demand traverses paths between an origin node to a destination node (or origin and
destination zone centroids). A path is a series of nodes connected by links such that no node
occurs more than once.
Trips induce congestion while traversing links according to a volume-delay function
(or a congestion function). Link travel time depends on link volume, and the volume-delay
function estimates congested travel time for given volume. The proposed model uses the
volume-delay function from the Bureau of Public Roads (BPR). Equation (3.52) shows the
functional form, where a, b, c are function parameters, t
0
is travel time at no volume on link
l . K is given parameter, known as practical capacity.

⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ + ⋅ =
c
K
v
b a t t
l
l
l l
0
····································································(3.52)

The network configuration and link capacities are exogenous. The transportation
network is public infrastructure that requires physical space. This capacity serves any travel
demand regardless of the origin and destination. This is why the state or federal
governments are implicitly assumed to be responsible for securing the network capacity via
construction and regular maintenance in this study. The responsibilities of local government
are restricted to spending to produce the worker amenity for local residents. Consequently,
the model requires developable land in each zone as input data, but not the land dedicated to
the transportation network.
The equilibrium condition in the transportation market consists of Wardrop’s first
principle of user equilibrium. According to this principle, travel times on any used routes

59
between a zone-pair are identical, and lower than on the unused ones in equilibrium (Sheffi
1985, p.22). A pair of origin and destination centroids is connected through many
alternative routes in general, and the drivers choose routes in such a way that the congested
travel time experienced by any individual traveler is minimized, and that all opportunities for
further improvement have been eliminated.
The zone-to-zone travel times and costs are calculated according to equations (3.53).
Travel time
k
pq
t on an arbitrary used route k between a zone-pair p-q is the sum of travel
time,
l
t on the links that make up the used route. Even though the travel time on used
routes are identical to each other, the distance between origin and destination on each route
may not be. Travel cost, c
pq
, is the out-of-pocket cost of driving calculated based on the
shortest distance between the origin and destination, according to an assumption that the
expense for gas, insurance, repair, and accidents is proportional to the driving distance. The
unit cost per mile driving, υ converts the shortest distance, which is the sum of length
l
d of
links in a used route between the zone-pair to cost. In the equations,
l , k
pq
δ is the
Kr őnecker’s delta, which is “1” if a link l is a part of the path k between zone-pair p and q,
and “0” otherwise.
l , k
pq
δ is Associated with link length
l
d ,
l , k
pq
δ

∑
⋅ = =
l
l
l
t t t
k
pq
k
pq pq
,
δ ·····················································································(3.53a)
⎟
⎠
⎞
⎜
⎝
⎛
⋅ ⋅ = =
∑
l
l
l
d c c
k
pq
k
k
pq
k
pq
,
min min δ υ ·····························································(3.53b)

Wardrop’s principle also applies to a network system with more than one mode.
Travel time and cost affect the household utility function according to equation (3.14).

60
Alternative travel modes are differentiated with respect to the time and cost. Therefore
household utility is distinguished according to travel modes, and the modal split is
determined from the household model. The endogenous modal travel demand establishes
the equilibrium conditions over the available routes given an exogenous modal capacity.

3.7. VARIABLES

While the behavior of economic agents and the implication of market equilibria are
straightforward, it may not be so clear what are the solutions that the proposed model
calculates for, and how to compute them. This section categorizes the endogenous variables
that the model estimates, followed by brief discussions about the existence and uniqueness
of the solution.

3.7.1. Model Solution

The demand equations for each economic agent in the model are continuous
functions of the prices of commodities, developed land, raw land, labor, and travel time and
cost as in equations (3.6), (3.23), (3.29), (3.30), (3.31), and (3.37), respectively. Supply is
also a function of price, as shown by equation (3.7), (3.17), (3.26), and (3.28) for labor,
commodity, and developed spaces. On the other hand, the equilibrium price of commodity
and the rent for developed space are also functions of prices, as equations (3.22) and (3.32)
illustrate. Equations (3.40) and (3.45) show that the rent for undeveloped land, and wages
rely on the optimal output (supply), which, in turn, are functions of prices. Travel time is a
function of travel demand, which also is a function of optimal demand.

61
In summary, all the quantity measures (demand, supply, or output) are functions of
prices and prices are interdependent. Therefore, all the endogenous variables can ultimately
be explained by prices, and thus, prices are the unknown variables in the urban CGE model.
The number of unknown variables depends on the number of zones (R), the number of
industries (I), the number of household types (H), and the number of travel modes (M).
Table 3.1 summarizes the six endogenous price measures in the model.

Table 3.1 Number of Variables to be Solved

Variable Type Number of unknowns Related Equation
Commodity Price I×R 3.22
Wage H×R 3.45
Raw Land Rent R (3×R)* 3.40 (3.41)
Travel Time M×R×R 3.53

*) When a land use plan is implemented, the market for raw land splits according to three land
use types – residential, industrial and governmental use.

3.7.2. Existence and Uniqueness of the Solution

According to Brouwer’s fixed-point theorem, a continuous function that maps a
compact convex surface on to the surface itself has at least one point where the function
maps the point itself (Halpern 1967, MacKinnon 1976). In mathematical form, a continuous
mapping S S f → : ,
n
S ℜ ⊂ , there exists at least one point of S x ∈ such that () x x f = ,
and such an x is a fixed point.

62
In a general equilibrium framework, the equilibrium price vector is a fixed point. A
rigorous proof of this assertion is provided by Browder (1967). As noted above, prices are
functions of prices themselves, while the other endogenous variables are also explained by
prices. This means that the whole system endogenous variables is simply described as
F(price) = price, where F denotes all of the equilibrium conditions. Thus the price is a fixed
point in general equilibrium.
However, to prove the existence of a price vector as a fixed point, the geometric
shape of the price surface must be examined. The existence and uniqueness of a fixed point
is based on the premise that the mapping surface should be compact (a region that is
bounded and includes the boundary), and convex (any two points on the surface can be
connected by a straight line that is defined within the boundary of the region).
For some of the numerical analyses performed, the urban CGE model was unable to
produce a solution for some exogenous inputs. Specifically, the model breaks down when
the exogenous network capacity is insufficient to accommodate the endogenous travel
demand, or the exogenous number of households is too great. As equation (3.52) shows, the
link volume-delay function is convex and unbounded for the standard BPR parameters,
a=1.0, b=0.15, and c=4.0. Consequently, zone-to-zone travel time, t
pq
, is not bounded either.
If network capacity is not sufficient or too many households are in the system, congestion is
severe, and the travel time for work trips,
pq
t in equation (3.10) for the household indirect
utility function exceeds the disposable income for households. In the equation, the
disposable income is calculated based on labor income, land rent income, less the travel cost
for working trips (
pq
h
q pq
h
pq
h
q
t w T c T D w E ⋅ ⋅ − ⋅ − + ⋅ ). Indirect utility is calculated for the
logarithm of disposable income. This utility function cannot be defined when disposable

63
income is negative, and without a utility function, numerical calculations of residential
allocation fails.
This phenomenon may have an implication in planning. If a regional network has
less capacity than its activity system requires, local residents and firms will pay more to
travel and ship than the residents of other regions do. Out migration to other places would
be encouraged, until the cost associated with congestion is reduced to the opportunity cost of
migration.
A similar observation is made by Anas (1984). He proved that enough land to
accommodate all activity is essential for the existence of a solution for his theoretical general
equilibrium model; otherwise the non-negativity condition is violated. However, in the
proposed model, the rent for developed spaces regulates demand for undeveloped land, and
prevents the rent from increasing not too much for numerical analysis.
Note that all the prices in the model are relative terms. Table (3.1) shows the
number of unknowns that should be solved and the related equations. Examining the
equations, the number of unknowns is identical to the number of conditions. According to
Walras’s law, if a system with k markets is in equilibrium as a whole and k-1 individual
markets are also in equilibrium, then the last market is in equilibrium too. Therefore one of
the equilibrium conditions is redundant, and the number of unknown is one more than the
number of equilibrium conditions (See Crouch (1972) for a rigorous proof of Walras’s law).
To overcome the mismatch between the number of unknowns and equilibrium conditions,
rent for undeveloped land in a specific zone is fixed, and all the other prices are calculated
relative to this one.

64
3.8. SOLUTION ALGORITHM

This section provides a solution algorithm for the urban CGE model. Although only
the prices are categorized as unknowns, the model includes many intermediate variables that
must be estimated to compute a price vector. For example, as equations (3.40) and (3.41)
show, output quantity is required to be able to calculate the rent for undeveloped land. For
this reason, the overall procedure is arranged so that as many endogenous variables as
possible are updated within limited number of steps together with the unknown prices.
The algorithm converges to a solution by using an iterative successive averaging
scheme. This averaging scheme as a solution method to the Walrasian fixed point problems
has been studied by Magnanti and Peraski (1977, 1988, and 2002). The idea is to combine
the auxiliary solution at the current iteration with the solution from the previous iteration by
using linear coefficients. In this study, the linear coefficients are defined by 1/n for the
auxiliary solution (n)-th iteration, and 1-1/n for the solution from (n-1)-th, respectively. The
first iteration, i.e., n=1, starts with assumed initial values, and calculates a set of auxiliary
solutions. The first auxiliary solutions and the initial values are combined at the end of the
first iteration with linear combination coefficients 1, and 0, respectively. Therefore, no
portion of the first set of solutions is derived directly from the assumed initial values. The
effect on the solutions for subsequent iterations is attenuated relative to the earlier solutions.
Unless the auxiliary solutions are extremely larger or smaller than previous solutions, the
gap between solutions from two consecutive iterations eventually becomes negligible, and
thus the process converges.
Within iteration, the 10-step procedure updates unknown variables from the
minimum number of known variables at any given step. Detail of the 10-step solution

65
procedure is provided below. Variables have a superscript, n, on left-hand side to specify
the iteration index at which the variable is updated.

Step 0: Initialization
• Set iteration index n=0

• Initialize major endogenous variables by assumption.
p p
n 0
= , X X
n 0
=
u u n
r r
0
= ,
v v n
r r
0
= , r r
n 0
=
w w
n 0
=

• Estimate initial travel cost and travel time by all-or-nothing assignment based on
free flow speed, i.e., travel time without congestion. The Dual-simplex minimum
path algorithm is used along with Moore-Pape algorithm.
l
v
n
=0
( )
pq
n n
v K t t
, l l
=
( )
∑
⋅ =
l
l l
l
t v t
n n k
pq
n
pq
n ,
δ
⎟
⎠
⎞
⎜
⎝
⎛
⋅ ⋅ =
∑
l
l
l
d c
k
pq
n
k
pq
n ,
min δ υ

• Initially distribute households evenly over the zones
R R
1
⋅
⋅ Θ = Ω
h h
pq
n

• Increase iteration index; n:=n+1

Step 1: Estimation of commodity prices
• Calculate the spatial input share coefficient for intermediate goods
( ) { }
() {}
∑
− −
− −
⋅ + ⋅
⋅ + ⋅
⋅ =
k
kq
n i i
k
n ij
rq
n i i
r
n ij
ij ij
rq
n
c p
c p
1 1
1 1
exp
exp
ξ µ
ξ µ
ϕ ϕ

• Calculate the auxiliary price vector for each commodity based on the price vector
from the previous iteration

66
() () ()
j
j j j
i
v
q
n
j
h
h
q
n
hj
r
rq
n i i
r
n
ij
rq
n
j
q
j
q
r w c p
p
θ
θ θ θ
σ ρ
ξ
ϕ
τ
−
−
−
−
−
−
− −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
⋅
−
=
∑∑ ∑
1
1
1
1
1
1
1
1 1
1
1
ˆ

• Update the commodity prices by combining the auxiliary price vector with the
vector from the previous iteration. The inverse of iteration index is use for the linear
combination coefficient
( )
j
q
n j
q
n n j
q
n
p p p ˆ 1
1
⋅ + ⋅ − =
−
π π ,
n
n
1
= π

Step 2: Residential development model
• Calculate the auxiliary floor rent for housing
()
⎪
⎩
⎪
⎨
⎧
⋅ + ⋅ ⋅
−
=
∑∑
−
−
ir
rp
n i i
r
n iu
rp
n
u
p
u
p
u
c p r
θ
ξ ϕ
τ
1
1
1
1
ˆ
+ () ( )
u
u u
p
n u
h
h
p
n hu
r w
θ
θ θ
σ ρ
−
−
−
−
−
⎭
⎬
⎫
⋅ + ⋅
∑
1
1
1
1
1
1

• Update the housing rent by combining the auxiliary rent with the previous value
( )
u
p
n u
p
n n u
p
n
r r r ˆ 1
1
⋅ + ⋅ − =
−
π π ,
n
n
1
= π

• Calculate household income from the land rent
( )
() {}
∑∑∑
∑
− − − −
− − − −
−
⋅ + ⋅ − ⋅
⋅ + ⋅ − ⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ =
hp q
pq
n h
q
n
pq
n h
q
n
pq
n h
q
n
pq
n h
q
n
p
p
n
p
h
pq
n
t w c T w E
t w c T w E
r A D
1 1 1 1
1 1 1 1
1

• Calculate the housing demand in quantity, based on the household income and
housing rent. The residential developers should supply this demand.
( )
u
p
n
pq
n h
q
n
pq
n h
pq
n h
q
n
h
h
h
pq
n
r
t w c T D w E
H
1
1 1 1 1
1
−
− − − −
⋅ + ⋅ − + ⋅
⋅
−
=
φ
β

∑∑
=
hq
h
pq
n
p
n
H H

• Calculate the final demand for intermediate inputs by residential developers

67
( )
u
rp
n i i
r
n
u
p
u
p
n
p
n iu
rp
n iu
rp
n
c p
r
H Z
θ
ξ
τ
ϕ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
− ⋅
⋅ ⋅ =
−1
1

• Calculate freight movement for delivery of intermediate inputs to residential
developers
∑
⋅ =
i
iu
rp
n t u t
rp
n
Z f γ
,

Step 3: Industrial development model
• Calculate the auxiliary floor rent for industrial space

()
⎩
⎨
⎧
⋅ + ⋅ ⋅
−
=
∑∑
−
−
ir
rq
n i i
r
n iv
rq
n
v
q
v
q
v
c p r
θ
ξ ϕ
τ
1
1
1
1
ˆ
+ () ( )
v
v v
q
n v
h
h
q
n hv
r w
θ
θ θ
σ ρ
−
−
−
−
−
⎭
⎬
⎫
⋅ + ⋅
∑
1
1
1
1
1
1

• Update the rent for industrial space by combining the auxiliary rent with the
previous value
( )
v
q
n v
q
n n v
q
n
r r r ˆ 1
1
⋅ + ⋅ − =
−
π π

• Calculate the quantity of industrial space demand, based on the firms’ outputs and
prices. The industrial developers should supply this demand.
( )
j
v
q
n
j
q
j
q
n
j
q
n j j
q
n
r
p
X F
θ
τ
σ
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ − ⋅
⋅ ⋅ =
−
1
1

∑
=
j
j
q
n
q
n
F F

• Calculate the final demand for intermediate inputs by industrial developers
( )
v
rp
n i i
r
n
v
q
v
q
n
q
n iv
rq
n iv
rq
n
c p
r
F Z
θ
ξ
τ
ϕ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ +
− ⋅
⋅ ⋅ =
−1
1

• Calculate freight movement for delivery of intermediate inputs to industrial
developers
∑
⋅ =
i
iv
rq
n t v t
rq
n
Z f γ
,

68

Step 4: Local government model
• Calculate the number of households in each local government’s jurisdiction
∑∑∑
∈
Ω
= Κ
g pq h
h
pq
n
g n
1

• Calculate tax revenue. Except for the output X, all variables for this calculation are
updated in this iteration
∑∑
∈
−
⎟
⎠
⎞
⎜
⎝
⎛
⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ = Γ
g pi
v
p p
n v
p
n u
p p
n u
p
n i
p
i
p
n i
p
n g n
F r H r X p τ τ τ
1

• Calculate the power mean of input prices
() () ()
∑∑∑ ∑
∈
−
−
−
−
−
−
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
+ +
⋅ +
= Λ
g pi r
p
n
g
h
h
p
n
hg
rp
n i i
p
n
ig
rp g n
g g g
r w c p
1
1
1
1
1
1
θ θ θ
σ ρ
ξ
ϕ

• Calculate the auxiliary worker amenity
()
1
1
−
Λ ⋅ Γ ⋅ Κ =
g
g n g n g n
p
n
G
θ

• Calculate final demand for intermediate inputs by local governments
ig ig
rp
n
ϕ ϕ = when ( )
kp
n i i
k
n
k
rp
n i i
r
n
c p c p
1 1
min
− −
⋅ + = ⋅ + ξ ξ
()
g
rp
n i i
p
n
ig
rp
n
g n
g n
ig
rp
n
c p
Z
θ
ξ
ϕ
1 −
⋅ +
⋅
Λ
Γ
=

• Calculate freight movement for delivery of intermediate inputs to the local
governments
∑
⋅ =
i
ig
rp
n t g t
rp
n
Z f γ
,

Step 5: Household model
• Calculate coefficient for commodity consumption
ih
rp
α
{ }
i
z
n
z
i
X X
1
min
−
=
(

69
() []
() []
∑
−
−
−
−
⋅ + ⋅ ⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + ⋅ ⋅ ⎟
⎠
⎞
⎜
⎝
⎛
⋅ =
z
pk
n i i
k
n ih
i
i
z
n
pk
n i i
k
n ih
i
i
k
n
ih ih
pk
n
c p
X
X
c p
X
X
ih
ih
1
1
1
1
exp
exp
ξ µ
ξ µ
α α
η
η
(
(

• Calculate the indirect utility
h
pq
n
V =() ( )( )
⎥
⎦
⎤
⎢
⎣
⎡
− − ⋅ ⋅ − ⋅ − + ⋅ ⋅ −
− − − − h
pq
n h
q
n
pq
n h
pq
n h
q
n h
t w T c T D w E φ φ 1 ln ln 1
1 1 1 1

( )
∑∑
− − −
⋅ ⋅ + ⋅ + ⋅ −
ik
pk
n i h
q
n
pk
n i i
r
n ih
pk
n
t w c p
1 1 1
ln ξ ξ α
( ) ( ) ( )
p
n h h
q
n h u
p
n h
G w r ln ln ln
1
⋅ + ⋅ − ⋅ −
−
φ δ β

• Distribute the given number of households onto the combination set of place of
residence p, and place of work q.
∑∑
⋅
⋅
⋅ Θ = Ω
pq
h
pq
n h
h
pq
n h
h h
pq
n
V
V
) exp(
) exp(
ψ
ψ

• Calculate household final demand for intermediate commodities
( )
pk
n i h
q
n
pk
n i i
k
n
pq
n h
q
n
pq
n h
pq
n h
q
n
h
ih
pk
n
ih
pqk
n
t w c p
t w c T D w E
Y
1 1 1
1 1 1 1
1
− − −
− − − −
⋅ ⋅ + ⋅ +
⋅ + ⋅ − + ⋅
⋅
−
=
ζ ζ φ
α

• Calculate home-based trips
work trips
∑
Ω =
h
h
pq
n w
pq
n
f
shopping trips
∑ ∑ ∑
⋅ =
hq i
i ih
pqk
n s
pk
n
Y f ζ

• Calculate the fraction of time endowment allocated to leisure
( )
h
q
n
pq
n h
q
n
pq
n h
pq
n h
q
n
h
h
h
pq
n
w
t w c T D w E
N
1
1 1 1 1
1
−
− − − −
⋅ + ⋅ − + ⋅
⋅
−
=
φ
δ

• Calculate the labor supply
∑∑∑
⎭
⎬
⎫
⎩
⎨
⎧
− ⋅ − ⋅ ⋅ − ⋅ Ω =
− −
p
h
pq
n
pq
n
pk
ik
n i ih
pqk
n h
pq
n h
q
n
N t T t Y E W
1 1
ζ

70
Step 6: Firm model
• Calculate the technical coefficient matrix
( ) { }
() {}
∑
∑
−
−
⋅ + ⋅
⋅ + ⋅
⋅ =
r
k
kq
n i i
k
n ij
rq
n i i
r
n ij
ij ij
rq
n
c p
c p
1
1
exp
exp
ξ µ
ξ µ
ϕ ϕ
() ( )
()
j
j j
rq
n i i
r
n
j
q
j
q
n
ij
rq
n ij
rq
n
c p
p
a
θ
θ θ
ξ
τ
ϕ
1
1
−
⋅ +
− ⋅
⋅ =

• Calculate the sum of final demands of intermediate inputs from households,
residential developers, industrial developers, and the local government.
∑ ∑ ∑ ∑ ∑∑∑
∈
+ + + ⋅ Ω =
gg p
ig
rp
n
q
iv
rq
n
p
iu
rp
n
hp q
ih
pqr
n h
pq
n i
r
Z Z Z Y FD

• Solve for output
j
q
X via the Gauss-Jordan method
i
r
jq
j
q
n ij
rq
n i
r
n
FD X a X = ⋅ −
∑∑

• Calculate freight movement for delivery of intermediate inputs between firms
( )
∑
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⋅ +
− ⋅
⋅ ⋅ =
−
j rq
n i i
r
n
j
q
j
q
n
j
q
n ij
rq
n i t
rq
n
j
c p
p
X f
θ
ξ
τ
ϕ
1
,
1

Step 7: Wage model
• Calculate the auxiliary wage from the equilibrium condition by using the Golden-
section method.

() ( )
()
∑
− ⋅
⋅ ⋅
j
h
q
j
q
j
q
n
j
q
n hj
j
j j
w
p
X
θ
θ θ
τ
ρ
ˆ
1
;labor demand from firms
+
() ( )
()
u
u u
h
q
u
q
u
q
n
q
n hm
w
r
H
θ
θ θ
τ
ρ
ˆ
1 − ⋅
⋅ ⋅ ;labor demand from residential developers
+
() ( )
()
v
v v
h
q
v
q
v
q
n
q
n hn
w
r
F
θ
θ θ
τ
ρ
ˆ
1 − ⋅
⋅ ⋅ ;labor demand from industrial developers

71
+
()
g
h
q
hg
g n
g n
w
θ
ρ
ˆ
⋅
Λ
Γ
; labor demand from the local government
–
h
q
n
W = 0 ; labor supply

• Update wage by combining the auxiliary wage with previous value
( )
h
q
n h
q
n n h
q
n
w w w ˆ 1
1
⋅ + ⋅ − =
−
π π

Step 8: Rents for undeveloped land
• Calculate the auxiliary rent for undeveloped land from the equilibrium condition by
using the Golden-section method

() ()
()
u
u
p
u
p
u
p
n
p
n u
r
r
H
θ
θ
τ
σ
ˆ
1 − ⋅
⋅ ; land demand from residential developers
+
() ()
()
v
v
p
v
p
v
p
n
p
n v
r
r
F
θ
θ
τ
σ
ˆ
1 − ⋅
⋅ ; land demand from industrial developers
+
()
g
p
g
g n
g n
r
θ
σ
ˆ
⋅
Λ
Γ
; land demand from the local government
– A
p
= 0 ; land supply available for development

• Update rent by combining the auxiliary rent with previous value
( )
p
n
p
n n
p
n
r r r ˆ 1
1
⋅ + ⋅ − =
−
π π

Step 9: Transportation model
• Identify the endogenous values for Kr őnecker’s delta,
l , k
pq
n
δ by using the Moore-
Pape, and the Dual-simplex minimum path algorithms, based on previous travel time,
l
t
n 1 −

•
• Aggregate travel demand
t
pq
n t s
pq
n s w
pq
n w
pq
n
f f f f ⋅ + ⋅ + ⋅ = γ γ γ

72
• Assign the demand on the path identified by Kr őnecker’s delta to calculate the
auxiliary link volume,
l
v ˆ (all-or-nothing assignment).

• Update link volume and link travel time. Instead of combining link travel times
between iterations, the link volumes were combined with previous value, and then
link travel time is calculated based one the combined volumes. It is to keep the
volumes on the routes identified from previous iterations.
• ( )
l l l
v v v
n n n n
ˆ 1
1
⋅ + ⋅ − =
−
π π
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ + ⋅ =
c
n
n
K
v
b a t t
l
l
l l
0

• Update zone-to-zone travel times and costs
∑
⋅ =
l
l
l
t t
n k
pq
n
pq
n ,
δ , and ⎟
⎠
⎞
⎜
⎝
⎛
⋅ ⋅ =
∑
l
l
l
d c
k
pq
n
k
pq
n ,
min δ υ

Step 10: Convergence check
• Calculate the maximum differences between prices in current and previous iterations

()
i
q
n
i
q
n i
q
n
q i
p
p p
p
1
1
,
max
max
−
−
−
= ∆
()
h
q
n
h
q
n h
q
n
q h
w
w w
w
1
1
,
max
max
−
−
−
= ∆
()
q
n
q
n
q
n
q
r
r r
r
1
1
max
max
−
−
−
= ∆

• If ε < ∆
max
p , ε < ∆
max
p , and ε < ∆
max
p , for a given tolerance ε >0, or n = N for
given maximum number of iterations N, then stop. Otherwise, update iteration index
n to n+1, and go to Step 1.

The proposed solution procedure employs several well-known algorithms to solve
for market equilibria. The golden section method is applied to solve for the equilibrium
wages and the rents for undeveloped land, which are given via non-linear power functions in
steps 7 and 8 respectively. The Gauss-Jordan algorithm is applied to solve the linear

73
equations for the input-output relations in step 6. In addition, two minimum path algorithms
are used in step 0, and 9, respectively, to establish the shortest paths between all zone-pairs.
The Moore-Pape algorithm is suggested by Sheffi (1985, pp.124-129), and applied to find
paths from an origin to all destinations. This path information is recycled to find paths from
other origins by the dual-simplex algorithm proposed by Florian et.al. (1982), and Pallottino
(1984). By applying the two minimum path algorithms, along with the internal network data
format (in computer memory) that was suggested by Dial, Glover, Karney, and Klingman
(1979), step 10 takes about 30 percent less time than when only the Moore-Pape algorithm is
applied.
Many prominent software packages for CGE models are on the market, but it is rare
that they support a (transportation) network data that is implemented in links-and-nodes.
Although it is possible to implement the network and associated endogenous variables in
matrices for the use of commercial software, eventually the results should be converted back
to a graph. This back-and-forth conversion process is computationally intensive. For this
reason the procedure was programmed by using C++ in Microsoft Visual Studio V5.0 to
provide an integrated development environment, rather than by using off-the-shelf software.
A personal computer with Pentium-4 2.0GHz CPU, and 1.0GB 333MHz RAM is used
throughout the numerical analyses.

In this chapter, the formulation of an urban CGE model was detailed. The behaviors
of households, firms, developers, and local governments were modeled to include
consumption preferences, and production technologies. The equilibrium conditions in
various markets were articulated so that competitive equilibrium prices can be determined.
Existence and uniqueness were briefly examined using intuitive description, followed by

74
discussion of a solution algorithm. In the following chapter, the characteristics of the
established model will be examined closely by use of a simple numerical example that
includes 11 zones, 3 types of households, and 3 types of industries.

75
4. CHARACTERISTICS OF THE URBAN CGE MODEL

4.1. OVERVIEW

In this chapter, the proposed urban CGE model is applied to a set of synthesized data
and the model characteristics are reviewed. Although the synthesized data is small in size,
it is a complete set to demonstrate model’s applicability to the complex real world problems.
The data is first used for a baseline analysis. Various perturbations are then applied to the
data to analyze model’s responses as a sensitivity analysis.
As a part of the baseline analysis, the model estimates various interrelated features
of an urban system, which are classified into four categories; 1) land use pattern; 2) spatial
production activity allocation; 3) residential allocation pattern; and 4) the performance of
transportation system.
To perform the sensitivity analysis, the parameters assumed for the sample data are
disturbed one by one, and then applied to the proposed model. This effort provides a basis
for the model to establish its applicability to real world problems. To be used as a planning
model, the marginal variations estimated by the model for the perturbed inputs must agree
with the common senses. Effects of the following parameters are analyzed in this sensitivity
analysis: 1) the elasticity of substitution in production functions; 2) the distance decay
coefficients for the intermediate inputs; 3) the number of households; and 4) network
capacity. In addition to the sensitivity analyses, comments on modeling behavior are
provided for issues including 1) the difference between the production functions of the

76
proposed model and the original CES production functions; and 2) the convergence and
stability of the model.
The sample data set in this chapter is referred to as “Linear-City”. The Linear-City
data set consists of a linear zone system with a simple transportation network system,
allowing easy review of the induced activity allocation pattern. The input share coefficients
for the production functions and the household utility coefficients are drawn based on a
transaction table for 1999 Southern California economy. A user of the model needs to
understand that using the real transaction table to draw the coefficients does not mean that
the example data will describe the actual economic system. This is because the unit of
transaction table is dollars, whereas the coefficients drawn from this transaction table is
applied to the quantity of goods and services in the proposed model.
The application will be thorough and none of the interactions between economic
agents (please see solution algorithm in section 3.8) in the proposed urban CGE model will
be suppressed. Analyses in this chapter were performed admitting all intermediate trades,
including intra-zonal trades between any industries. The intra-zonal trades were not
considered in the studies by Kim (1990), Anas and Kim (1996), and Anas and Xu (1999).
This chapter consists of five sections, including the overview section. The settings
for baseline analysis of linear-city data is summarized in section 4.2. The settings include
geometry of the region, transportation network configuration, and parameters for the
synthesized activity systems. Section 4.3 lists the results from the baseline application of the
linear-city. Section 4.4 discusses the sensitivity analyses for selective exogenous
parameters. Section 4.5 summarizes convergence and resilience of the algorithm. An
additional comment on the production function is provided in section 4.6.

77
4.2. THE SETTING FOR LINEAR-CITY APPLICATIONS

The Linear-City is a rectangular region of 1mile in north-south direction, and 11
miles in east-west direction, respectively. The region is divided evenly into eleven square
zones of 1mile×1mile, i.e., R=11. Figure 4.1(a) shows the arrangement of zones, with the
zone ID. The zone system is symmetric with respect to the center of region which is Zone 6,
in terms of its geometry, available land for development, as well as the network
configuration.
Although individual zones have same geometric area within each zone boundary, the
available land for development is not identical because of the different share of roadways in
each zone. Space for roadway is determined based the number of lanes [See Figure 4.1(d)]
and the assumed dimension. The required land for the road is calculated assuming the lane
of a congestible link to be 180ft wide. The unrealistically wide lane width is based on the
assumption that each link in Figure 4.1(c) represents a local network grid, which consists of
several links and roadway shoulders. In addition to the congestible links, 1 mile-by- 120ft
collector roads are included, and the corresponding land requirement is csalculated. The
total area allotted for roadway ranges from 7.4% (zone 1 and zone 11) to 22.7% (in zone 6)
of the total geometric area.
Transportation network for the linear-city includes 66 directional links (includes 22
centroid connectors), and 34 nodes (includes zone centroids). Figure 4.1(c) shows the
network is represented according to the conventional data format for transportation planning
models in terms of 1) zones are represented by a special node, zone centroid, and trips are
produced from and terminated to these nodes; 2) produced trips from centroids transmit to
congestible network through virtual, non-congestible links, centroid connectors; 3) links are

78
defined by two nodes one for each end, and have a convex congestion function to traffic
volumes [See equation (3.52)]; and 4) nodes present the locations where two or more links
adjoin.

(a) Zone System of the linear-city

1 2 3 4 5 6 7 8 9 10 11

(b) Available Land (10,000 sq-yd)

287 276 266 255 245 239 245 255 266 276 287

(c) Transportation Network

(d) Link capacity (number of lanes per direction)

1 2 2 3 3 4 4 5 5 6 6 6 6 5 5 4 4 3 3 2 2 1

Zone Centroids Nodes
Centroid Connectors (2-way) Links (2-way)

Figure 4.1 Zone System and Network Configuration of the Linear-City

1mile
1mile

79
Figure 4.1(d) shows the predefined number of lanes on each directional congestible
link. Free flow speed of the congestible link,
0
l
t , shown in equation (3.52) is assumed to be
40 mph, and the practical capacity
l
K per is assumed to be 1,000 Passenger-Car-Unit
(PCU). Free flow speed and total capacity of centroid connectors are assumed to be 25 mph
and 100,000 PCU respectively.
Nine distinct economic agents are included in the linear-city example. Those are
three types of households (H=3), firms of three industrial sectors (I=3), residential
developers (u), industrial developers (v), and the local governments (g). Refer to Appendix
A for the parameters that specify production technologies and consumption behaviors. 280
hours of endowment are given equally to all households over 20 working days in a month,
i.e., E=280, and T=20 in the budget constraint, from equation (3.2).
A total of 40,000 households reside in 11 square miles of the linear-city including
10,000 of Type 2 households. The household density is 3,636.4 per square mile. Comparing
this density to the density of City of Los Angeles which is 2,851.8 housing units per square
miles (US Census Bureau, 2000), the linear-city example depicts an extremely dense area.
Households are differentiated due to the preference in consumption and shopping
behavior (see Appendix A-a). 4,000 households of Type 1 prefer more share of housing
(
1
δ =0.350) for their utility, while the 26,000 households of Type 3 opt to consume more
commodities with utility coefficients of
13
α =0.206,
23
α =0.124, and
33
α =0.190. Type 3
households are most sensitive to the effective price (the sum of commodity price and
transport cost for shopping) in choosing the place of shopping for commodity 3, with the
decay coefficient
33
µ =-1.5.

80
The input coefficients for production functions were drawn from an estimated
transaction table for 1999 Southern California economy (Minnesota IMPLAN Group). The
initial transaction table was given with 528 sectors. It was aggregated into a 17-sector table
first, and further aggregated into a table for 6 sectors (3-industrial sectors, 2-developers, and
the local government) according to following aggregation scheme:

• Industry 1: Durable, and non-durable manufacturing.
• Industry 2: Wholesale and retail.
• Industry 3: F.I.R.E., business service, personal service, entertainment, health,
education, and professional related service.
• Developers: Construction
• Local government: Federal, state and local government.
• Agriculture, mining, and transportation in the 17-sector table are not included.

Even though the numerical values are based on real data, the input share coefficients
may not exactly reflect the actual industrial system of Southern California because of the
following two reasons. First, the original transaction table was in dollar unit, while the
production functions in the proposed CGE model requires coefficients comply with
quantities. Second, the input coefficients for undeveloped land (for residential developers,
industrial developers and local governments), or the developed space (for households and
firms) are arbitrarily assumed.
Note that quantity of land and developed space is measured by 10 square-yards. As
stated above, the utility functions and production functions are based on quantity of
consumptions or intermediate inputs, instead of the value of commodity. The input
coefficients in the production and utility functions should be changed according to the unit.
In the Linear-City example, the assumed utility coefficient for housing ranges 0.18~0.35

81
when land and space is measured per 10 square-yards. For the same amount of spaces, the
utility coefficients would be changed to 0.687~0.843 when “per square-yard” is applied
1
.
Input coefficients for production functions also need to be altered from 0.236~0.355 to
0.472~0.595. Since the units for other output quantity measurements were not explicitly
specified, it is not necessary to measure the land and space by known units. High input
share coefficients might generate a superfluous impression that the economy of linear-city
example relies only on land and space consumption. For this reason undeveloped land and
developed spaces are measured in “10 square- yards”, unless stated otherwise.
The parameters applied to estimate travel demand are summarized in Appendix A
(d). According to this, a household will make one trip to buy 4 units of commodities from
Industry 2, while it will make a trip for 20 units of product from Industry 3. Deliveries
between industries for the intermediate inputs are assumed to be 5~10 times less frequent
than household shopping trips. For example, 20 units of commodity from Industry 2 will be
delivered by one truck trip.
Trip production rates and return trip rates are based on 1996 Southern California
Association of Governments Travel Demand Survey (SCAG 1997) and 1997 Heavy-Duty
Truck Survey (SCAG 1998). According to the surveys, 63.4% of home-based working trips
concentrate for morning 3 hours (6AM to 9AM), and about 3.1% of trips are returning from
work to home in the same time period. Home-based other trips, including shopping trips,
have 28.4% of peak rate in the morning 3-hour peak, and 6.3% of trips are return to home
from the activities. The rates are converted for 2-hour peak time period for linear-city

1
For example, since the utility function is homogenous of degree 1, the utility coefficient 0.350 for
housing of Household Type 1 is changed to 0.84 when it is prorated to “square yard”, i.e., 0.84 =
(0.350 * 10) / (0.127 + 0.075 + 0.198 + 0.100 + 0.350 * 10 + 0.15).

82
example. The heavy-duty truck survey in SCAG area shows that 17.3% of truck trips take
place in the morning 3 peak hours. Assuming a truck is equivalent to 2.5 passenger cars in
contribution for congestion and constitute 10% of return trips, 25.9% of total daily truck
trips would be concentrated within the morning 2-hour peak period, and 2.6% of trucks will
return from its destination zones.
Elasticity of substitution φ, and decay coefficient, µ are assumed arbitrarily. The
effect from these arbitrary parameters is examined in Section 4.4. Note that the decay
coefficient for the local government is set to negative infinite. In the local government
model (Section 3.5), the intermediate input is purchased at a location where the effective
price is the lowest [Equation (3.39)] and this behavior is modeled by the negative infinite
decay.
The solution procedure starts with following initial values:
• Land rent,
0
r = $10.0 / 10 sq-yd / month
• Residential rent,
0
r
u
= $10.0 / 10 sq-yd / month
• Industrial rent,
0
r
u
= $10.0 / 10 sq-yd / month
• Commodity price,
0
p = $10.0 / unit
• Wage,
0
w = $10.0 / hour
• Output ,
0
X = 10
6
units / month

In addition to the initial values, the land rent in Zone 1 that is used for residential space is
fixed at $10.0 per 10 square yard for a month. Therefore all the endogenous prices including
output prices per unit, hourly wages, and monthly rents for developed spaces is presented in
terms of relative prices to this land rent under the Walrasian Law.

83
4.3. BASELINE OF THE LINEAR-CITY

This section reviews the baseline of the Linear-City. The proposed urban CGE
model produces various aspects of activities in the Linear-City, and the results are
categorized into four groups. These are as follows: 1) land use pattern; 2) household
allocation pattern; 3) industrial activity allocation; and 4) distribution travel demand and the
performance of transportation network. This section lists all the estimates produced by the
model without detailing reasons for such baseline estimates.
Two types of graphs are used to show the activity patterns over zones in the Linear-
City throughout the section. Since the zones and transportation network are arranged
symmetric to the center of the region, which is Zone 6, all activity patterns are also
symmetric, unless any localized perturbations, such as zonal land use restriction is applied.
Because of the symmetric configuration, a pair of graphs over two halves of the Linear-City
show the results. One half represents the pattern estimated by the model, and the other half
shows percent of zonal values to the maximum estimations. For example, the left-hand of
Figure 4.2 (a) shows the zonal land rent estimated by the model over Zone 1 through 6,
while the right-hand shows the ratio to the maximum rent in percentage over Zone 6 to 11.

4.3.1. Land Use Pattern and Floor Development

Housing, office developers, and local government compete in the land market at
each zone, and realize the market clearing rent for land. Relative to the given $10 in Zone 1,
the proposed model estimates $12.84 for a 10 sq-yd a month in the center of region, Zone 6.
Figure 4.2 (a) shows 22% of difference on the rent gradient over the 5.5 mile-distance
between center to the edge of Linear-City. Note that no predefined land use pattern is

84
applied in baseline, so that the rent is determined only by competition in the market. If land
use is predefined for any reason, each zone would have separate land markets for different
uses as land use plan determines supply of land for the specific uses. See Section 5.4 for an
example analysis with predefined land use.
Figure 4.2 (b) depicts endogenous land use for residences, industrial activities, and
the local governments. In the baseline, about 90% of developable land is about equally
shared by residential and industrial purposes in each zone, and the remaining 10% is used by
the local government. Industrial land use shows a peak in the center, while the share for
residential buildings peaks at Zone 3 or 9.
Combining with land share in percentage with total, land use pattern in sq-yd is
shown by Figure 4.2(c). The actual lot size that is allocated for each land use type is
different from the land share in percentage as shown by Figure 4.2 (b). This is because
zones at the edge of region have more developable land then inside zones. As Figure 4.1 (b)
shows, Zone 6 has 13% less available land than Zone 1 or 11. For housing, 21% less land is
used in the center than is used than the fringe zones, while 14% less land is used in the
center for industrial use than the edges of the Linear-City.
Figure 4.2 (d) shows the developed floor spaces for housing and industrial use. An
underlying assumption for the figure is that the developers’ production technologies comply
with the floor space in 10-square yard as its output. More residential buildings can be found
outside the region then inside. In floor size, 48.9×10
6
sq-yd of housing space was developed
(and thus consumed) at the outskirt, while 44.6×10
6
sq-yd was developed at the center.
Developed floor space for industrial use, however, peaks at both of the center and the edge
with different quantities of 49.4×10
6
sq-yd, and 46.5×10
6
sq-yd respectively.

85

(a) Rent for developable land

(b) Land use (share in %)

(c) Land use (quantity in sq-yd)

Figure 4.2 Land use Pattern in Baseline of Linear-city

88%
90%
92%
94%
96%
98%
100%
11 Zone 10 9 8 7 6
Residential
Industrial
Government
0
5
10
15
20
25
30
35
40
45
50
Zone 1 2 3 4 5 6
Ratio to available land (%)
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
0
2
4
6
8
10
12
14
Zone 1 2 3 4 5 6
$ / 10 sq-yd / month
60%
65%
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Residential
Industrial
Governmental
0
20
40
60
80
100
120
140
Zone 1 2 3 4 5 6
100,000 sq-yd

86
(d) Floor space

(e) Rent for developed floor space

(f) Building height *

*) Story is calculated based on assumption that the building to land ratio is 1.0 for all types of
building uses.

Figure 4.2 Land use pattern in baseline of Linear-city (continued)
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Residential
Industrial
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
Zone 1 2 3 4 5 6
Story
86%
88%
90%
92%
94%
96%
98%
100%
11 Zone 10 9 8 7 6
Residential
Industrial
420
430
440
450
460
470
480
490
500
Zone 1 2 3 4 5 6
100,000 sq-yd
84%
86%
88%
90%
92%
94%
96%
98%
100%
11 Zone 10 9 8 7 6
Residential
Industrial
13.5
14.0
14.5
15.0
15.5
16.0
16.5
Zone 1 2 3 4 5 6
$ / 10 sq-yd / month

87
The market clearing rents for developed floor spaces in the region is illustrated in
Figure 4.2 (e). According to this figure, rents increase strictly over the distance toward the
center. Housing rent for a month ranges from $14.4 in the edge zones to $16.0 per bundle of
10 sq-yds at the center of Linear-City. Rent for industrial space is about 1.2~1.6% higher
than that of residential space throughout the region.
Building height is inferred as the ratio of the developed floor area, to the land shared
for the development of the spaces, based on an assumption that the building footprint is same
as the lot size. The skyline of the linear-city would look like Figure 4.2 (f). In the fringe, the
average height of residential buildings is 3.71 stories, and 1.5% higher than industrial
buildings of 3.66 stories. However, at the center, industrial buildings are 4.8% higher than
residential buildings.

4.3.2. Household Allocation Pattern

Households actualize their utility through consumption of goods and services,
including housing floor spaces, and maximize the utility by choosing appropriate places of
residence and work, under endogenous income constraint. In the model, household earning
consists of labor income and land rent, and its spatial variation is determined only by labor
income, because the income from land rent is redistributed to the households according to
the labor income per household by Equation (3.3).
Figure 4.3 (a) shows the distribution of household total income, i.e., the sum of labor
and land rent income, for the three types of households in the Linear-City, by the place of
residence. Based on the figure, among Type 1, the households who reside in the center of
the region earn about $5,122 per month, or 4 weeks, over 20 working days in average, which

88
is about 2.3 times higher income than household Type 3. Average monthly income for
household Type 3 is $2,237 in the same zone of the region. Average income difference
between the center and the outskirt is about 20% across all household types.
Workers in the Linear-City share significant portion of the endowment to participate
in labor and maintain the earning. According to the Household utility model in Section 3.2,
time endowment, 280 hours per month, is given to households as a discretionary resource. It
is also assumed that each household supplies one fulltime worker. So the number of fulltime
workers is identical to the number of households. In the baseline, 57% ~59% of endowment
is shared for labor, so that the average weekly working hours is 40.97 hours, 39.92 hours and
40.12 hours, respectively for Type 1, 2, and 3 households, at the hourly wages as shown in
Figure 4.4 (a).
Worker amenity is one of the household utility factors that contribute to residential
location choice. In the proposed model, the local governments produce amenity based on
the revenues from sales tax on industrial output, and the spending for input. And the
produced amenity is distributed to households evenly in each zone. Therefore, the quantity
of amenity that each household consumes is directly proportional to the industrial output,
and inversely proportion to number of households. These two quantities output (see
explanation Section 4.3.3) and households are not proportional to each other, so the ratio of
these quantities in the Linear-City example is not monotonic over distance from the center.
In the equilibrium state, households at the center of the region enjoy 2.7% more amenity
than the edges zones, and 3.4% more than the zone with minimum worker amenity (See
Zone 3 or 9 in Figure 4.3 (b)).

89

(a) Household income (labor + land rent income)

(b) Amenity

(c) Size of average house

Figure 4.3 Household Allocation Pattern in Baseline of Linear-city
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Household 1
Household 2
Household 3
0
1,000
2,000
3,000
4,000
5,000
6,000
Zone 1 2 3 4 5 6
$ / household / month
90%
91%
92%
93%
94%
95%
96%
97%
98%
99%
100%
11 Zone 10 9 8 7 6
0
10
20
30
40
50
60
Zone 1 2 3 4 5 6
Amenity / household
84%
86%
88%
90%
92%
94%
96%
98%
100%
11 Zone 10 9 8 7 6
1000
1050
1100
1150
1200
1250
Zone 1 2 3 4 5 6
Sq-Ft / household

90

(d) Indirect utility

(e) Allocated number of households

(f) Employment

Figure 4.3 Household allocation pattern in baseline of Linear-city (continued)

95.0%
95.5%
96.0%
96.5%
97.0%
97.5%
98.0%
98.5%
99.0%
99.5%
100.0%
11 Zone 10 9 8 7 6
Household 1
Household 2
Household 3
0
500
1,000
1,500
2,000
2,500
3,000
Zone 1 2 3 4 5 6
Households
99.60%
99.65%
99.70%
99.75%
99.80%
99.85%
99.90%
99.95%
100.00%
11 Zone 10 9 8 7 6
Household 1
Household 2
Household 3
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
Zone 1 2 3 4 5 6
Indirect Utility / household
84%
86%
88%
90%
92%
94%
96%
98%
100%
11 Zone 10 9 8 7 6
Household 1
Household 2
Household 3
0
500
1,000
1,500
2,000
2,500
3,000
Zone 1 2 3 4 5 6
Employment (households)

91
Households in the center of the Linear-City consume less housing floor spaces than
residences in the edge zones. The average housing size is calculated based on housing floor
space [Figure 4.2 (c)] and the estimated number of households [Figure 4.3 (e)]. According
to the Figure 4.3 (c), the average housing size per household at the center of the region is
1,077 sq-ft, which is 10% less than the average size in edge zones or 1,217 sq-ft.
Figure 4.3 (e) shows the household allocation pattern estimated by the proposed
CGE model. The spatial variation of households’ indirect utility is trivial, so the difference
between highest and lowest utility is less than 0.3% [Figure 4.3 (d)]. As negligible
variations of indirect utility is amplified by the dispersion factors
h
ψ in Equation (3.11),
resulting household allocation pattern varies up to 3.3% between the center and the outskirt
of Linear-City [Figure 4.3 (e)]. According to the figure, Type 3 households have a
considerable difference in allocation pattern when compared to the other household types.
While households of Type 1 or Type 2 prefer the outside zones for their residence, by 3.2%,
and 1.0%, respectively, more Type 3 households choose the center by 2.5%.
Using Equation (3.11), the proposed urban CGE model determines the places of
working, along with the places of residence. Since the model assumes one full-time
employee from each household, the total number of households by working place is identical
to the employment number. Figure 4.3 (f) shows the distribution of employment by place of
work that consists of peak at the center. The difference between allocation patterns that
Figure 4.3 (e) and (f) show respectively represents the job–housing mismatch. In the
proposed model, the mismatch is an inevitable phenomenon as long as the model is based on
idiosyncratic preferences. As Guilano and Small (1993) found, policies aimed for reducing

92
distributional difference between employment and residence might not be so effective,
unless polices are designed to intervene with households’ preferences.

4.3.3. Industrial Activity Allocation Pattern

As the production technologies postulate, firms require resources to produce goods
and services. In the proposed urban CGE model, the required resources include developed
floor space, commodities from other industries as well as its own output, and labor of
various skills. In the Linear-City example, the three types of households supply labors with
respective skills at the wages shown in Figure 4.4 (a). Over the distance, the highest wage at
the center differs by 2.8% to the lowest in Zones 3 and 9 for Labors 2 and 3. The average
wage for Labor 1, $26.31/hour is more than twice as much as the average of Labor 3 which
is $12.32/hour. So, wages are more classifiable by the labor skill, rather than job location in
this example.
Figure 4.4 (b) shows the spatial price by the place of production. Generally the
prices vary by 2.2%~3.8% over the region, and become expensive toward the center. At the
equilibrium state, Commodity 2 is most expensive in terms of unit price, as it is sold in
$19.34 at the center and $18.90 at the edge zones, while the prices of Commodity 1 and 3
range from $17.42 to $18.40 per unit.
In this study, industrial activity is measured by the quantity of produced goods and
services, or the value of the output. The estimated output in quantity shows considerably
deviated spatial distribution pattern. Industry 1 is relatively flat while Industry 2
concentrates at the outskirt, and Industry 3 peaks at the center. Figure 4.4 (c) shows these
patterns. The production of Industry 1, input coefficients of which are based on

93
manufacturing of Southern California, is reduced gradually toward the center of the Linear-
City. The lowest output produced in the center is 14.6 % less than the output from Zone 1 or
11, and about 400,000 units are produced from each zone. The lowest zonal productions of
Industry 2 and 3 are less than 33.2% of output from outskirt, and 37.4% of output from the
center. Note that the input coefficients for Industry 2 in this analysis were driven from the
trades of wholesale and retail sectors in Southern California, and coefficients for Industry 3
were based on service industries. Generally, the proposed model estimates concentrations of
service industry at the center, manufacturing toward outside, while commercial sectors
locate throughout the region.
Dollar values of products were also calculated based on the output quantity and the
price. Because the measurement of output quantity is arbitrary and not compatible,
comparison between industries by quantity is not relevant. According to Figure 4.4 (d), the
spatial distribution of output (in values) is not much different from that of quantities
excepting the higher proportions of outputs at the center due to higher prices. In total, output
$26.97 million from Zone 6 is 11.8% more than the output $23.80 million from Zone 1 or
Zone 11.
Figure 4.4 (e) shows the office floor space used in production of $1,000. In the
center, more office floor space is available (Figure 4.2 d), with higher rent than outskirt
(Figure 4.2 e), while 11.8% more output is produced. Considering this results, 6.3% less
space is used to produce $1,000 value in the center than the firms in the edge zones.

94
(a) Wage

(b) Output price

(c) Output in quantity

Figure 4.4 Firm Allocation Pattern in Baseline of Linear-city

94%
95%
96%
97%
98%
99%
100%
11 Zone 10 9 8 7 6
Commodity 1
Commodity 2
Commodity 3
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
Zone 1 2 3 4 5 6
$ / unit
95.5%
96.0%
96.5%
97.0%
97.5%
98.0%
98.5%
99.0%
99.5%
100.0%
11 Zone 10 9 8 7 6
Labor 1
Labor 2
Labor 3
0
5
10
15
20
25
30
Zone 1 2 3 4 5 6
$ / hour
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Industry 1
Industry 2
Industry 3
0
100
200
300
400
500
600
700
800
900
Zone 1 2 3 4 5 6
1,000 units / month

95
(d) Output in dollar value

(e) Average office space per $1,000 of production

Figure 4.4 Firm allocation pattern in baseline of Linear-city (continued)

Small data size of the Linear-City example allows to visualize the ideal (Isard, 1958)
interregional transaction table. Figure 4.5 shows the transaction. Intermediate transactions
between industries in all zones are calculated based on the endogenous intermediate input
coefficients,
ij
rq
ϕ from Equation (3.24), and the estimated value of outputs,
i
r
i
r
X p ⋅ .
The intermediate transaction table is divided into 9 sub-matrices, according to the
three industries in supply (row) and demand (column) sides. Each sub-matrix consists of 11-
by-11 transactions between the origin zones (row) and destination zones (column).

Industry j 1 2 3
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Industry 1
Industry 2
Industry 3
0
2
4
6
8
10
12
14
16
18
Zone 1 2 3 4 5 6
$Million / month
90%
91%
92%
93%
94%
95%
96%
97%
98%
99%
100%
11 Zone 10 9 8 7 6
160
164
168
172
176
180
Zone 1 2 3 4 5 6
10 sq-yd / $1,000

96
ZONE q1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11
i r1
2
3
4
5
1 6
7
8
9
10
11
1
2
3
4
5
2 6
7
8
9
10
11
1
2
3
4
5
3 6
7
8
9
10
11

↑ ↑ ↑ ↑
$700 $7,000 $70,000 $700,000

Figure 4.5 Intermediate Transactions in Baseline of the Linear-city

Transaction values range from $1,173 (from Industry 2 in Zone 1 to Zone 11), to
$646,367 (from Industry 3 in Zone 6, to Industry 2 in the same zone). To visualize the
values of this wide spectrum, i.e., order of 10
3
, the values in the transaction table were

97
modified for semi-log scale, and standardized to 32 gradual black-and-white colors. The
darker the cell is, the more the transaction is taking place.
From Figure 4.5, three significant point are observed. First, diagonal cells, or cells
adjacent diagonal in each sub-matrix are generally darker, i.e., more transactions are taking
place between these cells than between off-diagonals. The diagonal or near diagonal
elements in the nine sub-matrices represent the intra-zonal transaction, i.e., trades between
industries in a zone, or adjacent zones. Transport cost is low when the origin and destination
zones of intermediate movement are closely located, and the cost of intra-zonal delivery is
always lower than delivery to other zones. Firms prefer purchasing more from where
delivery cost is lower, if the prices are same. Consequently, Figure 4.5 shows relatively
darker diagonals. One exception is the intermediate trade between Industry 1, i.e., the sub-
matrix of upper-left corner of table. According to Appendix A, the distance decay of
intermediate input between Industry 1 is given positive,
11
µ =2, i.e., Industry 1 purchases
more intermediate inputs from Industry 1 from longer distance. Although it is not realistic,
the proposed model responded to the input coefficient as expected.
Second, the sub-matrices are symmetric with respect to the elements in the middle of
the sub-matrices, i.e., the cells that represent intra-zonal transactions in Zone 6. Due to the
symmetric nature of zone and network configurations, spatial distributions of all the
activities are symmetric to the center, including intermediate transactions. Therefore,
ij
q r
ij
rq
A A
− −
=
12 , 12
, I ∈ ∀ j i, , R ∈ ∀ q r, for the Linear-City example, where
ij
rq
A represents
transaction from industry i in zone r to industry j in zone q.
Third, transactions originating from Industry 3 have somewhat different pattern from
other industries. The most of transactions originating from Industry 1 and 2 show darker

98
elements near upper-left and lower-right corner of the corresponding sub-matrices, while the
intermediated goods of Industry 3 show horizontal strips. Darker elements near corners
correspond to the substantial prodution of Commodity 1 and 2 in the edges of the region [
Figure 4.4 (d)]. The bottom sub-matrices are noticeably dark, and it reflects that Commodity
3 is an essential input to all industries in the example. Once Commodity 3 is produced from
each zone, it is preferred from everywhere in the system, due to low distance decay
coefficients,
3 i
µ , I ∈ ∀i .

4.3.4. Travel Demand and Congestion in the Transportation System

Overall, 36,767.6 trips in passenger car unit (PCU) are generated in the Linear-City
example, over a 2-hour analysis time period, including 2,334 PCU of trucks for freight
movement which is 6.3% of total . Table 4.1 summarizes trip production and the network
performance. While shopping trips and freight movement are endogenous, the total number
of work trips is given by the exogenous number of households. According to Appendix A
(d), 44.3% of working trips are concentrated for the peak 2 hour time span. Since 40,000
households are given, and one fulltime worker is assigned to each household, 17,720 PCU
(48.2% of total) of working trips are generated in the peak 2 hours.
In the transportation system, passengers and trucks spend total of 4,004.8
PCU ·Hours to travels over 146,016.4 PCU ·Miles. On average, freight movement is
relatively shorter in both of travel distance and time, than passenger trips. In the system,
passengers drive 3.99 miles, whereas freight movements take 3.68 miles, 8% less than
passenger travel distance. And this difference is also reflected to the average travel time.

99

Table 4.1 Baseline Transportation Network Performance in the Linear-City

Passenger Freight Average Total
Travel demand (PCU) for 2 hours
Home based working 17,720.0
Home based shopping 16,713.8
Sub total 34,433.8 2333.8 36,767.6
Travel time (PCU · Hour) 3,769.2 235.6 4,004.8
Travel distance (PCU · Mile) 137,421.1 8,595.2 146,016.4
Average travel time (Min / Trip) 6.57 6.06 6.54
Average trip distance (Mile / Trip) 3.99 3.68 3.97
Average speed (mph) 36.46 36.48 36.46

Spatial distribution of travel demand, in terms of the origin and destination to each
zone, is closely correlated to activity allocation patterns. Figure 4.6 (a) through (c) show
zonal trip production and attraction by trip purposes. As the spatial variation of household
allocation pattern is insignificant [Figure 4.3 (e)], the production of home-based trips, i.e.,
working and shopping trips, distributes with relatively small variations over the region.
Furthermore, working trip production pattern is very similar to the household allocation
pattern, due to 1-to-1 relation between the number of households and the labor supply.
The destinations of home-based trips are shaped by household consumption and
industrial production activity patterns. According to Appendix A (e), households shop more
frequently for Commodity 1 and 2, i.e., households makes one trip to purchase 4 and 8 units
of Commodity 1 and 2, respectively, while one trip is made for 20 units of Commodity 3. In
addition to the shopping trip frequencies, households consume significant quantities of

100
Commodity 1 and 2. Due to these two factors, more shopping trips destine to outskirt of the
Linear-City then the centers, following the industrial activity allocation patterns, especially
of Industry 1 and 2.
Industrial activity allocation pattern is also closely related to freight trip generation.
In general, more trucks are generated from outskirt of region, and delivered to whole area
evenly with a small variation, as Figure 4.6 (c) shows. In the Linear-City example, all trucks
are unique, and one truck is equivalent to 2 passenger cars in road congestion. During the
peak 2 hours, 231.3 trucks in PCU are generated from each edge zones, i.e., Zone 1 and 11,
where more of freight-prone industries – Industry 1 and 2 – are located, and 190.4 PCU are
generated from the center. In destination side, 210~215 trucks in PCU are destined in each
zone, with a peak at the center.
The center of region is advantageous in terms of access. Figure 4.6 (d) and (e)
show that the average travel time and cost to access the center is about 40% less than that
traveling to the edges in the equilibrium state. Drivers of both of passenger cars and trucks,
on average, spend around 8.1 minutes of time and $2 to access the edges, while they spend
only 4.8 minutes with $1.2 per trip to access the center.
In contrast to the advantageous access travel time and cost to the center, the driving
speed to traverse each zone shows somehow distinct gradient. Although the difference
between the highest speed and lowest is less than 5%, Figure 4.6 (f) shows the unique U-
shape curves for travel speed. The figure implies that the links traversing Zone 3 and 9 are
most congested, and drivers spend more time to pass these zones in the Linear-City example.

101

(a) Home based working trip

(b) Home based shopping trip

(c) Intermediate input delivery (trucks)

Figure 4.6 Transportation System in Baseline of Linear-city
82%
84%
86%
88%
90%
92%
94%
96%
98%
100%
11 Zone 10 9 8 7 6
Production
Attraction
1,300
1,350
1,400
1,450
1,500
1,550
1,600
1,650
Zone 1 2 3 4 5 6
PCU / peak 2 hours
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Production
Attraction
0
50
100
150
200
250
Zone 1 2 3 4 5 6
PCU / peak 2 hours
88%
90%
92%
94%
96%
98%
100%
11 Zone 10 9 8 7 6
Production
Attraction
1,480
1,520
1,560
1,600
1,640
1,680
1,720
Zone 1 2 3 4 5 6
PCU / peak 2-hour

102

(d) Average access travel time

(e) Average access travel cost

(f) Travel speed to traverse each zone (mph)

Figure 4.6 Transportation System in baseline of Linear-city (continued)
92%
93%
94%
95%
96%
97%
98%
99%
100%
11 Zone 10 9 8 7 6
35.5
36.0
36.5
37.0
37.5
38.0
38.5
Zone 1 2 3 4 5 6
Miles per hour
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
Zone 1 2 3 4 5 6
Minutes / trip
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
0.0
0.5
1.0
1.5
2.0
2.5
Zone 1 2 3 4 5 6
$ / trip

103
This section reviews the various aspects of the baseline of Linear-City example.
The results are categorized into 1) the land use pattern; 2) the household allocation pattern;
3) production activity allocation pattern; and 4) the performance of the transportation
system. However, this is not the only way to categorize the results. For example, job
allocation pattern, classified into the household allocation pattern, shown by Figure 4.3 (f), is
also related to the production activity. All the pieces are interrelated within a grand
equilibrium process, similar to the actual urban systems.

4.4. SENSITIVITY ANALYSES FOR SELECTIVE PARAMETERS

This section reviews the responses of the proposed CGE model to marginal
fluctuation imposed to selected input parameters. In numerical analysis section, Kim (1990)
analyzed the effects of congestion toll to the formation of urban structure, economy of scale
in consumption, and shopping travel impedance. Those shopping preferences are also
included in the proposed CGE model via 1) shopping distance decay coefficients
ih
µ ; and 2)
coefficients for economy of scale in shopping
ih
η , respectively. Effects of these coefficients
are identical to what Kim had found. In addition to Kim’s numerical analyses, this section
reviews following factors: 1) the effect of elasticity of substitution in regional production; 2)
the effect of distance decay in intermediate goods movement to industrial activity allocation;
3) the effect of growing population and household allocation pattern; and 4) the effect of
network configuration to travel time.

104
4.4.1. Effect of Elasticity of Substitution in Regional Production

Beside the formal definition, the elasticity of substitution in a production function
defines a firm’s agility in searching cheaper inputs that can substitute expensive inputs.
When the production technology allows, firms, which are governed by constant elasticity of
substitution in the proposed model, will use greater amount of cheaper inputs in production
as long as no additional cost is required in substitution. As input prices are lower by
substitution, the outputs also become cheaper than the products from firms that use inflexible
production technology in substitution. In general equilibrium, the lowered output price of a
firm cascades throughout the industrial linkage, and eventually induces increased production
quantity. The proposed CGE model captures this tendency from the Linear-City example.
Table 4.2 summarizes the results from applications of the proposed CGE model with
various elasticity of substitution of Industry 3 of the Linear-City example. The elasticity of
substitution for Industry 3,
3
θ varies from 0.4 to 2.8 with increments of 0.8. Average price
of Commodity 3, $24.13 per unit with
3
θ =0.4, goes 35.9% down to $16.83 per unit with
3
θ =2.8. It is 5.2% cheaper then the baseline case (
3
θ =2.0). The prices of other products
are also declined with similar changing ratio to Commodity 3. By varying the elasticity of
substitution for an industry, prices of all the commodities are affected.
The value of production shows a composite effect from decreasing prices, and
increasing quantity produced. As Table 4.2 also shows, total produced quantity is increasing
by 1.4% as the elasticity of substitution
3
θ changes to 2.8 fro 2.0. In production value,
however, relative production is less than the baseline by -3.9% for the same elasticity of
substitution.

105
Table 4.2 Regional Production of the Linear-City
by Elasticity of Substitution of Industry 3
(Difference to Baseline,
3
θ =2.0 in percent)

Elasticity of substitution,
3
θ

0.4 1.2 2.0 2.8
Industry 1 23.44 (29.9) 19.55 ( 8.3) 18.05 17.17 (-4.9)
Industry 2 26.22 (37.6) 20.96 (10.0) 19.06 17.98 (-5.7)
Industry 3 24.13 (35.9) 19.43 ( 9.4) 17.76 16.83 (-5.2)
Average
Price
$/unit
Average 24.34 (34.3) 19.79 ( 9.2) 18.13 17.19 (-5.2)
Industry 1 4,144.6 (-12.9) 4,582.0 (-3.7) 4,760.4 4,850.1 ( 1.9)
Industry 2 2,645.4 (-17.5) 3,044.0 (-5.1) 3,208.3 3,208.3 ( 0.0)
Industry 3 6,034.5 (-12.8) 6,726.0 (-2.8) 6,920.3 6,951.3 ( 0.4)
Output in
1,000 units
Total 12,824.5 (-13.9) 14,352.0 (-3.6) 14,889.0 15,091.7 (1.4)
Industry 1 97.2 (13.2) 89.6 (4.3) 85.9 83.3 (-3.0)
Industry 2 69.4 (13.4) 63.8 (4.2) 61.2 59.2 (-3.3)
Industry 3 145.6 (18.5) 130.7 (6.3) 122.9 117.0 (-4.8)
Output in
$M
Total 312.2 (15.6) 284.1 (5.2) 270.0 259.5 (-3.9)

In addition to the aggregated output and price, the elasticity of substitution also
affects industrial allocation pattern over the space. In the proposed CGE model, firms
purchase from near and far. Therefore substitution is happened not only between different
input commodities, but it is also happening between different locations. With a positive
elasticity of substitution, a firm is looking for cheap inputs from vicinity, up to the distance
from which the delivery cost of the input exceeds the spatial price difference.
To firms, however, not only the movable commodities, but limited office space and
labor are also essential inputs. With high elasticity of substitution, thus with low input
prices, firms might produce more outputs. In this case, two opposite impacts maybe

106
imposed on rent and wage. Due to high elasticity of substitution, cheaper inputs can replace
the use of labor and office, so the general trend might show reducing wage and rent for
higher elasticity of substitution. On the contrary, as production increases with higher
elasticity of substitution, more labor and office space might be required. Note that labor is
from the exogenous number of households, and office space is based on the given
developable lane, so the supply would be relatively inflexible in the example.
In the Linear-City example, the later effect is prominent, i.e., the office rent, and
wage gradually increase relative to commodity price, along with increasing elasticity of
substitution. According to Figure 4.7, office rent is about 72% ~77% of the average
commodity price when
3
θ = 0.4. With
3
θ = 2.8, however, relative office rent to average
commodity price is raised to around 78% ~85%. Similar pattern is shown to wage.
As rent and wage are becoming relatively expensive to the commodity price with
higher elasticity of substitution for Industry 3, more of Industry 3 choose the outskirts of the
region for their business location in the Linear-City example, where rent and wage are still
relatively cheap. Figure 4.7 (c) shows the output from Industry 3 in each zone, with varying
elasticity of substitution,
3
θ . According to the figure, with
3
θ =0.4, firms of Industry 3 in
the edge zones produce only 46% of the output that the firms in the center produce, and this
ratio increase to 63% with
3
θ =2.8. In spited of this shift, the center seems still
advantageous overall so that the gradient peaks at the center for the production of
Commodity 3.
As production pattern of Industry 3 changes with varying elasticity of substitution
3
θ , the gradients of other industrial productions are also affected. Like the production of
Commodity 3 is becoming flatter over the Linear-City, production of Commodity 2 and 3

107
are becoming less steep too. In the baseline, as Figure 4.4 (c) and (d) show, Industry 1 and 2
have different allocation patterns from Industry 3 – more output is produced from outskirt
then the center. In spite of this different pattern, the effect of higher elasticity of substitution
of an industry is resulted in the same way, i.e., less spatial variation of production. The
difference of the zonal outputs produced from center and edge of the region is 21% when
3
θ =0.4, for Industry 1. With
3
θ =2.8, this difference is reduced to 10%. Similar changes
were found from Industry 2, so the difference between outputs from center and outskirt is
reduced from 54% to 27% when
3
θ increase from 0.4 to 2.8. It is not clear, though, if an
extremely higher elasticity of substitution can create a reversed gradient – peak at outskirt of
Industry 3, and peak at center of Industry 1 and 2.
(a)
3
θ = 0.4 (b)
3
θ = 1.2

(c)
3
θ = 2.0 (d)
3
θ = 2.8

Figure 4.7. Relative Office Rent and Wage to Commodity Price
by Varying Elasticity of Substitution,
3
θ
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Zone 1 234 56
Ratio to commodity price
Wage
Rent
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Zone 1 2 34 56
Ratio to commodity price
Wage
Rent
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Zone 1 2 3 4 5 6
Ratio to commodity price
Wage
Rent
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Zone 1 2 3 4 5 6
Ratio to commodity price
Wage
Rent

108

(a) Industry 1

(b) Industry 2

(c) Industry 3

Figure 4.8 Industrial Allocation Pattern by Varying Elasticity of Substitution,
3
θ

75%
80%
85%
90%
95%
100%
Zone 11 10 9 8 7 6
Elasticity = 0.4
Elasticity = 1.2
Elasticity = 2.0
Elasticity = 2.8
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Zone 1 2 3 4 5 6
$Million / month
40%
50%
60%
70%
80%
90%
100%
Zone 11 10 9 8 7 6
Elasticity = 0.4
Elasticity = 1.2
Elasticity = 2.0
Elasticity = 2.8
3.0
8.0
13.0
18.0
23.0
28.0
Zone 1 2 3 4 5 6
$Million / month
40%
50%
60%
70%
80%
90%
100%
Zone 11 10 9 8 7 6
Elasticity = 0.4
Elasticity = 1.2
Elasticity = 2.0
Elasticity = 2.8
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
Zone 1 2 3 4 5 6
$Million / month

109
By rationalization, firms become free from restrictions that are imposed by space
and locations. Progressing from the era of massive production, and massive consumption,
capitals under Post-Fordism are much flexible then before (Lee, 1992). To survive in the
market that is filled with rapidly shifting demand in terms of preferences, firms should
optimize their corporate structures by eliminating inefficiencies from production process,
and rigidity from administration. Light-weight production facility is operated by on-call
workers to produce various customized commodity in small amount, and deliver to customs
as it is produced just-in-time. Advanced management and delivery system reduces the need
for stock management and storage. On the other hand, as externality, such as congestion,
piles up, CBD losses its spatial advantage and the firms’ tenacity to the center fades away.
With high mobility that comes from expanded freeways and advance transportation modes,
production facilities spread out toward outskirt. A simple test with varying elasticity of
substitution describes the effect of flexibility in production technology in the Linear-City
example on spatial distribution of firms.

4.4.2. Effect of Distance Decay in Intermediate Goods Movement

Firms interact with each other by exchanging intermediate goods. When
commodities in different locations supplied at identical price, the majority of firms purchase
more intermediate inputs from near suppliers, because it can be delivered with lower
transport cost. Through active interaction with near business, firms eventually establish
geographic clusters over short distance for optimal trading conditions. On the other hand,
there is a possibility that firms may push away each. Competing business for a market share

110
in proximity may not trade, and in this case, more intermediate inputs are purchased from
with businesses in distances.
In the model, the distance decay coefficients for intermediate input, µ, describe this
pulling or pushing forces between firms. The initially purpose of using the coefficients in
the proposed model was to distribute demand for intermediate input over the space. Once
total demand for an input is determined by input and output prices, the input share
coefficient φ is distributed spatially by the distance decay coefficient in Equation (3.24). In
spite of its name, decay coefficient, it is not restricted in the model, so - ∞ < µ < ∞. If the
coefficient is zero, demand is allocated into all zones regardless of the price or transport cost.
However, with negative coefficients, more quantity of input is purchased from suppliers
whose effective price (price and transport cost) is cheaper.
This section provides a summary of the model responses to the various distance
decay coefficients. In the baseline of Linear-City application, the distance decay coefficients
are ranged from -2 to 2 as Appendix A (b) shows. In the model, the total number of distance
decay coefficients are I × (I+3), where I is the number of industries, or number of different
commodities. The decay coefficients for the local government is not included in this
analysis, because it is modeled to always purchase the input for the minimum effective
prices. Following list summarizes the five analysis scenarios;

• µ = 0 : No distance decay is assumed in any intermediate transaction
• µ =-10: For all the intermediate transactions, the coefficient is assumed –10.
• µ = 10: For all the intermediate transactions, the coefficient is assumed +10.

111
•
ii
µ =-10: For the intermediate transactions within an industry (i = j), operating
different locations, the coefficient is assumed –10. For other transaction,
no distance decay is assumed.
•
ii
µ = 10: For the intermediate transactions between one industry (i = j), operating
different locations, the coefficient is assumed +10. For other transaction,
no distance decay is assumed.

As Table 4.3 summarizes, the aggregated production from the Linear-City is
practically invariable to the various distance decay coefficients, excepting one case. For all
of three indices – total output in quantities, total output in dollars, and average output prices
– the proposed model estimates less than 4% of differences from the baseline case.

Table 4.3 Regional Production in the Linear-City by the Distance Decay Coefficients
for Intermediate Inputs.

Distance Decay

µ = 0 µ = -10 µ = 10
ii
µ = -10
ii
µ = 10 Baseline
Industry 1 18.03 17.67 14.68 17.80 18.00 18.05
Industry 2 19.03 18.67 16.91 18.80 19.01 19.06
Industry 3 17.72 17.34 17.56 17.47 18.85 17.76
Average
Price
$/unit
Average 18.10 17.73 16.37 17.86 18.61 18.15
Industry 1 4,750.2 4,678.7 3,456.1 4,691.3 4,848.4 4,760.4
Industry 2 3,203.2 3,157.2 2,183.8 3,165.0 3,246.9 3,208.3
Industry 3 6,917.0 6,820.6 3,937.7 6,841.2 6,536.4 6,920.3
Output
in
1,000
units
Total 14,870.4 14,656.5 9,577.7 14,697.6 14,631.8 14,889.0
Industry 1 85.6 82.7 50.7 83.5 87.3 85.9
Industry 2 61.0 58.9 36.9 59.5 61.7 61.1
Industry 3 122.5 118.3 69.1 119.5 123.2 122.9
Output
in $M
Total 269.1 259.9 156.7 262.6 272.2 269.9

112
When all the given distance decay coefficients are positive, µ=10, the average price,
$16.37, is 10% lower than the baseline case. Along with the low prices, the proposed CGE
model normally estimates more output quantities, so the value of produced goods stays
relatively constant in general equilibrium states. However, in this case with unusual distance
decay coefficients, the model estimates significantly less output than the baseline. Along
with the lower equilibrium prices, the estimated difference is 42% in quantities, and 36% in
dollar to the baseline, respectively. In short, the set of distance decay coefficients creates an
inactive economic system than the baseline case.
A comparison between the model’s equilibration processes for the baseline, and
µ=10 case depicts what has happened more clearly. Figure 4.9 shows the average price of
all commodity, and the total output in dollars at the end of each iteration. As Figure 4.9 (a)
shows, µ=10 case starts same average price to the baseline due to the identical initial values.
After 3rd iteration, the average price is higher than baseline, keeps increasing up to 22nd
iteration. After then it starts decreasing, and eventually the average price in µ=10 case is
lower than the baseline after 29th iteration throughout the equilibrium process. On the other
hand, the total output value estimated in µ=10 case, shown by Figure 4.9 (b), is almost
identical to the baseline case. After then, the output rapidly drops.
Given positive distance decay coefficients, firms unnaturally prefer purchasing
expensive inputs, or purchasing from farther locations where more delivery cost is required.
As a result, all the industry in this scenario is experiencing high input and output prices at
the early stage of partial equilibrium. However, the high output prices from all industry
does not result in increased production, because abundant output does not mean lower input

113
price, but it means more delivery cost. As the effective price is unreasonably higher than the
production technology supports, the production activity is limited, and eventually
commodity price drops.

(a) Average Commodity Price (b) Total Output Value

Figure 4.9 Price and Output Value of µ=10 Case and Baseline by Iteration

The distance decay coefficients determines the spatial distribution of intermediate
input transactions. As reviewed above, the distance decay coefficients represent pulling, or
pushing interactions between firms. With negative distance decay, firms are pulling each
other, thus more interaction is taking place over shorter distance. In a inter-regional
transaction table for the Linear-City example, zones in near distances are aligned along the
diagonal elements. Consequently, with negative distance decay coefficients, the cells near
diagonal elements of the transaction table will be populated with large values.
Figure 4.10 shows the endogenous transactions between industries located over the
Linear-City, for various distance decay coefficients settings. As Figure 4.5 was developed,
inter-regional transaction tables in the Linear-City for various distance decay coefficient
settings are presented with 3×3 sub-matrices for origin and destination industries. Each sub-
150
180
210
240
270
0 10203040
Ite ration
$M
µ = 10
Baseline
10
12
14
16
18
20
0 10 203040
Ite ration
$ / unit
µ = 10
Baseline

114
matrix consists of 11×11 geographic zones. Semi-log of the ratio of transaction to the
maximum value originated from an industry determines the grey scale. So, each row of 3×3
sub-matrix has its own grey scale. It is to enhance the display of transactions from the wide
variation of quantities transacted.
Figure 4.10 (a) shows transactions take place when no distance decay effect is
applied. According to Equation 3.24, the input share coefficient,
ij
ϕ is evenly distributed
over the zones with µ =0. Consequently, each of 3×3 sub-matrices shows uniform
distribution of transactions. As expected, diagonals and near diagonal elements are darker,
with negative distance decay, as depicted by Figure 4.10 (b). However, elasticity of
substitution, push Industry 3 to outskirt of the region, so darker cells are originated from near
Zone 1 and Zone 11. Figure 4.10 (d) shows a combination between Figure 4.10 (a) and (b).
In this case, only the intra-industrial transactions are governed by negative distance decay, so
the off-diagonal sub-matrices show uniform transactions, i.e., uniform grey scale.
The peculiar case with µ=10 produces also atypical intermediate transaction
patterns, as Figure 4.10 (c) shows. All of firms are demanding for only expensive
commodities in terms of effective prices in this case, and the outskirt zones are only eligible
places from which the intermediate inputs are purchased with extreme delivery cost. The
transaction table for this case captures this pattern with a few darker horizontal lines. Each
of the lines represents intermediate goods movement originated from either of Zone 1 or
Zone 11. Industries also produce outputs from internal zones, i.e., Zone 2 to Zone 10, but
only a small fraction of output is used as intermediate inputs.

115
(a) µ = 0 (b) µ = -10

(c) µ = 10 (d)
ii
µ = -10

(e)
ii
µ = 10 (f) Baseline

↑ ↑ ↑ ↑
0% 1% 10% 100%=MAX
Figure 4.10 Intermediate Transactions by Distance Decay Coefficients

116
In the case with
ii
µ = 10, most of intra-industrial transactions are originated from
the center. For non-intra-industrial trades, no distance decay is applied in this case. Thus,
for an industry, one intermediate input from same industry is purchased from where the
effective price is expensive, while two inputs are delivered from anywhere without
considering the effective prices.
Thus, at least a certain portion of production activity is allocated in each zone.
Given the inter-industrial intermediate transactions in everywhere, the equilibrium price
behave just like as in the baseline case that Figure 4.4 (b) shows. Therefore the intra-
industrial trades originated from each industry are concentrated in the center, where the price
is highest in the Linear-City example.
Industries may overcome externality by clustering (Head, et.al., 1995, Henderson,
et.al., 1995). In the proposed model, various distance decay coefficients result
distinguishable clustering patterns. Figure 4.11 shows industrial mixture in each zone of the
Linear-City for the various distance decay settings.
With negative decay coefficients, each industry has own location to concentrate as
shown by Figure 4.11 (b) and (d). However, with positive distance decay for all
intermediate transactions, i.e., in µ=10 case, all industrial sectors are concentrated in the
edge, as Figure 4.11 (a). When only intra-industrial transactions are affected by positive
decay coefficients, some industries concentrate at the center of the region. Of course,
without distance decay, i.e. µ=0, there is not concentration of production, and not shown this
case. Even though the model produced distinctive clustering patterns in this simple test,
model’s applicability to studies on agglomeration patterns can be judged based on empirical
evidence for the existence of positive distance decay coefficients.

117
(a) µ = 10 (b) µ = -10

(c)
ii
µ= 10 (d)
ii
µ = -10

Figure 4.11 Zonal Production Compositions by Distance Decays

4.4.3. Effect of Growing Population in Urban Structure

For the proposed CGE model, population is indirectly given by the number of
households. In terms of number of households, the model does not impose any explicit
limitation. However, ever-increasing population and households will bring a significant
unbalance between the capacity of infrastructure and the demand for it. In worst case, the
urban system may not sustain with extreme externalities caused by population explosion.
In this section, the proposed model is test with various household sizes. For the
baseline analysis 40,000 (40K) households are given, and the proportion of Type 1, 2, and 3
0%
20%
40%
60%
80%
100%
Zone 1 2 3 4 5 6
Industry 1
Industry 2
Industry 3
0%
20%
40%
60%
80%
100%
Zone 1 23456
Industry 1
Industry 2
Industry 3
0%
20%
40%
60%
80%
100%
Zone 1 2 3 4 5 6
Industry 1
Industry 2
Industry 3
0%
20%
40%
60%
80%
100%
Zone 1 23456
Industry 1
Industry 2
Industry 3

118
Households is 1: 2.5: 6.5 respectively. Maintaining this proportion unchanged, the total
number of households is increased to 80K, 160K, 200K, and 240K. The settings remain also
same to the baseline, including network configuration and available land.
Up to a certain state, the increasing number of households boosts the regional
production. According to Table 4.4, as population doubles up from the baseline, the wage
declines to $9.30/hour from $15.77/hour on average. With the exogenous number of
households 200K, it goes down to $6.26/hour, or only 40% of the baseline case. More
household is regarded as addition supply of labor to firms. Increasing labor supply causes
wage cheaper, and in turn, general input price goes down. Due to low input price, the output
quantity increases more than 8 time of the baseline output in 200K case with 122.8 million
units. In spite of this significant increase in output quantity, the production value increases
only 42% from the baseline for low commodity price.
The model estimates somewhat counterintuitive rents. As Table 4.4 also shows, the
rent for developable land is increasing only slight fractions. Furthermore, the rent for
developed floor space declines as population increases, with rather significant rate. A
possible reality is that the excessive demand for housing will cause rent increase, and then
the developers will produce more floor space to clear the excessive demand. So high
housing rent will bring up the rent for developable land, and then the rent for office floor
space. On the contrary, the absolute rent for developable land at Zone 1 in the Linear-City
example is fixed for the Walras’ rule, regardless of the demand for the land. Consequently
land rent is independent from population size. With fixed land rent, abundant labor, and
cheaper inputs, the developers are able to produce more floor spaces in lower rent.

119
Table 4.4 Regional Production of the Linear-City by Number of Households.

Number of Households (1,000)

40 80 160 200 240
Average Wage ($/hour) 15.77 9.30 6.51 6.26 11.65
Average Price ($/unit) 18.13 12.94 10.39 10.24 8.46
Output (Million units) 14.9 93.6 111.3 122.8 210.3
Output ($M) 269.9 292.2 347.4 383.8 696.8
Land 11.18 11.21 11.65 11.97 9.93
Housing 15.11 11.98 10.54 10.56 13.16
Avg. Rent
($/10yd
2
)
Office 15.37 12.41 10.91 10.91 10.06
Housing 3.9 5.1 6.8 7.5 12.3
Avg. Height
(Story)
Office 4.0 5.5 7.8 8.8 16.8
Household 1 4,726 2,727 1,816 1,710 4,199
Household 2 3,561 2,059 1,368 1,286 3,540
Household 3 2,062 1,247 868 829 1,020
Income
($/month)
Average 2,703 1,598 1,088 1,032 1,968
Household 1 44.90 44.91 43.15 42.26 39.82
Household 2 43.39 43.37 41.50 40.43 39.16
Weekly
Labor Hours
Household 3 41.83 42.10 41.56 41.47 50.31
Time (min) 6.54 9.38 29.60 43.40 38.9
Distance
(mile)
3.97 3.84 3.59 3.42 2.60
Average
Travel
Speed
(mile/hr)
36.46 24.60 7.29 4.72 4.01

To acquire additional space, developers may have two choices. When the
technology and regulations permit, they will build high-rise buildings at given developable
land. Or urbanized area will be expanded until the land rent at the boundary excesses the
rent for the surrounding undeveloped land. However, for the proposed CGE model,
geographical extent is one of the exogenous inputs, and the boundary of Linear-City is fixed.

120
To accommodate the increasing population, under given geographic extent, developers build
taller structures. According to the table, the average height of office buildings is taller to 8
stories in 200K, from 4 stories of the baseline.
Declining wage means less income to households. Overall, the monthly income is
reduced to 60% of baseline when the number of households is doubled to 80K. And, with
200K households, the total income decreases to %40 of the baseline or $1,032. Lower wage
contributes to this income reduction. In reality, loosing opportunity to participate labor
might be another reason that causes income reduction. However, the proposed model
assumes one fulltime worker per household will participate, there is no unemployment in the
system. That is why the proposed model estimates insignificant changes in labor hours.
According to Table 4.4, each household participate in labor for 41 to 45 hours a week on
average, for all cases. Basically the proposed model does not allow fundamental shift of the
time endowment to leisure from labor, unless new household preference is assumed.
Travel time in the Linear-City shows a radical increment with increasing population.
Competing over given network capacity by increasing population naturally provokes
congestion. On average, residents in the Linear-City used to spend 6.5 minutes for a trip
when 40,000 households reside. With 80K households, it increase to 1.4 times of baseline,
and eventually it ends up to 6.6 time of baseline in 200K case. As travel time increase, the
average travel distance is gradually decreases. Shorter travel distance is achieved by moving
more residence and business toward the center of region.
Zone 3 and 9 are experiencing more severe congestion from increasing population.
More or less, 1 minute is enough to traverse each zone in the baseline. In the case with
160K households, however, drivers need 2.9 to 4.5 times more travel time than the baseline

121
to across the zones. Furthermore, 5.6 times more travel time is required to traverse Zone 3
or 9, as Figure 4.12 shows. In the 200K case, the ratio of travel time to the baseline
increases more than 10 times. It means that drivers should pay higher cost to travel
anywhere in the Linear-City example, while their income declines as population increases.
Especially the drivers who travel between the center and the outskirt of region need pay
extremely high cost to across Zone 3 or 6, and these zones are barriers to them.

(Ratio to the baseline)
Figure 4.12 Traversing Travel Time by Number of Households

The barriers actually split the Linear-City into three sub-regions as the number of
households increases to 240K, and the proposed model behaves somewhat differently for
this case. As above Table 4.4 shows, results are consistent over the growing population up
to the 200K case. However, in the case with 240K households, the average income, $1,968
is not consistent to the trend over varying households, and even more, it is comparable to the
baseline, in spite of the huge difference of the number of households. This discontinuity is
0
1
2
3
4
5
6
7
8
9
Zone 1 2 3 4 5 6
Traversing Time (minutes)
40K 80K 160K 200K
(5.9)
(8.8)
(10.3)
(9.3)
(7.1)
(6.0)
(4.0)
(5.7)
(6.6)
(5.9)
(4.6)
(3.9)

122
due to incomparable household allocation patterns. Figure 4.13 compares the ratio of
households in each zone to the maximum for the 200K case and the 240K case respectively.
According to Figure 4.13 (a), all of the three types of households in the Linear-City are
distributed over the whole region in 200K case. In the 240K case, however, Household 3
reside only to the zones near the center, i.e., Zone 4 through 8. When Household 3 choose
Zone 1,2, 10, or 11, their indirect utility is not defined because the income less travel cost,
pq
h
q pq
h
pq
h
q
t w T c T D w E ⋅ ⋅ − ⋅ − + ⋅ , from Equation 3.10 is negative. It means that if there is
any Household 3 reside in the outskirt, their total income is not even enough to compensate
the travel costs for shopping and working trips. To avoid this condition, Household 3 is
inevitably concentrated only in the center of the region, and push away Household 1 and 2,
as well as most of the production activities to outskirt of the region.

(a) 200K (b) 240K

Figure 4.13 Household Allocation Patterns in 200K and 240K Cases

As the Linear-City is divided into three sub-region by household allocation pattern,
the proposed CGE model estimates unique economic features in this case. Figure 4.14
20%
40%
60%
80%
100%
Zone 1 2 3 4 5 6
Household 1 Household 2 Household 3
20%
40%
60%
80%
100%
Zone 1 2 3 4 5 6
Household 1 Household 2 Household 3

123
depicts the distinctive output and income in each zone. Overall, the sub-region in the center,
shaded with grey color, is occupied Household 3, and not much of land is used for industrial
activity. Thus the center can be characterized as high population density, low income, and
low industrial activity. On the other hand, industries in the sub-regions near two edges
manage production without labors from Household 3, on limited developable land. Labors
from Household 1 and 2 substitute the Labor 3 at high wages. Thus the outskirt sub-regions
are characterized with less population density, high income, and high industrial production.

(a) Output in dollar value

Edge Center Edge
(b) Income

Edge Center Edge
Figure 4.14 Output and Income Distribution in 240K Case

0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
11 Zone 10 9 8 7 6
Industry 1
Industry 2
Industry 3
0
10
20
30
40
50
60
70
Zone 1 2 34 56
$M / month
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Household 1
Household 2
Household 3
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
4,500
5,000
Zone 1 2 3 4 5 6
$ / month

124
The proposed CGE model was able to show splitting of an urbanized area with
increasing population. When transportation capacity is not expanding properly to
accommodate increasing travel demand, travel time to work and shop might exceed the
disposable income for some of low income households. For them, the given study area is
divided into sub-regions in the model. On the other hand, relying on the fixed study
boundary is a limitation of the model that should released to study the effects from
geographical changes. The average the travel time increases in the Linear-City example,
while the driving distance becomes shorter with increasing population. In reality, however,
urban boundary should expand for population influx, and the travel time increment might be
due to congestion, as well as longer travel distance.

4.4.4. Effect of Network Configuration in Travel Time to CBD

The Linear-City example is controversial to the reality regarding the estimated travel
time to the center. Many researchers have consistently, that the travel time to the downtown,
especially travel time for journey to work is longer than that of trips to other sub-centers.
Some examples are found from Giuliano and Small (1991), Pisarski (1996), and Gordon et al
(2004). In the Linear-City example, the geographic center is also the activity center where
the most of economic activity is taking place. Different from the findings, the access travel
time to center is lower than the travel time to other zones, as Figure 4.6 (d) shows.
This section reviews the effect of various network configurations, including changes
in access travel time to each zone. Travel time is determined as an equilibrium price
between travel demand and given network configuration. And the proposed model estimated
the travel demand through the endogenous activity allocation. If the travel time estimation

125
does not agree to common sense, then faulty exogenous data would be the possible reason
for it, which is the network configuration in this case.
Since all congestible links in the example have identical capacity per lane (1,000
PCE per lane), the number of lanes is an index that is equivalent to the link capacity. Figure
4.1 (d) shows that links at the center, i.e., Zone 6 have six lanes, while only one lane was
assumed for the links at the boundary of the region in the baseline. Besides the baseline,
settings with 3 different network configurations are analyzed in this section. Table 4.5
summarizes the link capacity for each configuration. Each zone consists of 4 half-mile
directional links, and the table shows two numbers of lanes in each zone assuming the
opposite links have identical capacity. In the 6-6 Case, all links are designed with 6 lanes,
while in the 2-2 Case consists with 2-lane links. The 6-1 Case represents an opposite
configuration to the baseline, i.e., 6-lane links at outskirt, and 1-links at the center. As link
capacity is changed, available land in each zone is also adjusted according to the scheme
explained in Section 4.2. In 6-1 Case, more land is available in the center than Zone 1 or 11.

Table 4.5. Network Configuration Scenarios for the Linear-City
(the number of lanes for direction links)

Scenario Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6
Baseline 1 2 2 3 3 4 4 5 5 6 6 6
6-6 6 6 6 6 6 6 6 6 6 6 6 6
2-2 2 2 2 2 2 2 2 2 2 2 2 2
6-1 6 5 5 4 4 3 3 2 2 1 1 1

126
The average travel times to access and egress to each zone are ranged from 4
minutes to 35 minutes, depending on network configuration scenarios. Figure 4.15 depicts
the travel time gradients over zones for the cases. As one might expect, more network
capacity leads less travel time. According to Figure 4.15 (a), drivers in the Linear-City
spend 4.6 ~ 6.8 minutes to travel if every street segments consist of 6 lanes. With 2 lanes for
every links, the average travel time to access to each zone increases to 29~35 minutes per
trip. In either cases, the gradients still show lowest at the center, like the baseline case has.
The proposed model estimates a peak of access time to the center with the 6-1 Case.
As Figure 4.15 (c) shows, drivers spend 11.5 minutes to access Zone 1 (or 11) and Zone 4
(or 8) on average, and 9.8 minutes to access Zone 4 (or 9) respectively. For the trips to the
center, on the other hand, they drive 14.9 minutes. The egress traveling time (the average of
travel time weight by trip production) also has similar profile over the region, i.e., showing a
peak at the center.
Although it is not a generalized case, at least this simple test shows that the observed
longer travel time to the activity center maybe related to an imbalance between network
capacity and allocated activity in the Linear-City. In all the case of network configurations
used in this test, the model estimates that the most production activity would happen at the
center as Figure 4.16 shows. Even in the 6-1 Case, in which the links in the center have only
one lane, firms produce $25 million per month from the center, while they are producing $21
million from the outskirts.

127

(a) 6-6 Case

(b) 2-2 Case

(c) 6-1 Case

Figure 4.15 Average Travel Time by Network Configurations

50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Egress
Access
4.0
4.5
5.0
5.5
6.0
6.5
7.0
Zone 1 2 3 4 5 6
Minutes / trip
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
11 Zone 10 9 8 7 6
Egress
Access
4
6
8
10
12
14
16
Zone 1 2 34 56
Minutes / trip
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
Egress
Access
20
24
28
32
36
40
Zone 1 2 34 56
Minutes / trip

128

Figure 4.16 Output Values Produced by Network Configurations

Comparing to the baseline case, in which the links in the center have 6 lanes, the
output produced from the center is only $1.9 million, or 7% less in the 6-1 Case. Although
only 1/6 of baseline link capacity is supporting similar amount of production activity of the
baseline in the 6-1 Case, still the center is spatially advantageous so the more activity is
taking place than the other zones. This localized mismatch between network capacity and
concentration of activity should causes the longer travel time to the center in the 6-1 Case.
Ever since the time of the standard mono-centric model, which has dominated the
urban economies and urban models within regional science (Richardson, 1988), travel time
has been considered as one of the most important driving forces that shape the urban
structures. Negative rent gradient, along with various allocation patterns, is resulted from
the given travel time to the CBD. As become more sophisticated, many models use
endogenous travel time to estimate allocation patterns (see Southworth, 1995 for the use of
travel time in various urban models). For this, the importance of travel time estimation
seems never have been faded out in all urban models, including the proposed urban CGE
model, that have a functionality for estimating allocation patterns. And from the simple test
70%
75%
80%
85%
90%
95%
100%
11 Zone 10 9 8 7 6
6-6
2-2
6-1
Baseline
15
17
19
21
23
25
27
29
Zone 1 2 34 56
$Million / month

129
in this section, the model shows a reasonable travel time variation for given network
configurations.
The proposed model was scrutinized through sensitivity analyses for selected inputs
– elasticity of substitution, distance decay coefficients, population, and network
configurations. Even with unrealistic inputs, such as extremely high population density, the
model results are explicable. It is true, however, the model may need even more rigorous
tests to before assert its applicability to real-life conditions. Geometry of zone system (see
Anas and Xu 1999 for alternatives zone systems tested), assumption on endowment, labor
supply per household, and the unit transportation cost per driving distance are the exogenous
parameters need to be studied in near future.

4.5. CONVERGENCE AND RESILIENCE OF THE ALGORITHM

This section provides a brief review about the model’s convergence and resilience
against disturbed initial values. The algorithm adjusts endogenous values over many
iterations until an equilibrium state is achieved, in which the solution vectors from
consecutive iterations are identical within a predefined tolerance. More specifically, two
solution vectors from successive iterations are compared to find the maximum difference,
and the algorithm stops when this maximum is smaller than the convergence tolerance. Of
course, the time to achieve a converged solution is related to number of endogenous
variables (or size of solution vector), given tolerance, as well as the efficiency of the
algorithm.
Figure 4.17 (a) exhibits the diminishing maximum differences of solutions from the
Linear-City example over iterations. The maximum difference between solution vectors

130
from consecutive iterations is less than 1% after 12-th iteration, according to the figure.
With R=11, I=3, H=3, M=1, and 66 links, the algorithm solves the model for total 197
variables in this case (see Section 3.7.1, and Table 3.1 for the number of variables to be
solved). This difference is reduced to less than 0.1% after 64-th iteration. The algorithm
takes 0.721 second for 100 iterations.
In a larger system, however, the maximum gap is not monotonic, and the
convergence rate is much slower than the Linear-City case. To test the proposed model for
practical applications in Chapter 5, another set of data, named SCAG was developed based
on Southern California statistics. In terms of size of the system, the SCAG data is much
bigger than the Linear-City example, with R=266, I=3, H=3, M=1, and 15,896 links, so the
total 26,069 variables are unknowns. As Figure 4.17 (b) shows, the algorithm estimates
diverging states in the first few iterations. Especially the wage estimated at the end of 4-th
iteration was 590% different to the estimation from the previous iteration. But, the
maximum difference is eventually reduced below to 0.5% after 199 iterations, for 65.77
minutes of running time.
Note that the step size that Figure 4.17 shows is used to adjust the solutions by a
linear combination at the end of each iteration. In the successive averaging scheme that the
algorithm is based on, the inverse of iteration index n is used as the step size α , i.e., α = 1/n.
At the end of an iteration n, the auxiliary solution y
n
is combined with the solution from
previous iteration x
n-1
, and the difference of the two solutions, d
n
is as given as Equation 4-1.
At the equilibrium state, the solution is converged, and two consecutive results are identical.
Therefore d
n
=0, and y
n
=x
n-1
.

131

(a) R=11, I=3, H=3 M=1, and 66 links (197 variables)

(b) R=266, I=3, H=3, M=1, and 15,896 links (72,617 variables)

Figure 4.17 Convergence of the Model

0.0001
0.001
0.01
0.1
1
10
0 1020 304050 6070 8090 100
Iteration
Maximum Difference (log scale)
Commodity Price
Wage
Land Rent
Step Size (1/N)
0.0001
0.001
0.01
0.1
1
10
100
1000
0 20 40 60 80 100 120 140 160 180 200
Iteration
Maximum Difference (log scale)
Commodity Price
Wage
Land Rent
Step Size (1/N)

132
()
1
1 1
1
−
− −
− ⋅ + ⋅ −
=
n
n n n
n
x
x y x
d
α α

=
1
1
−
−
−
⋅
n
n n
x
x y
α ·······························································(4.1)

As Figure 4.17, in the Linear-City, the maximum difference is always than the step
size, α = 1/n. In SCAG case, even though it is very close, the step size is still smaller than
the maximum difference of a variable (the wage, in this case). Therefore, these examples
show d
n
< α before the model converges. Then the auxiliary solutions in any iteration is
bounded to the upper according to Equation (4.2) Over the iteration, if the auxiliary solution
is not twice bigger than previous solution, than the diminishing step size will guarantee
model’s convergence, at least for the two example cases.
1 1 − −
⋅ < − ⋅
n n n
x x y α α
1
2 0
−
< <
n n
x y ·················································································(4.2)

Besides the convergence, variability of the model to arbitrary given initial values is
an important concern too. The proposed model starts with a set of initial guesses for the
major variables – land rent, wage, and commodity price, and output quantities. And the
model should produce unique results for slightly agitated initial values. To verify this
expectation, initial price for Commodity 3 in the Linear-City example has been this modified
in two different levels of agitation. In the baseline, the initial commodity prices are $10/unit.
First the price of the Commodity 3 produced from the centers was perturbed with ±10%.
And then initial price for the same commodity from all zone was changed ±10%.
The first perturbation to the initial price does not leave any significant effects to the
result. In spite of ±10% agitation to the price 3 at the center, the equilibrium price of

133
Commodity 3 at Zone 6 is only different to the baseline by –0.039% ~ 0.094% after 100
iterations. Since the perturbed initial value looses its effect over the iteration, all of the
endogenous variables are also returned to the baseline results. For example, the total output
from the center varies 0.090%, and –0.036% for +10% and –10% variations, respectively.
As Figure 4.17 (a) shows, the model converges for less than ±0.1% of tolerance at 100-th
iteration for the baseline of Linear-City. Therefore the result obtained from the perturbed
input is identical to the baseline for the given convergence tolerance.
The proposed model produces somewhat different solution from the second
perturbation to the baseline. When the price of Commodity 3 is agitated by ±10% from all
zones, the endogenous output prices after 100 iterations are differ by –0.620% ~ 0.553% to
the baseline. Although these variations are relatively smaller than the applied disturbance,
±10%, still higher than the convergence tolerance ±0.1% at the end of 100-th iteration.
These variations can be permanent even after infinite number of iterations, so that the
disturbed initial values cause a new equilibrium state which is not same to the baseline.
Series of brief tests in this section show that the proposed CGE model is able to
produce converged results for reasonable number of iterations and computation time. Since
the model is solving vast number of unknown variables, sometimes the model is vulnerable
to the arbitrarily given initial values. Small agitations in the initial values are transferred to
the solutions. Although uniqueness of solution is guaranteed for a fixed-point problem with
compact and convex price surface, the CGE model is generating distorted results
corresponding to significant alteration of initial values. Creation of relevant initial values
will be an important matter to enhance the model’s applicability.

134
4.6. REVIEW ON THE CES PRODUCTION FUNCTIONS IN
MULTI-REGIONAL CONTEXT

The production functions in the proposed model differ from the standard constant
elasticity of substitution (CES) production function, regarding the character of input share
coefficients. More specifically the sum of input share coefficients is 1 without the elasticity
of substitution,
j
θ as power to coefficients, i.e., 1 = + +
∑ ∑ ∑
j
h
hj
ir
ij
rq
σ ρ ϕ . The
standard CES production function, however, the sum of coefficients is 1 with the elasticity of
substitution, i.e., () () () 1
1 1 1
= + +
∑ ∑∑
j j j j
h
hj
ir
ij
rq
θ θ θ
σ ρ ϕ . With this condition, the
economy of scale is easily separated from the elasticity of substitution in the standard CES
function. However, due to this condition, the standard CES function may not be suitable for
multi-regional modeling. And the modified CES function is a means to overcome the
problem without sacrificing any property of the standard CES function. Using an inductive
approach, this section details the rationale of the modification. Referring Equation (3.22)
once more, the equilibrium price is calculated as follows.
j
q
p =
j
q
τ − 1
l
····························································································· (3.22)
=
() () ()
j
j j j
i
v
q
j
h
h
q
hj
r
rq
i i
r
ij
rq
j
q r w c p
θ
θ θ θ
σ ρ
ξ
ϕ
τ
−
− − −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
⋅
−
∑∑ ∑
1
1
1 1 1
1
1

Equation (3.22) shows that the output price is homogeneous of degree zero with
respect to the number of inputs, i.e., number of zones, number of industries, and household
types. Otherwise, the endogenous output price would be a function of the data size, rather
than any economic factors. The zone system is one of the external inputs those might be

135
determined based on the region-specific characteristics, and the objectives of study. If a
study requires detailed analysis, the study region can be presented with a finely divided zone
system. A detailed zone system, thus more number of zones in the system, may requires
additional computational burden, while the model would produce subtle variations in zone-
to-zone transportation costs, and the subsequent allocation patterns. And the result should be
comparable to the result analyzed with coarse zone system. Same argument is applicable to
classification of industries.
On the other hand, the homogeneity of degree zero of price is a premises for model’s
transferability to different urban systems. The proposed CGE model was not developed for a
specific region, but it is aimed for general application to anywhere with any economic and
spatial characteristics. To be able to so, the model should behave reasonably against vast
range of different industries, and indefinite number zones.
It turns out, however, the output price calculated based on the standard CES
function, in which the unity sum is satisfied with the elasticity of substitution, is sensitive to
the number of zones and number of industries. More specifically, the output price is
proportional to the number of input suppliers, i.e., R×I. The output price calculated by
Equation (3.22) is a partial equilibrium price that maximizes firms’ profit at the given input
prices, and it is properly working with any given set of positive input share coefficients,
regardless of the elasticity of substitution in partial equilibrium condition. In general
equilibrium framework, however, the input share coefficients in the very same equation
work differently depending on including the elasticity of substitution,
j
θ for the unity sum
conditions.

136
To demonstrate different responses caused by
j
θ in the output price calculation,
Equation (3.22) is applied to an output from a simple system with a zone, and an industry,
then applied to a detailed the system with multiple zones. Let’s assume a system with only
one zone (R=1) and one industry (I=1) in it. No sales tax is imposed for simplicity. The
industry in the region uses only one intermediate input produced by the industry itself.
Given initial price p
0
, and the travel cost c
0
, the model calculates the endogenous output
price p according to Equation (3.22), and in the first iteration, the calculation will be as
Equation (4.3);
() ()
()
0 0
1
1
0 0
1
1
1
0 0
1
1
1
0 0
1 1
c p
c p
c p c p
p + =
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
=
−
−
−
−
−
−
θ
θ
θ
θ
θ
θ
ϕ
. ···(4.3)
Since only one factor is used in the production, the input coefficient 1
1
= =
θ
ϕ ϕ . The
output price is obviously same to the initial effective price, which is the sum of output price
and delivery cost in this simple system.
Let’s further assume a detailed system that is exactly same to the previous example,
excepting the number of zones. The region in this case is divided into R identical zones
(R=R), and travel costs between any zone pairs are identical at the first iteration, i.e., c
rq
=c
0
.
The initial value for the commodity price is assumed equally, i.e.
0
p p
rq
= . Since the
effective price (price of intermediate input plus delivery cost) is identical, firms in every
zone in the system have same potential to supply their products to other’s production process
in anywhere. Therefore, according to Equation (3.24), the input share coefficient ϕ will be
divided also evenly for all suppliers from all zones. Therefore
R
ϕ =1/R , when θ is not
applied to the unity sum for the modified production function. Or ()
θ
ϕ
1
R
= 1/R with θ for

137
the standard CES function. When no elasticity of substitution is applied to the input
coefficients, i.e.,
∑
R
R
ϕ =1, with the evenly distributed input share coefficients and
effective prices, the output price from a zone is clearly identical to the effective price, as
Equation (4.4) shows;
p =
()
θ
θ
ϕ
−
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
∑
1
1
1
0 0
R
R
c p

=
()
θ
θ
θ
ϕ
−
−
− ⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
∑
1
1
1
1
1
0 0
1
R
R
c p
············································(4.4)

=
0 0
c p +

However, when ()
∑
R
R
θ ϕ
1
=1 as in the standard CES function, the output price
calculated by the same Equation (3.22) does not recover to the effective input price. For the
elasticity of substitution, θ > 0, and θ ≠ 1, the input share coefficient
R
ϕ = ()
θ
R / 1 .
Substituting ϕ in Equation (3.22) with ( )
θ
R / 1 , the output price p is R times higher than
effective input price, as Equation (4.5) shows. Same conclusion can be drawn with a case, in
which disaggregated industries are operating in a zone. In that case, the output price is
proportional to the number of industries, I.
p =
()
θ
θ
ϕ
−
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
∑
1
1
1
0 0
R
R
c p

=
()
θ
θ
θ
ϕ
−
−
− ⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
∑
1
1
1
1
1
0 0
1
R
R
c p
.

138
p = ()
θ
θ
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
⋅ ⋅ +
1
1
0 0
1
R
R c p
= () () []
θ
θ
−
−
⋅ +
1
1
1 0 0
R c p = ( ) R c p ⋅ +
0 0
·················································(4.5)

Equation (4.5) shows that the output price will keep increasing as long as a firm
purchases more than one intermediate inputs, under the standard CES production function.
The number of zones, R, acts as if a multiplier to the output price estimation. The number of
industries, I, also is functioning same way. The solution algorithm summarized in Chapter
3.8 starts with updating the output price based on given initial prices, and according to
Equation (4.5) the output prices ever increase as iterations go on for systems with R > 1, and
I > 1. As a result, the solution vector is diverging over iterations, and the algorithm can not
be continued. More specifically, the output prices overflow the limit of a conventional
double-precision variable (64 bits), i.e., 1.79×10
308
in the first few iterations. Because of
Equation (4.5), the model, with elasticity of substitution in unity sum of input coefficients, is
incomputable.
The problem in the standard CES function might be prevented with some restriction
on the firms’ behavior. Since the input price is increased by the number of input suppliers,
i.e., I and R, output price will be stable if only a few intermediate inputs are used. A simple
method is using negative infinite distance decay coefficients for intermediate transaction, µ
in Equation (3.24), and limiting firms to use only the cheapest input. Or predefined
interaction rules can be used for the firm model. However, unless an observation supports,
arbitrary restriction will reduce model’s degree of freedom, and will distort the subsequent

139
calculations. Instead, the modification implemented for the proposed CGE model allows the
model determines only the endogenous conditions.
The similar problem has been identified from the Kim’s original work (1990), and
the subsequent studies by Anas and Kim (1996). In the studies, the firms’ behavior is
modeled based on the profit maximization scheme, with the Cobb-Douglass production
function of constant return to scale. The equilibrium price is derived from zero-profit
condition in a perfectly competitive market. By applying the notations used in this study to
Kim’s work, his spatial equilibrium price is restated as Equation (4.6).

() ( ) ( )
() ( ) ( )
j hj ij
rq
j hj ij
rq
j
h
hj
ir
ij
rq
v
q
h
h
q
ir
rq
i i
r
j
q
r w c p
p
σ ρ ϕ
σ ρ ϕ
σ ρ ϕ
ξ
⋅ ⋅
⋅ ⋅ ⋅ +
=
∏ ∏∏
∏ ∏∏
·······································(4.6)

where, 1 = + +
∑ ∑∑
j
h
hj
ir
ij
rq
σ ρ ϕ for constant return to scale. By applying the
settings from the above example of multi-zonal system to Equation (4.6), the output
price resulted is also proportional to the number of zones, R;
p =
()
()
()
∏
∏
∏
∏
⎟
⎠
⎞
⎜
⎝
⎛
+
=
+
R
R
R
R
R
R
R
R
c p c p
R
R
1
1
0 0 0 0
1
ϕ
ϕ
ϕ
= () [ ]
∏
⋅ +
R
R
R c p
1
0 0

= ( ) R c p ⋅ +
0 0

The Cobb-Douglass production function is a special form of the CES function when
the elasticity of substitution, θ = 1, and Equation (3.22) is a generalized form of output price,
since it is driven from CES function. The special case of output price for the Cobb-Douglass

140
function is driven based on L’Hospital’s, as Equation (4.7) shows. See Appendix C for
detail procedure.

j
q
p
j
1
lim
→ θ
= () ( ) ( )
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅ +
∏ ∏∏
j hj ij
rq v
q
h
h
q
ir
rq
i i
r
r w c p
σ ρ ϕ
ξ ························(4.7)

( ) ( ) ( ) 1 = + +
∑ ∑∑
j
h
hj
ir
ij
rq
σ ρ ϕ .

This chapter scrutinizes the proposed CGE model in various aspects. Applying the
model to a synthesized data provides a detailed cross-sectional view of a urbanized area,
including land use, households, industries, and transportation. The following sensitivity
analyses exhibit the proposed model’s relevant responses to the perturbed exogenous
parameters and inputs. In spite of the guaranteed convergence, model’s resilience from
disturbed initial values is an issue that calls for another rigorous study. In the next chapter,
the proposed model will be applied to another synthesized data, the SCAG to test the effects
of selected regional policy alternatives.

141
5. MODEL APPLICATION

5.1. OVERVIEW

This chapter details the application results from a series of fictitious policy
alternatives for a synthesized metropolitan area. In Chapter 4, we examined the application
results of the proposed model with perturbed parameters. This chapter assumes that there is
no changes in consumption preferences, production technologies, and any parameterized
travel behavior for both of passengers and trucks. With given parameters, the proposed
model analyzes the urban system for policy options that might alter the physical aspect of the
system, such as changes in the size of developable land, changes in land use plans, and
enhanced transportation network capacity. These attempt to demonstrate applicability of the
proposed model to real world planning problems considering a practical problem size.
Three sets of fictitious policy options are analyzed in this chapter. First, regional
impacts from alternative land use plans are evaluated. Land use data is optional for the
proposed model, so the model takes exogenous land use data when it is given, or internally
estimates it as a share of available land to each use in a zone. Three land use plans are tested
with the model by the optional land use data requirement. The land use scenarios are as
follows: 1) increasing the share of residential land in a suburb zone to simulate housing
development; 2) increasing share of industrial land use in a downtown zone to model a
business complex development project; and 3) allowing the model determines land use
internally, without using external land use data. The impacts from each scenario are

142
categorized in the level of production activity, household utility, land use change, and
performance of transportation system, and compared to the status-quo.
Second, the impacts from fictitious region-wide disasters are analyzed. It is
assumed that unexpected events curtail land availability in damaged area proportional to
severity of the event. Reduced land input to the developers results in less housing and office
floor spaces in damaged zones. Any interaction with agents in the damaged zones is
affected indirectly from the unexpected events. In summary, the model simulates area-wide
activity interruption from reduction in land due to disasters. Two fictitious disasters with
different severities is specified. Note that this study does not specify whether the disasters
are manmade, or natural. Furthermore, model does not simulate any non-economic factors,
such as panic, anticipation of price increase, or short-term disequilibrium due to rapid
disruption in the system.
Under the Transportation Equity Act for the 21st Century (TEA-21), relatively small
projects to manage congestion and mobility become more acceptable than massive
investments in urbanized area due to the high land price and construction cost (Weiner,
1997; U.S. Senate, 2002). Complying with this trend in transportation system management,
three types of options to improve the existing network are analyzed. These are 1) increase
free flow speed of all links for 5%; 2) increase capacity of all link for 5%; and 3) complete a
freeway segment, which represents I-710 extension in Pasadena in California. This study
does not detail the engineering specifications applicable to improve free flow speed and
street capacity independently. This study assumes current engineering technology can
improve the two parameters separately.

143
The purpose of the policy analysis is to demonstrate model’s applicability to real-
world planning problems, not to draw any practical implications from the options. The
parameters such as consumption preferences and production technologies are exactly similar
to the ones used in the Linear-City example. Therefore the results exhibited in this chapter
should not reflect actual conditions. In fact, model estimated baseline is significantly
different from observation. This is summarized in Section 5.3.
The physical setting in this chapter is practical with respect to the data size and
number of endogenous variables. The extent of the study area is identical to the TAZ system
for the 1996 SCAG OD survey which covers most of the 5-County area (Los Angeles,
Orange, Ventura, and part of Riverside, and part of San Bernardino) within the Southern
California Association of Governments (SCAG) planning area. Imperial County although a
part of the SCAG region is not a part of the study. Within the boundary, only the developed
areas are identified according to 2000 land use survey,
2
and divided into 266 square zones.
Some of zones in sparse part of region are not adjoining geographically, but the
transportation network connects them logically. The transportation network is created based
on the topology from National Highway Planning Network (NHPN and the attributes drawn
from Highway Performance Monitoring System (HPMS).
This chapter consists of five sections, including this overview section. Section 5.2
describes the settings for SCAG data. Section 5.3 presents selected outputs from the
baseline analysis. The Section 5.4 describes the scenarios which are: alternative land use
plans, regional disasters, and transportation system management options. The estimated
differences associated with the scenarios to the baseline are also discussed.

2
http://www.scag.ca.gov/landuse/index.htm

144
5.2. SETTING FOR SCAG APPLICATIONS

The SCAG 5-County area is represented by publicly available survey-based data,
along with several assumptions for the applications in this chapter. Identical to the Linear-
City example in Chapter 4, the economic agents in the SCAG economy are 3 household
types (H=3), 3 industries (I=3), housing developers, office developers and the local
governments. Based on 1996 SCAG regional transportation plan, 5,160,000 households in
the planning region are divided in 1:2:3, for Household Type 1, 2, and 3 respectively.
The coefficients are taken from the Linear-City application for the production
functions, household utility functions, shopping preferences, and distance decay coefficients
for intermediate inputs, and adjusted for large scale applications. As stated in Section 4.2
this is fictitious, even though the production function coefficients for the Linear-City were
drawn from a SCAG regional transaction table. The positive distance decay coefficients,
which are unrealistic, are replaced with negative ones.
In addition, parameters such as household dispersion factors are adjusted for a wide
geographic extent. The estimated travel costs between some outskirt zones are too high to
make the commuting costs between the zone-pairs exceed the household income, and no
activity is allocated in those zone-pairs. For this reason, households tend to allocate only in
the middle of the region, rather than spreading throughout the modeling area. This tendency
is adjusted by the household dispersion factor. Appendix B shows the coefficients, and
parameters used for SCAG application.
Excluding the vacant, and agricultural land out of 38,000 square miles of the 5-
County SCAG region, 3,995 square-miles of area is modeled by 266 zones in the
application. SCAG developed a TAZ system with 3191 zones (and 26 external zones), in

145
Los Angeles, Orange, Ventura counties, and some part of Riverside and San Bernardino
counties, as the light gray area shown in Figure 5.1 (SCAG, 1997). SCAG had used it for
regional transportation plan, and related socio-economic data archive, including 2000
Census. However, it was not suitable for the application with the proposed CGE model
because of different zone sizes. Size of the largest TAZ, which locates at the eastern edge of
San Bernardino County, is 76,400 times larger than the smallest one in the system, located in
Los Angeles downtown. Using this data without modification, the proposed model is likely
to estimate more economic activities at the fringe of the area where huge amount of land is
available in each zone.
A zone system with relatively uniform zone size is used to prevent the problem of
heavy production activity at the outskirt of the region. Following is the procedure to create
the zone system. The procedure identifies 266 of 5 km×5 km cells with developed land (see
Figure 5.1).

• Divide the 5-County area with a 61 columns × 33 rows grid. The grid creates 1813
cells, and each sizes 5 km × 5 km.
• Overlay the 2000 SCAG Land Use Map for whole 5-County area on the grid,
• Identify the composition of land use in each 5 km × 5 km cell. The land use types
are commercial, industrial, residential (low, high, and rural density), open space &
recreation, and public facilities & institutions, agriculture, and vacant.
• Identify cells with developed land use. A cell is assumed developed if land allocated
for agriculture and vacant is no more than 22.5 km
2
, i.e., 90% of cell’s geometric
area.
• Select only the grid cells with developed land use.

146

Figure 5.1 Analysis Zone System for SCAG Applications

The land use which identifies developed area complies the endogenous land use
types in the proposed CGE model. Table 5.1 summarizes the 2000 SCAG Land Use Map
that categorizes lots into 109 detailed land use types and 13 aggregated types. It was further
aggregated into four land use types – residential, industrial, public, and others (including
agriculture and vacant). The first three land uses are the types that the proposed urban CGE
model uses as exogenous inputs for the baseline analysis. And any cells in which the
amount of land for the three land use types is not more than 10% of 25 km
2
were dropped
from the zone system. From the criteria, the largest zone, regarding land availability is only
9 times larger than the smallest zone, theoretically. As Figure 5.2 shows, zones with more
developable land are concentrated on north of Orange County to south of Los Angeles
County, which is overlapping on the Interstate 5 corridor. Also, some of zones along I-10
toward San Bernardino, and north of Ventura have high land availability.
Analysis Zone
1996 SCAG RTP Area
SCAG 5-County

147
Table 5.1 Land Use Included in the SCAG Zone System

5-County SCAG 266-Zone System
Land use
Area
(Sq-mile)
Area
(Sq-mile)
Ratio to
5-County (%)
Modeled
Land use
Agriculture 1734.56383.62 22.1
Commercial 212.42199.06 93.7 Industrial
Extraction 127.4136.91 29.0
Former Military 32.06 12.61 39.3
Industrial 210.87190.44 90.3 Industrial
Low Density Residential 1,312.69 1,213.66 92.5 Residential
Medium to High Density
Residential
261.28 245.69 94.0 Residential
Open Space & Recreation 1,483.09 371.64 25.1 Public
Public Facilities & Institutions 264.59 155.70 58.8 Public
Rural Density Residential 249.67 127.63 51.1 Residential
Transportation & Utilities 477.81 225.01 47.1
Vacant 31,693.392,335.19 7.4
Water & Floodways 489.63 71.22 14.5
No Data 79.49 15.80 19.9
Total sum 38,628.95 5,584.18 14.5
Sum of modeled land use area 3,994.61 2,503.83 62.7

The modeled area by the zone system is only 14.5% of 5-County region, as
summarized in Table 5.1. However, excluding the land used for agriculture, extraction,
military (former), vacant, water & floodway, and transportation & utility within the region,

148
62.7% of land is included by the zone system. In addition, if only the commercial, industrial,
and residential land are considered, the selected 266 zones cover more than 90% of
developed land. Land for transportation facility was excluded because the proposed model
relies on given transportation network data, and does not explicitly allocate land for the use.

Figure 5.2 Ratio of Developed Land in SCAG Application

A transportation network data was created by based on two federal databases. First,
the network topology, i.e., node and street coordinates, is from the NHPN database (FHWA
a, 2005). The NHPN includes not only freeways, but also major local roadways like
arterials, and delivers them in GIS format (ESRI uncompressed e00 format). The links and
nodes within the 1996 SCAG planning region were selected from the state-wide NHPN
database.
Even though NHPN well represents the link geometry by series of points as a GIS
database, the link-node connectivity is not guaranteed. For example, two links are not meet
Land Ratio
85 % to 100 %
70 % to 85 %
55 % to 70 %
40 % to 55 %
25 % to 40 %
10 % to 25 %

149
on a location of common node to represent an intersection. This problem was dissolved by
re-creating link map according to From-Node, and To-Node attributes of each link, on top of
the coordinate information from node table in NHPN. In addition, center points of the 266
zones were added to the node data to represent zone centroids. 8-unidirectional links per
centroid are added into the link data to connect the centroids to nearest four nodes from each
corners of the zones. Figure 5.3 shows the resulted network.

*) Centroids and centroid connectors are not shown for visibility

Figure 5.3 Network data created for SCAG Application

The parameters for link congestion function were obtained from the HPMS (FHWA
b, 2005). According to Equation 3.52, link congestion is modeled by length, free flow speed
(or free flow travel time,
0
l
t ), number of lanes, and the practical capacity per lane (for link
capacity,
l
K ). The length and the number of lanes were taken directly from the HPMS,

150
while the free flow speed, and the capacity per lane were assumed according to designated
link functional classes in the HPMS. As summarized in Table 5.2, the free flow speed is
generally corresponding to common speed limits . For capacity, freeways were assigned
with 2,200 PCU/land/hour, and 1,500 PCU/land/hour was assumed for local streets.

Table 5.2 Free Flow Speed and Capacity for SCAG Applications

HPMS Link Functional Class
Free Flow Speed
(Miles / Hour)
Capacity
(PCU / Lane / Hour)
Interstate 65
Other Freeways and Expressways 60
2,200
Other Principal Arterial 50
Minor Arterial 45
Major Collector 40
Minor Collector 35
Collector 35
Local 30
1,500
Centroid Connectors 30 3,400

Besides the links from NHPN and HPMS, the centroid connectors were modeled as
slow free flow speed, 30 mph, yet less congestible capacity of 3,400 PCU/lane/hour, and 10
lanes. Including 2,128 centroid connectors from 266 zones (= 266 zones × 4 corners × 2
directions), total 15,896 unidirectional links represent the roadway network system in the
SCAG application. Table 5.3 summarizes the distribution of links for number of lanes and
free flow speed.

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Table 5.3 Distribution of Links by the Attributes

Link Type

Centroid
Connectors
Local
Streets
Freeways
Sum
Number of Links 2,128 10,004 3,764 15,896
2 7,942 28 7,970
3 2,002 1,330 3,332
4 60 1,314 1,374
5 954 954
6 138 138

Lanes
10 2,128 2,128
30 2,128 808 2,936
35 1,122 1,122
40 7,680 7,680
45 232 232
50 162 162
60 130 130

Speed
(miles/hour)
65 3,634 3,634
1,500 10,004 10,004
2,200 3,764 3,764

Capacity
(PCU/Lane)
3,400 2,128 2,128

The initial outputs,
0
X , are assumed according to the industrial land use, along with
an arbitrary assigned scale factor. Basically, the initial output,
0
X is assumed to be
proportional to the industrial land use identified from 2000 SCAG survey. It is because that
more land supply to industrial activity should result in more output production due to low
office rent. In addition to the industrial land use, a spatial factor is considered for the initial
output values to roughly represent the distribution of the industrial activity allocated within

152
the Los Angeles basin. The factor ranges from 0.1 to 1.0 as Figure 5.4 shows. Any actual
observations that explain the spatial distribution of industrial activity, such as zonal
employment, should replace this arbitrarily assumed factor in the future.

*) Residential rent in the circled zone is fixed to be $20 / 10 sq-yd / month.
Figure 5.4 Distribution of the Scale Factor for Initial Output Value

For assumed initial values, 1 million units of output from each industry in a zone is
scaled by the ratio of industrial land to the geometric area – 25 km
2
, and the assumed spatial
factor shown in Figure 5.4. In this scheme, the possible initial output ranges from 1 million
units to 0. Including the initial outputs, all the initial values required for SCAG applications
were provided as followings;

• Land rent,
0
r = $20 / 10 sq-yd / month
• Residential floor rent,
0
r
u
= $50 / 10 sq-yd / month
• Industrial floor rent,
0
r
u
= $50 / 10 sq-yd / month
• Commodity price,
0
p = $10 / unit
• Wage,
0
w = $10, $8, $6 / hour for Labor Type 1, 2, and 3, respectively
Factor
1.0
0.8
0.7
0.6
0.5
0.2
0.1

153
• 2000 Land use survey land share for residential, industrial, and public uses,
• Output ,
0
X = industrial land ratio × initial factor × 10
6
units / month
• Residential land rent in Zone 8 (circled in Figure 5.4) is fixed to be $20 / 10 sq-yd
per month

5.3. SCAG BASELINE

This section presents the baseline of SCAG 5-County, analyzed based on the given
settings and the initial values described previous section. Table 5.4 summarizes selected
measures from the snap-shot of baseline. Appendix D provides a thorough list of thematic
maps, regarding the distribution of land use pattern, household activity, industrial activity,
and performance of transportation system.
For two reasons, the analysis result should reflect the reality of the region. First, all
the estimated prices of land rent, space rent, commodity price, and travel cost are relative to
the arbitrary rent of residential land in a zone near Santa Barbara, circled in Figure 5.4.
Consequently, the estimated quantities of production, developed spaces, and trips are
consistent with the arbitrary price, not to the actual observable price. Second, the parameters
were just borrowed from synthesized example used in Chapter 4. Therefore, the behavior of
economic agents should not represent actuality that can be observed.
Acknowledging that the results should not be believed as actual, comparing the
model results to observations might still serve as a benchmarking to judge the relevance of
the analysis. Remainder of this section provide a brief comparison between model result and
some observable statistics. This effort will provide a solid background for model calibration
and validation in near future too.

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Table 5.4 SCAG Baseline Estimation

Household 1 Household 2 Household 3 Sum / Avg
Household
Number of
Households
860,000 1,720,000 2,580,000 5,160,000
Utility / household 5.050 5.038 4.859 4.950
Wage ($ / hour) 11.623 8.714 8.321 9.002

Labor Income
($ /household /month)
2,021.4 1,274.5 1,116.1 1319.8

Rent Income
($ /household /month)
423.6 267.1 233.9 276.6
Weekly work hour 43.64 38.19 32.10 36.05
Industry 1 Industry 2 Industry 3 Sum / Avg
Production
Output Quantity
(million units)
307.8 212.5 374.5 894.8
Output Value ($ M) 3,910 2,747 4,755 11,411
Price ($ / unit) 12.70 12.92 12.70 12.75
Residential Industrial Public Sum / Avg
Land Use Land share (%) 63.38 15.56 21.06 100.00

Land rent
($ /10 sq-yd)
20.08 26.40 14.20 19.96

Developed Floor space
(10 million sq –yd)
124.01 66.24 190.25

Floor rent
($ / 10 sq-yd)
15.39 16.63 15.82
Passenger

Working Shopping
Freight Sum / Avg
Trips
Trip Production
(PCU / 4Hour)
2,285,880 648,569 109,822 3,044,271

Travel distance
(10
6
PCU*Mile / Day)
208.64 8.48 217.12

Travel time
(10
3
PCU*Hour/Day)
4,960.6 200.2 5,160.9

Avg Travel Distance
(Miles)
101.4 109.4 101.7

Avg Travel Time
(minutes)
71.10 77.25 71.32

Avg Speed
(miles / hour)
42.06 42.37 42.07

155
Relative to statistics, the proposed CGE model described the economy in SCAG 5-
County region with 1) low income households, 2) low industrial activity, 3) low rent, and 4)
excessive travel distance. According to the snap-shot of baseline in Table 5.4, the model
estimated the average household income would be $1,596.4 per month (or 20 working days),
which includes $1,319.8 from labor income, and $276.6 from land rent. The annual
equivalent income $19,157 (= 1,596.2 × 12) is compared to only 45% of Los Angeles
County median household income surveyed for 2000 Census, which was reported as
$42,189
3
. Because the comparison was made between the mean estimate and the median
observation, the ratio of estimated income, i.e., 45% may not be exactly valid. However,
according the rule of big number, the actual mean income should not be far from the
reported median income. Therefore, it is safe to conclude that the model estimates about half
of actual household income. Out of many latent reasons, assumption of one full time
employment per household should be the one, and eventually this assumption need to be
released.
The model estimated that industrial system in the 5-County SCAG would produce
$11.4 billion per month, or $137.9 billion annually. Due to several modeling assumptions,
comparison of this estimation to observation is ambiguous. First the economy is closed to
outside world so no interaction – import and export of goods, or capital transfer is included
in the model. For this reason, it is not clear whether the observation should include export or
not, to be fair comparison. Second, construction sector in the proposed model is described by
developers, who contribute to the economy by final demand, but by intermediate
transactions. For this reason, the commodity market of which equilibrium condition follows

3
State and County QuickFact, http://quickfacts.census.gov/qfd/states/06/06037.html

156
Equation 3.48, does not include developers, or construction activity, and treatment for this
structural difference is not available yet. Third, a consistent gross regional production for
the 5-County area was not readily available for this study.
In spite of the ambiguity, some of available statistics show that the estimation is
relatively less than actual figure. According to federal statistics, California gross statewide
production marked as $1,519.2 billion (BEA, 2004, current dollar), and for Southern
California region only, the industrial output is 6.7 times higher than the estimation, or $924.3
billion annually according to the 1999 regional transaction table (IMPLAN Group;
recalculated for only the sectors corresponding to this application). The output is calculated
from relative price to an arbitrarily determined residential land rent, and the corresponding
quantity produced. Along with capriciously determined parameters, such as elasticity of
substitution, this significant difference should be resolved in near future.
According to survey, rent for an sq-ft of Class A or B office space in Los Angeles
area ranges from $2.0 to $2.2 per month (Grubb & Ellis Research, 2006)
4
. In contrast, the
model estimation ranges from $10.64 to $26.83 / sq-yd, or $0.118 ~ $0.298 / sq-ft over all of
266 zones. Again, the price measure is relative term, and should be adjusted to reflect actual
rent.
Also, the estimated travel pattern is somehow inconsistent with observation. The
proposed model estimated 217.1 million vehicle-miles traveled per day, while 368.5 million
VMT per day was reported (SCAG 2004, Table 3.3-11). However, using the lower estimate
in VMT, the estimated travel distance turns out to be 101.7 miles per trip, and travel time is
71.3 minutes per trip. These represent much longer trips than actual average which are 12.6

4
For the first quarter of 2006.

157
miles and 21.6 minutes, (ibid, Appendix C-30; for home-to-work trips, year 2000). One
possible explanation of this mismatch is that higher travel cost is not properly reflected into
the activity allocation pattern. As mentioned above, among the parameters from Chapter 4,
household dispersion factors were adjusted for large study area, and this adjustment allows
the model to allocate households throughout the study region, but also it prevents the model
from allocating the jobs to closer to residence. Relevant dispersion factors should be
calibrated according to proper trips distance measure in the future.
Unlike the total or average measurements reviewed above, the model estimated
generally coherent activity allocation patterns. As Figure 5.5 and 5.6 show, darker cells are
concentrated around dense freeway network. It means more households, and industries are
located along side of more accessible locations via freeway.
The intermediate transactions between 3 industries in 266 zones are traced and
plotted in semi-log scale, as Figure 5.7 shows. The transaction values range from $17,000 to
$17 for a typical month (for 4weeks of 5 business days). The Darker the cell is, more the
transactions occurred. But, it is hard to identify any significance in transaction from this plot,
unlike the simple system in Chapter 4. However, this type of visualization will be helpful in
diagnosis of the difference from baseline to any policy implemented conditions.
This section does not provide a rigorous statistical analysis to validate the model
estimation. It is because the purpose of this Chapter is to demonstrate the proposed urban
CGE model is producing stable results for a large system. This section summarizes the
baseline result. The quality of the results should be guaranteed with properly calibrated input
data.

158

Figure 5.5 Household Allocation Pattern in Baseline of SCAG

Figure 5.6 Industrial Production Allocation Pattern in Baseline of SCAG

Households
25,370 to 26,790
24,050 to 25,370
22,380 to 24,050
20,570 to 22,380
19,020 to 20,570
16,690 to 19,020
10,750 to 16,690
130 to 10,750
Freeway
$M
42.0 to 50.4
35.3 to 42.0
30.1 to 35.3
26.5 to 30.1
23.6 to 26.5
20.7 to 23.6
16.0 to 20.7
6.3 to 16.0
Freeway

159

Industry j 1 2 3
Zone 266 ↓

1

2

3

↑ ↑ ↑ ↑
$17 $170 $1,700 $17,000

Figure 5.7 Intermediate Transactions in Baseline of SCAG Application

160
5.4. POLICY OPTION ANALYSES

In the previous section, the estimated SCAG baseline shows significant differences
from observations. Therefore, further policy analyses using same parameters from baseline
analysis will not be relevant if the findings from the analyses are used to evaluate the policy
options. However, the purpose of this section is limited to show the proposed model’s
applicability to practical planning questions. To accomplish the purpose, this section
provides brief reviews for selected policy options, focusing on how to use the proposed
model to analyze the options and relative difference with the baseline.

5.4.1. Land Use Control

Given baseline land use is modified in three ways. First, residential land in selected
zones in North Los Angeles county (marked by a circle in Figure 5.8), is increased from
16.77 km
2
to 59.23 km
2
. In the baseline, the total available land was 22.29 km
2
. A fictitious
development plan increases the available land to the geometric maximum in each zone, i.e.,
75 km
2
in three zones, assuming the allocated proportion between residential and industrial
land to the total land remains fixed. Since more than 75% of the total was allocated to
residential purpose in the baseline, this option is characterized as a housing development
project. Table 5.5 (a) shows the land allocated for the baseline and after the plan
implementation.
Second, industrial land use in Los Angeles downtown is increased while the
residential land is reduced. Unlike the housing development option, this option does not
increase the total land availability because the selected zones are already fully development

161
in the baseline. As Table 5.5 (b) shows, the land allocated to residential and industrial uses
are simply exchanged, so the industrial land increases from 31% in baseline to 59% after the
plan implemented. Since this option increases industrial land in downtown to almost double,
it is typified as a downtown development.

Table 5.5 Land Use Options for SCAG Application

(a) Land use under Housing Development Option (km
2
)
Baseline Land Use Option Land Use
Zone ID
Residential Industrial Total Residential Industrial Total
40 7.05 2.41 10.74 16.40 5.60 25.00
46 6.28 0.81 7.73 20.33 2.63 25.00
52 3.44 0.19 3.82 22.50 1.25 25.00
Sum 16.77 3.41 22.29 59.23 9.48 75.00

(b) Land Use under Downtown Development Option (km
2
)
Baseline Land Use Option Land Use
Zone ID
Residential Industrial Total Residential Industrial Total
65 12.08 7.65 22.24 7.65 12.08 22.24
66 15.04 7.11 24.02 7.11 15.04 24.02
Sum 27.12 14.56 46.26 14.56 27.12 46.26

Third option is highly theoretical. Instead of relying on exogenous land use input,
the proposed urban CGE model determines the land use as a byproduct of equilibrium

162
process. According to Feldman’s affirmation (1989), orders can be achieved without design,
and marketplace is an example that demonstrates highly ordered system without centralized
authority. Land market should also be able to create such an order, at least in theory by the
proposed model. No assumptions, though, were made for the additional externality that
might arise from mixed land use. This option is named “No land use plan.”
The detailed results are summarized in Appendix E.1 to E.3, and Figure 5.8 through
5.10. Appendix E.1 shows that the housing development option increases about 1.5% of
housing space, and 0.5% office space, while it reduces 0.8% and 0.3% of rents, respectively.
The positive impacts of housing development propagates to industry, so regional production
increases about 0.6%. Increasing household income ends up in more shopping trips, by
0.66%. As Figure 5.8 shows, almost all zones are experience positive production increase
through the region.

Figure 5.8 Output Changes Due to Housing Development Option

M$ Increased
(# of Zones)
4,990 to 5,091 (3)
262 to 4,990 (41)
224 to 262 (48)
188 to 224 (55)
154 to 188 (57)
117 to 154 (34)
-14 to 117 (28)

163
Unlike the housing development option, the impact from downtown development
option is negligibly small. Generally, housing rent increases 0.23%, while the office rent
drops 0.46%, in response to reduced residential space 0.21%, and increase industrial space
by 0.65%. These changes do not cause much of regional production, or transportation
demand changes. Appendix E.2 summarizes the results. Looking at the spatial variation that
Figure 5.9 depicts, the production activity is actually shifted toward the downtown, so only
the two zones in downtown have positive increase in production, while the others experience
negative levels of production.

Figure 5.9 Output Changes Due to Downtown Development Option

‘No land use plan’ option generates quite remarkable difference from the baseline.
In the equilibrium, the proposed model suggests 82.5% more land should be allocated to
industrial activity, while residential and public purposes are reduced to 9.6% and 43.7%
respectively. From these change, the regional economy produces 14.9% more commodity in
quantity, or 49% more in the value. Labor participation is increased 48% in weekly labor
$M Increase
(# of Zones)
4,000 to 8,000 (2)
0 to 4,000 (13)
-10 to 0 (5)
-20 to -10 (16)
-30 to -20 (32)
-40 to -30 (72)
-50 to -40 (56)
-120 to -50 (70)

164
hours, and bring 38.2% more income. Overall 3.3% more trips are produced, and trip length
becomes 0.17% longer than baseline. See Appendix E.3 for details. Almost all zones are
showing increase production in Figure 5.10. If there is no additional externality,such as
degradation of residential amenity, which arises from allowing mixed development, an order
not only exists without design, but also the order is much more efficient than the designed
order.

Figure 5.10 Output Changes Due to ‘No Land Use’ option

5.4.2. Impacts of Unexpected Events

Impacts from disasters are also modeled by adjusting land availability. In loss
estimation tools such as HAZUS
5
, the impacts from unexpected events are estimated by the
following steps: 1) potential hazards modeling to estimate frequency or magnitude of

5
http://www.fema.gov/plan/prevent/hazus/
$M Increase
(# of Zones)
32,000 to 40,000 (12)
24,000 to 32,000 (76)
16,000 to 24,000 (138)
8,000 to 16,000 (28)
0 to 8,000 (6)
-8,000 to 0 (6)

165
unexpected (natural) events; 2) vulnerability modeling to estimate direct physical damage
through fragility analysis of structures; 3) indirect physical damage to such as inundation,
fire, land slide; and 4) direct economic and / or social loss estimation (FEMA 2000, Chapter
1). The calculated direct physical damage in step 2 is translated to reduction in land supply,
and the proposed model estimates the indirect economic impacts.
Two sets of event scenarios are analyzed. First, a zone is selected, and 90% of land
is not usable for a month due to an unspecified event. The zone faces the west coast, and by
chance, it is where the Los Angeles International airport is located. The circle in Figure 5.11
identifies the zone. According to the figure, the reduced activity from the damage zone
relocates zones in its east bound along side of Interstate 10 freeway. Appendix E.4 shows
that the region-wide developed space rent increases about 0.3% due to the event, and 0.01%
of regional production is reduced in value. However, this ratio is not significant for the
convergence rate is around 0.02% after 200 iterations.

Figure 5.11 Output Changes Due to Unexpected Event in a Coastal Zone
$M Increase
(# of Zones)
90 to 120 (3)
70 to 90 (20)
60 to 70 (28)
50 to 60 (38)
30 to 50 (99)
-260 to 30 (77)
-11,850 to -260 (1)

166
The second scenario is about an eminent earthquake event. Scientists found a blind
(hidden) fault under central Los Angeles area, and named it Puente-Hill Blind Thrust Fault
(Field et. al. 2005). The polygon on Figure 5.12 shows its projected fault plane to the
surface. Earthquake events on this fault is not frequent – about around once in 3,000 year,
but the expected magnitude ranges 7.2 to 7.5. When an earthquake happens with magnitude
7.0 at normal work day afternoon, the expected direct economic damage would around $130
billion including devastating amount of fatalities.
The ratio of damaged buildings at 6 months from the earthquake event is estimated
by HAZUS-99 (SR2). According to the building occupancy types, the damage ratio was
aggregated into reduction in land supply for the proposed model. And Figure 5.12 and
Appendix E.5 summarize the estimation result. Contrary to expectation, the region-wide
production is increased by 0.68%, due to increased price for 2.64%, while the quantity
produced is decreased 1.9%. Increased commodity price, and residential rent by 7.75%
should affect the household utility, and decreased by 0.83%.

Figure 5.12 Output Changes Due to Puente-Hill Earthquake Scenario
$M Increase
(# of Zones)
2,500 to 4,700 (84)
2,200 to 2,500 (25)
1,800 to 2,200 (23)
900 to 1,800 (25)
-200 to 900 (21)
-1,400 to -200 (27)
-2,600 to -1,400 (34)
-6,400 to -2,600 (27)

167
5.4.3. Transportation System Management Options
From the various available alternatives to increase mobility, three options are
selected and analyzed. As the first option, it is assumed that any link in the transportation
network system has 5% faster free flow speed than baseline. An increased regional
production was expected, for improve travel time and cost, the model generates rather
puzzling results. Appendix E.6 summarizes the result. As the region-wide travel speed is
improved 3.7% at the equilibrium state, the commodity price decreases 1.26%. Unlike the
common sense, the lower price was coupled with less production of 0.78% in quantity, and -
2.03% in produced value, in this case. Figure 5.13 shows the spatial distribution of zonal
production changes, and according it, only some of zones located near outskirt get positive
production increase. Travel time increases exponentially with higher volume-capacity ratio,
so marginally improved free flow speed does not affect to the regional activity system. Only
a few less congested zones would experience relatively improved travel speed.

Figure 5.13 Output Changes Due to 5% Free-Flow Speed Increase Option
$M Increase
(# of Zones)
1,500 to 3,760 (4)
-310 to 1,500 (21)
-610 to -310 (38)
-860 to -610 (54)
-1,060 to -860 (45)
-1,250 to -1,060 (36)
-1,440 to -1,250 (43)
-1,860 to -1,440 (25)

168
The second option is also about improving link performance with 5% more practical
capacity. According to the BPR function shown equation (3.52), increased capacity may not
be so effective in reducing travel time when traffic volume is relatively small. But as
volume approaches or even exceeds the link capacity, even a marginal capacity increment
would be significant, since the travel time is exponential to the volume-capacity ratio. The
effect of 5% increased capacity is characterized as ‘increased’ regional production. As
Appendix E.7 shows, the commodity quantity increase only 0.16%, while the value increases
1.89%, with 1.73% price increase. Wage and income also increase more than 2% region-
wide. The effect looks like propagated via freeway network. As Figure 5.14 shows, more
production increases in central Los Angeles and North of Orange county, where freeway
network is dense. And only a few outside zones in San Bernardino County are experiencing
negative production increase.

Figure 5.14 Output Changes Due to 5% Capacity Increase Option

$M Increase
(# of Zones)
2,270 to 5,450 (4)
1,050 to 2,270 (38)
870 to 1,050 (44)
750 to 870 (64)
670 to 750 (46)
520 to 670 (50)
230 to 520 (7)
-980 to 230 (13)

169
The last option is about the completion of I-710. This freeway connects massive
amounts of freight traffic from Los Angeles/Long Beach Ports to freeway routes, but the
connection ends just east of Los Angeles downtown. And the suggestion was made to
complete the route [Figure 5.15 (b)]. Groups that support this suggestion
6
argues that the
cost associated with detouring the missing link would hinder the regional as well as the
national economy. Local residence group
7
against this suggestion argues that the new
freeway segment would dissect the local environment, without improving too much of travel
cost.
According to the model result, very minor region economic impact would be
produced from the additional I-710 segment. As Appendix E.8 summarizes, changes relative
to the baseline in almost all aspects are smaller than the convergence rate. Unfortunately,
the proposed model is lacks the capability of dealing with the both of the opposing interest
groups’ arguments. On one hand, since the model analyzes a closed economy, no export or
import is included. Therefore special consideration for concentrated freight traffic from the
ports is not part of the model. On the other hand, the model does not consider environmental
aspect in current version, therefore its degraded residential environment is not valued in the
model properly. These limitations should be released in future version.
Even though the change is very small, its spatial variation shown in Figure 5.15
makes sense. As I-710 is connected to I-210 to its north end, activity is shifted from around
I-5 to zones around I-210. Also, as more alternatives are available in the network system,
central Los Angeles area would have more production activity.

6
http://www.710gap.com
7
http://www.710freeway.org

170

(a) Regional Impact

(b) Impact Near the Extended I-710

Figure 5.15 Output Changes Due to I-710 Completion Option
$M Increase
(# of Zones)
70 to 150 (17)
60 to 70 (17)
50 to 60 (23)
40 to 50 (35)
30 to 40 (63)
20 to 30 (55)
-50 to 20 (56)
$M Increase
(# of Zones)
70 to 150 (17)
60 to 70 (17)
50 to 60 (23)
40 to 50 (35)
30 to 40 (63)
20 to 30 (55)
-50 to 20 (56)
I-210
I-5
I-710

171
This chapter demonstrates applicability of the proposed model to synthesized, yet
practical planning scenarios, in terms of the problem size. Since the parameters used in this
chapter were not calibrated, baseline analysis for SCAG region shows significant gaps to
observations. Also, for some policy alternatives, such as analysis of I-710, the proposed
model is not able to provide a ‘full story’ to support for any of interest groups involved in
the project. However, the proposed urban CGE model still produces proper results in
response to policies that invoke marginal changes. In near future, the model would be
applied to a calibrated system and the applicability will be validated.

172
6. CONCLUSION

6.1. SUMMARY AND CONCLUSION

The objective of this study is to establish an alternative modeling scheme for
comprehensive urban transportation studies. The model developed in this study is based on
Anas-Kim’s computable general equilibrium model and expanded in applicability to address
real world problems existent in complex urban systems.
The early conceptualization of a comprehensive general equilibrium (Anas, 1980)
has turned into an operational model by Anas and Kim (1990, 1996), Anas and Xu (2000),
and has been applied to small theoretical examples. Despite the sound theoretical
background on which their model was developed, the equilibrium price formulation has an
undesirable property of depending on number of factors, i.e., combination of input
commodities and input locations in multi-regional context. A numerical example proves the
model may not be computable, unless input data is strictly manipulated.
This study updates three aspects of the early comprehensive general equilibrium
model. First, the model has been expanded to accommodate for multiple households and
industries. This allows for diverse preference and production technologies. Second,
developers and local government are included as additional economic agents. By including
developers, land use pattern is explicitly modeled. Finally, a CES function replaces the
production function in the model, and a stable equilibrium price formulation is derived. The

173
CES function used is constant return to scale, yet the unity sum of input share coefficients is
held without any exponent on the coefficients.
The proposed model is applied two distinctive data sets. A small data with 11 zones
is created, and applied to evaluate characteristics of proposed model for selected parameters.
The model estimates that the tendency to concentrate at the center of region is reduced, as
firms become agile with higher elasticity of substitution. The model responds suitably to
distance decay inputs, even for extreme ones. In response to ever growing population, and
prohibitively high travel cost, model estimates formation of sub-regions. And finally, the
small data set shows that spatial mismatch between network capacity and level of activity
causes high travel cost in downtown.
The proposed model demonstrates its applicability to real-world planning options
through a Southern California dataset. Impacts from changes in land use policies are
analyzed with respect to regional production, household utility, and transportation system
performance. Increased housing land brings positive economic impact throughout the region,
yet the option that converts residential land to industrial in downtown area drags more the
economic activity into downtown without net impact. The proposed model analyzes a
situation where no pre-determined land use plan is implemented, and shows the “order
without design” condition exists at least in the model estimation, and more efficient
regarding regional economy. Impacts from unexpected events are calculated by adjusting
land availability input data. In testing transportation system improvement options, the model
shows capacity is of greater importance than free flow speed to improve regional mobility.
Demonstration of the estimated results with a small synthesized data set and a real-world one
fulfill the objective of the study.

174
6.2. SUGGESTION FOR FUTURE RESEARCH

With large dataset, the model reveals its limitations in analyses. For example,
analyzing the impact of additional freeway segment on I-710, the model estimates negligibly
small impacts. To measure the impact properly from the completed freeway route,
throughput traffic should be included. The model considers only closed systems and no
nationwide traffic is modeled. To resolve this limitation and extend the model developed in
this study, some research is suggested as follows.
To better represent the urban system, the proposed model should include interaction
with outside of the study area. Unlike national economy where clear barriers exist, a
regional economy is highly interrelated. Interaction with outside, such as export / import
should be incorporated to properly model a regional economy. In addition, throughput
traffic i.e. traffic between inside and outside, as well as outside to outside that pass through
the network system in the region should be considered to better estimate the network
performance.
To be efficient, the model needs a better solution algorithm than the current
successive iterative scheme. As reviewed in Chapter 3, the model gradually converges, but
the convergence rate is highly related to the pre-determined step size, especially when large
number of unknown variables is considered. Results from two consecutive iterations are
combined with a step-size, which is the inverse of iteration index, eventually slow the
convergence rate after a few iteration. For example, at 50
th
iteration, the step size is only
0.02, which means the only 2% of the 50
th
estimation is used to update the global solution.
Investigation for a more efficient algorithm will possibly provide a better alternative.

175
To demonstrate the usability, the model should be applied to a calibrated dataset,
and validated against actual statistics. This study only tested whether the proposed model
can handle large data sets, without too much consideration on whether the model can
replicate the reality in the modeling world. To be used as a robust planning analysis tool, the
model should be evaluated with calibrated data.

176
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181
APPENDIX A. PARAMETERS FOR “LINEAR-CITY” APPLICATION

Household
A.1 Parameters for Households
1 2 3
Number of households(
h
Θ , Equation 3.11)
4,000 10,000 26,000
Dispersion factor (
h
ψ , Equation 3.11) 2.0 1.0 3.0
Commodity1 0.127 0.192 0.206
Commodity2 0.075 0.077 0.124
Commodity3 0.198 0.161 0.190
Leisure 0.100 0.120 0.150
Housing 0.350 0.300 0.180
Utility coefficient
(
ih
pk
α ,
h
β ,
h
δ ,
h
φ ,
Equation 3.1)
Amenity 0.150 0.150 0.150
Commodity1 0.3 0.2 0.1
Commodity2 1.0 1.0 1.0
Shopping preference to level of
output (
ih
η , Equation 3.8)
Commodity3 1.5 1.2 0.8
Commodity1 -1.0 -0.5 0.0
Commodity2 -0.5 -0.5 -0.5
Shopping preference to effective
price (
ih
µ , Equation 3.8)
Commodity3 -0.5 -0.5 1.5
Industry
A.2 Parameters for Firms
1 2 3
Elasticity of Substitution (
j
θ , Equation 3.17)
1.20 0.60 2.00
Tax rate (
j
q
τ , Equation 3.16) 8.0% 8.0% 8.0%
Commodity1 0.032 0.017 0.082
Commodity2 0.048 0.028 0.110
Commodity3 0.184 0.297 0.140
Labor1 0.088 0.175 0.075
Labor2 0.233 0.156 0.156
Labor3 0.128 0.091 0.082
Input coefficients
j hj ij
σ ρ ϕ , , , Equation 3.17
Space/Land 0.287 0.236 0.355
Commodity1 2.00 -0.50 -0.50
Commodity2 -0.50 -2.00 -1.00
Distance Decay
(
ij
µ , Equation 3.24)
Commodity3 -0.10 -0.10 0.00

182

Developer
A.3 Parameters for Developers and Local
Governments
Residential Industrial
Local
Government
Elasticity of Substitution(
j
θ )
0.80 1.20 0.70
Tax rate (
j
q
τ ) 8.0% 8.0% -
Commodity1 0.068 0.136 0.024
Commodity2 0.042 0.084 0.082
Commodity3 0.079 0.157 0.130
Labor1 0.071 0.035 0.075
Labor2 0.111 0.082 0.271
Labor3 0.174 0.101 0.193
Input coefficients
j hj ij
σ ρ ϕ , ,
Space/Land 0.455 0.405 0.225
Commodity1 -1.0 -0.1 - ∞
Commodity2 -1.0 -0.1 - ∞ Distance Decay (
ij
µ )
Commodity3 -0.1 -1.0 - ∞
A.4 Parameters for Transportation
Market
Trip per quantity of
intermediate input
Trip per quantity of
consumption
Commodity 1 1/40 1/8
Commodity 2 1/20 1/4
Frequency of shopping /
delivery trip
( ζ
i
, Equation 3.2)
( ξ
i
, Equation 3.16)
Commodity 3 1/200 1/20
Trip production rate Return trip rate
Working trips, γ
w
0.422 0.021
Shopping trips, γ
s
0.284 0.063
Peak rate
(Equation 3.42)
Delivery by truck, γ
t
0.259 0.029

183
APPENDIX B. PARAMETERS FOR SCAG APPLICATION

Household
B.1 Parameters for Households
1 2 3
Number of households(
h
Θ , Equation 3.11)
860,000 176,000 2580,000
Dispersion factor (
h
ψ , Equation 3.11) 2.0 0.5 1.0
Commodity1 0.127 0.192 0.190
Commodity2 0.075 0.077 0.124
Commodity3 0.198 0.161 0.206
Leisure 0.050 0.120 0.200
Housing 0.350 0.300 0.180
Utility coefficient
(
ih
pk
α ,
h
β ,
h
δ ,
h
φ ,
Equation 3.1)
Amenity 0.200 0.150 0.100
Commodity1 0.3 0.1 0.2
Commodity2 0.5 1.0 0.8
Shopping preference to level of
output (
ih
η , Equation 3.8)
Commodity3 0.3 0.1 0.2
Commodity1 -0.2 -0.8 -0.5
Commodity2 -0.4 -0.6 -0.8
Shopping preference to effective
price (
ih
µ , Equation 3.8)
Commodity3 -0.2 -1.0 -0.1
Industry
B.2 Parameters for Firms
1 2 3
Elasticity of Substitution (
j
θ , Equation 3.17)
1.50 0.70 2.00
Tax rate (
j
q
τ , Equation 3.16) 8.0% 8.0% 8.0%
Commodity1 0.084 0.087 0.113
Commodity2 0.066 0.047 0.095
Commodity3 0.144 0.208 0.158
Labor1 0.186 0.156 0.131
Labor2 0.155 0.165 0.142
Labor3 0.108 0.101 0.096
Input coefficients
j hj ij
σ ρ ϕ , , , Equation 3.17
Space/Land 0.257 0.236 0.265
Commodity1 -1.50 -0.25 -0.25
Commodity2 -0.25 -2.00 -0.25
Distance Decay
(
ij
µ , Equation 3.24)
Commodity3 -0.25 -0.25 -1.00

184

Developer
B.3 Parameters for Developers and Local
Governments
Residential Industrial
Local
Government
Elasticity of Substitution(
j
θ )
0.80 1.20 0.70
Tax rate (
j
q
τ ) 8.0% 8.0% -
Commodity1 0.098 0.103 0.024
Commodity2 0.075 0.084 0.082
Commodity3 0.112 0.152 0.130
Labor1 0.091 0.085 0.155
Labor2 0.102 0.095 0.193
Labor3 0.137 0.125 0.201
Input coefficients
j hj ij
σ ρ ϕ , ,
Space/Land 0.385 0.356 0.215
Commodity1 -0.20 -0.20 - ∞
Commodity2 -0.20 -0.20 - ∞ Distance Decay (
ij
µ )
Commodity3 -0.20 -0.20 - ∞
B.4 Parameters for Transportation
Market
Trip per quantity of
intermediate input
Trip per quantity of
consumption
Commodity 1 1/80 1/16
Commodity 2 1/200 1/40
Frequency of shopping /
delivery trip
( ζ
i
, Equation 3.2)
( ξ
i
, Equation 3.16)
Commodity 3 1/40 1/8
Trip production rate Return trip rate
Working trips, γ
w
0.422 0.021
Shopping trips, γ
s
0.284 0.063
Peak rate
(Equation 3.42)
Delivery by truck, γ
t
0.259 0.029

185

APPENDIX C. PRICE WITH ELASTICITY OF SUBSTITUTION,
j
θ =1

The equilibrium price of the commodity i from zone q is
() () ()
j
j j j
i
v
q
j
h
h
q
hj
r
rq
i i
r
ij
rq
j
q
j
q
r w c p
p
θ
θ θ θ
σ ρ
ξ
ϕ
τ
−
− − −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
⋅
−
=
∑∑ ∑
1
1
1 1 1
1
1

with 1 = + +
∑ ∑∑
j
h
hj
ir
ij
rq
σ ρ ϕ .
Due to the power, 1/(
j
θ − 1 ), the price equation is not defined when
j
θ =1. To avoid this
condition, both sides of equality are modified with logarithm.

( ) = − + ∴ ) 1 log( log
j
q
j
q
p τ
() () ()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
⋅
−
∑∑ ∑
− − −
i
v
q
j
h
h
q
hj
r
rq
i i
r
ij
rq
j
j j j
r w c p
1 1 1
log
1
1
θ θ θ
σ ρ
ξ
ϕ
θ
····················(C.1)

When
j
θ Æ 1, the denominator of equation (C.1),
j
θ − 1 =0. Also the numerator approaches
to 0,
() () ()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
∑∑ ∑
− − −
→
i
v
q
j
h
h
q
hj
r
rq
i i
r
ij
rq
j j j
j
r w c p
1 1 1
1
log lim
θ θ θ
θ
σ ρ
ξ
ϕ

=
⎥
⎦
⎤
⎢
⎣
⎡
+ +
∑∑ ∑
i
j
h
hj
r
ij
rq
σ ρ ϕ log
= 0 .

The L’Hospital’s rule can be applied to the log of price equation. The rule specifies that

186
) (
) (
lim
) (
) (
lim
0 0
x g
x f
x g
x f
x x x x
′
′
=
→ →
when 0 ) ( lim
0
=
→
x f
x x
and 0 ) ( lim
0
=
→
x g
x x
for continuous f(x) and
g(x). Both of denominator and numerator approach to 0 when
j
θ goes to 1.
( ) [ ] ) 1 log( log lim
1
j
q
j
q
p
j
τ
θ
− +
→
=
() () ()
()
′
−
′
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
∑∑ ∑
− − −
→
j
i
v
q
j
h
h
q
hj
r
rq
i i
r
ij
rq
j j j
j
r w c p
θ
σ ρ
ξ
ϕ
θ θ θ
θ
1
log
lim
1 1 1
1
··················(C.2)

The derivative of numerator with respect to
j
θ is,
() () ()
′
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ +
⋅ +
∑∑ ∑
− − −
i
v
q
j
h
h
q
hj
r
rq
i i
r
ij
rq
j j j
r w c p
1 1 1
log
θ θ θ
σ ρ
ξ
ϕ
=

–
() () ()
1
1 1 1
−
− − −
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+ +
⋅ +
∑∑ ∑
i
v
q
j
h
h
q
hj
r
rq
i i
r
ij
rq
j j j
r w c p
θ θ θ
σ ρ
ξ
ϕ

×
()
()
( )
()
()
()
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⋅
+
⋅
+
⋅ +
⋅ + ⋅
∑∑ ∑
− − −
i
v
q
v
q
j
h
h
q
h
q
hj
r
rq
i i
r
rq
i i
r
ij
rq
j j j
r
r
w
w
c p
c p
1 1 1
log log log
θ θ θ
σ ρ
ξ
ξ ϕ
.

And when
j
θ approaches to 1, the derivative of numerator is
–
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+ +
∑∑ ∑
i
j
h
hj
r
ij
rq
σ ρ ϕ
1

× () () () ⎟
⎠
⎞
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
⋅ +
∑∑ ∑
i
v
q
h
h
q
r
rq
i i
r
j hj ij
rq
r w c p
σ ρ ϕ
ξ log log log

= – () () ()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅ +
∏ ∏∏
j hj ij
rq v
q
h
h
q
ir
rq
i i
r
r w c p
σ ρ ϕ
ξ log .

187
The derivative of denominator of Equation (C.2) with respect to
j
θ is – 1. Then, according
to the L’Hospital’s rule, Equation (C.2) turns as follows,

( ) ) 1 log( log
j
q
j
q
p τ − +
= () () ()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅ +
∏ ∏∏
j hj ij
rq v
q
h
h
q
ir
rq
i i
r
r w c p
σ ρ ϕ
ξ log .

Therefore, when elasticity of substitution is
j
θ Æ 1, the price of commodity is
j
q
p = () () ()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅ + ⋅
−
∏ ∏∏
j hj ij
rq v
q
h
h
q
ir
rq
i i
r j
q
r w c p
σ ρ ϕ
ξ
τ 1
1
··············(C.3)

188
APPENDIX D. SCAG BASELINE
D.1 Land Use

Residential Land Rent

Industrial Land Rent

Public land rent

$ / 10 Sq-Yd
594.9 to 821.5
73.7 to 594.9
41.7 to 73.7
27.1 to 41.7
18.2 to 27.1
12.9 to 18.2
3.5 to 12.9
$ / 10 Sq-Yd
131.7 to 175.4
93.0 to 131.7
62.6 to 93.0
44.7 to 62.6
30.3 to 44.7
19.1 to 30.3
7.8 to 19.1
$ / 10 Sq-Yd
247.5 to 423.0
157.6 to 247.5
71.9 to 157.6
42.2 to 71.9
25.1 to 42.2
11.1 to 25.1
0.0 to 11.1

189

Land Use for Residential Structures

Land Use for Industrial Structures

Land Use for Local Government

Share (%)
85.5 to 88.8
77.9 to 85.5
68.7 to 77.9
59.2 to 68.7
45.2 to 59.2
19.6 to 45.2
4.5 to 19.6
Share (%)
40.7 to 74.9
26.6 to 40.7
19.4 to 26.6
14.2 to 19.4
9.1 to 14.2
7.0 to 9.1
6.6 to 7.0
Share (%)
83.7 to 88.2
41.8 to 83.7
21.5 to 41.8
13.5 to 21.5
9.5 to 13.5
6.5 to 9.5
3.8 to 6.5

190

Housing Rent

Office Rent

Housing Floor Space

$ / 10 Sq-Yd
80.04 to 96.53
28.49 to 80.04
21.30 to 28.49
17.51 to 21.30
14.71 to 17.51
12.73 to 14.71
8.03 to 12.73
$ / 10 Sq-Yd
23.11 to 25.31
20.81 to 23.11
19.10 to 20.81
17.55 to 19.10
15.76 to 17.55
14.12 to 15.76
11.58 to 14.12
10,000 Sq-Yd
1,520 to 1,904
1,282 to 1,520
1,031 to 1,282
793 to 1,031
601 to 793
400 to 601
73 to 400

191

Office Floor Space

D.2 Households

Amenity

Utility – Type 1 Household

10,000 Sq-Yd
904 to 1,332
673 to 904
499 to 673
396 to 499
308 to 396
239 to 308
103 to 239
per Household
30.9 to 62.1
23.8 to 30.9
21.0 to 23.8
18.8 to 21.0
15.7 to 18.8
12.3 to 15.7
7.9 to 12.3
per Household
5.18 to 5.31
5.10 to 5.18
5.02 to 5.10
4.94 to 5.02
4.87 to 4.94
4.75 to 4.87
4.40 to 4.75

192

Utility – Type 2 Household

Utility – Type 3 Household

Labor Income – Type 1 Household

per Household
5.03 to 5.12
4.94 to 5.03
4.84 to 4.94
4.76 to 4.84
4.69 to 4.76
4.55 to 4.69
4.24 to 4.55
per Household
5.17 to 5.33
5.10 to 5.17
5.02 to 5.10
4.95 to 5.02
4.88 to 4.95
4.79 to 4.88
4.56 to 4.79
$ / Month
2,173 to 2,221
2,110 to 2,173
2,018 to 2,110
1,904 to 2,018
1,767 to 1,904
1,634 to 1,767
1,429 to 1,634

193

Labor income – Type 2 Household

Labor income – Typ3 Household

Average Labor Income Per Household

$ / Month
1,407 to 1,440
1,366 to 1,407
1,314 to 1,366
1,242 to 1,314
1,134 to 1,242
976 to 1,134
811 to 976
$ / Month
1,221 to 1,251
1,184 to 1,221
1,141 to 1,184
1,075 to 1,141
986 to 1,075
866 to 986
724 to 866
$ / Month
1,464 to 1,511
1,422 to 1,464
1,359 to 1,422
1,279 to 1,359
1,163 to 1,279
1,008 to 1,163
834 to 1,008

194

Number Of Zonal Households – Type 1

Number Of Zonal Households – Type 2

Number Of Zonal Households – Type 3

Households
4,836 to 5,497
4,219 to 4,836
3,507 to 4,219
2,878 to 3,507
2,321 to 2,878
1,582 to 2,321
394 to 1,582
Households
7,396 to 7,673
7,174 to 7,396
6,806 to 7,174
6,465 to 6,806
6,034 to 6,465
5,190 to 6,034
628 to 5,190
Households
11,794 to 12,550
11,225 to 11,794
10,359 to 11,225
9,397 to 10,359
8,206 to 9,397
6,296 to 8,206
1,063 to 6,296

195

Number Of Households – All Type

D.3 Industrial Activity

Wage – Type 1 Labor

Wage – Type 2 Labor

Households
24,172 to 25,719
22,839 to 24,172
20,552 to 22,839
18,496 to 20,552
16,631 to 18,496
13,001 to 16,631
2,087 to 13,001
$ / Hour
17.70 to 19.10
12.66 to 17.70
11.93 to 12.66
11.54 to 11.93
11.19 to 11.54
10.80 to 11.19
10.20 to 10.80
$ / Hour
12.18 to 19.41
9.27 to 12.18
8.66 to 9.27
8.22 to 8.66
7.89 to 8.22
7.57 to 7.89
7.21 to 7.57

196

Wage – Type 3 Labor

Number Of Workers – Type 1 Labor

Number Of Workers – Type 2 Labor

$ / Hour
16.19 to 30.52
9.88 to 16.19
9.21 to 9.88
8.78 to 9.21
8.42 to 8.78
8.06 to 8.42
7.65 to 8.06
Workers
4,005 to 4,399
3,652 to 4,005
3,296 to 3,652
3,028 to 3,296
2,678 to 3,028
2,204 to 2,678
576 to 2,204
Workers
7,215 to 7,575
6,986 to 7,215
6,700 to 6,986
6,381 to 6,700
5,839 to 6,381
4,664 to 5,839
805 to 4,664

197

Number Of Workers – Type 3 Labor

Number Of Workers – All Types

Price – Commodity 1

$ / Unit
15.63 to 18.78
13.45 to 15.63
12.96 to 13.45
12.75 to 12.96
12.61 to 12.75
12.48 to 12.61
12.28 to 12.48
Workers
11,480 to 12,257
10,847 to 11,480
10,169 to 10,847
9,330 to 10,169
8,066 to 9,330
6,253 to 8,066
1,814 to 6,253
Workers
22,746 to 24,231
21,587 to 22,746
20,289 to 21,587
18,888 to 20,289
16,658 to 18,888
12,960 to 16,658
3,196 to 12,960

198

Price – Commodity 2

Price – Commodity 3

Output Quantity – Commodity 1

$ / Unit
13.62 to 19.22
13.47 to 13.62
13.17 to 13.47
12.97 to 13.17
12.79 to 12.97
12.58 to 12.79
12.36 to 12.58
$ / Unit
13.89 to 17.46
12.96 to 13.89
12.74 to 12.96
12.62 to 12.74
12.56 to 12.62
12.47 to 12.56
12.30 to 12.47
1,000 Units
1,507 to 1,753
1,383 to 1,507
1,247 to 1,383
1,121 to 1,247
944 to 1,121
707 to 944
71 to 707

199

Output Quantity – Commodity 2

Output Quantity – Commodity 3

Output Quantity – Total

1,000 Units
1,318 to 1,838
1,044 to 1,318
828 to 1,044
690 to 828
547 to 690
396 to 547
62 to 396
1,000 Units
1,783 to 2,021
1,652 to 1,783
1,526 to 1,652
1,392 to 1,526
1,227 to 1,392
921 to 1,227
201 to 921
1,000 Units
4,557 to 5,610
3,980 to 4,557
3,555 to 3,980
3,115 to 3,555
2,700 to 3,115
2,107 to 2,700
335 to 2,107

200

Output Value – Commodity 1

Output Value – Commodity 2

Output Value – Commodity 3

$M
16.3 to 22.8
12.5 to 16.3
9.9 to 12.5
7.9 to 9.9
6.5 to 7.9
5.1 to 6.5
1.1 to 5.1
$M
18.7 to 21.6
17.4 to 18.7
15.7 to 17.4
14.1 to 15.7
12.2 to 14.1
9.4 to 12.2
1.3 to 9.4
$M
22.2 to 24.9
20.7 to 22.2
19.3 to 20.7
17.7 to 19.3
15.7 to 17.7
12.2 to 15.7
3.5 to 12.2

201

Output Value – Total

D.4 Transportation

Trip Production – Working Trip

Trip Attraction – Working Trip

$M
56.6 to 69.1
50.1 to 56.6
45.1 to 50.1
39.6 to 45.1
34.9 to 39.6
28.0 to 34.9
6.0 to 28.0
PCE / 4 Hours
10,696 to 11,338
10,045 to 10,696
8,976 to 10,045
8,157 to 8,976
7,396 to 8,157
5,779 to 7,396
947 to 5,779
PCE / 4 Hours
10,069 to 10,700
9,518 to 10,069
8,877 to 9,518
8,205 to 8,877
7,269 to 8,205
5,704 to 7,269
1,392 to 5,704

202

Trip Production – Shopping Trip

Trip Attraction – Shopping Trip

Trip Production – Freight

PCE / 4Hours
3,403 to 3,678
3,071 to 3,403
2,645 to 3,071
2,268 to 2,645
1,821 to 2,268
1,252 to 1,821
272 to 1,252
PCE / 4Hours
3,389 to 3,873
3,025 to 3,389
2,621 to 3,025
2,291 to 2,621
1,865 to 2,291
1,343 to 1,865
235 to 1,343
PCE / 4Hours
505 to 583
465 to 505
430 to 465
396 to 430
362 to 396
287 to 362
61 to 287

203

Trip Attraction – Freight

Trip Production – Total Trip

Trip Attraction - Total Trip

PCE / 4Hours
14,583 to 15,510
13,696 to 14,583
12,177 to 13,696
10,888 to 12,177
9,538 to 10,888
7,359 to 9,538
1,281 to 7,359
PCE / 4Hours
14,053 to 15,174
13,209 to 14,053
12,194 to 13,209
11,381 to 12,194
10,273 to 11,381
8,000 to 10,273
1,691 to 8,000
PCE / 4Hours
553 to 637
497 to 553
436 to 497
376 to 436
322 to 376
249 to 322
64 to 249

204

Average Egress Travel Time

Average Access Travel Time

Average Egress Travel Cost

Hours / Trip
2.49 to 3.05
2.08 to 2.49
1.74 to 2.08
1.54 to 1.74
1.43 to 1.54
1.31 to 1.43
1.21 to 1.31
Hours / Trip
2.91 to 3.43
2.19 to 2.91
1.83 to 2.19
1.58 to 1.83
1.37 to 1.58
1.24 to 1.37
1.14 to 1.24
$ / Trip
10.78 to 13.84
9.31 to 10.78
8.27 to 9.31
7.53 to 8.27
6.99 to 7.53
6.45 to 6.99
5.9 to 6.45

205

Average Access Travel Cost

Travel Time For Traversing The Zone

$ / Trip
11.25 to 13.91
9.56 to 11.25
8.25 to 9.56
7.44 to 8.25
6.78 to 7.44
6.28 to 6.78
5.79 to 6.28
Minutes
23.1 to 61.0
12.5 to 23.1
7.4 to 12.5
3.8 to 7.4
1.9 to 3.8
0.8 to 1.9
0.1 to 0.8
206

Appendix E. RESULTS FROM SCAG POLICY OPTION ANALYSES

E.1 Land Use – Housing Development Option

Model Estimation Ratio to Baseline (%)
Household Household 1 Household 2 Household 3 Sum / Avg Household 1 Household 2 Household 3 Sum / Avg
Utility / household 5.058 5.046 4.868 4.959 0.164 0.202 0.161 0.182
Wage ($/hour) 11.635 8.720 8.327 9.009 0.106 0.067 0.072 0.078
Income ($ /household) 2441.0 1537.0 1343.5 1590.9 -0.166 -0.299 -0.484 -0.343
Weekly work hour 43.56 38.09 31.96 35.93 -0.172 -0.278 -0.458 -0.337
Production Industry 1 Industry 2 Industry 3 Sum / Avg Industry 1 Industry 2 Industry 3 Sum / Avg
Output (million units) 309.8 213.9 376.8 900.6 0.654 0.655 0.621 0.640
Output ($ M) 3,934 2,763 4,782 11,478 0.599 0.582 0.569 0.583
Price ($ / unit) 12.70 12.91 12.69 12.75 -0.054 -0.073 -0.051 -0.057
Land Use Residential Industrial Public Sum / Avg Residential Industrial Public Sum / Avg
Land share (%) 63.47 15.50 21.03 100.00 0.197 -0.344 -0.086 -
Land rent ($ /sq-yd) 19.91 26.26 14.04 19.79 -0.828 -0.548 -1.131 -0.821
Floor space (M Sqyd) 125.83 66.59 192.42 1.469 0.524 1.140
Floor rent ($ / sq-yd) 15.26 16.58 15.72 -0.794 -0.301 -0.631

207
E.1 Land use – Housing Development Option (continued)

Trips Working Shopping Freight Sum / Avg Working Shopping Freight Sum / Avg
Trip (PCU / 4Hour) 2,285,880 652,862 110,494 3,049,236 0.000 0.662 0.612 0.163
VMT (Million mile) 208.71 8.50 217.21 0.036 0.180 0.042
VTT (1000 Hour) 4,960.9 200.5 5,161.5 0.006 0.148 0.012
Avg Distance (Miles) 71.022 76.923 71.236 -0.110 -0.429 -0.121
Avg Time (minutes) 101.3 108.9 101.6 -0.140 -0.461 -0.151
Avg Speed (MPH) 42.07 42.38 42.08 0.030 0.032 0.030

208
E.2 Land Use – Downtown Development Option
Model Estimation Ratio to Baseline (%)
Household Household 1 Household 2 Household 3 Sum / Avg Household 1 Household 2 Household 3 Sum / Avg
Utility / household 5.049 5.038 4.859 4.950 -0.004 0.004 0.009 0.004
Wage ($/hour) 11.623 8.713 8.319 9.001 -0.003 -0.016 -0.017 -0.014
Income ($ /household) 2444.4 1541.1 1349.4 1595.8 -0.027 -0.033 -0.047 -0.037
Weekly work hour 43.63 38.19 32.09 36.05 -0.021 -0.010 -0.025 -0.019
Production Industry 1 Industry 2 Industry 3 Sum / Avg Industry 1 Industry 2 Industry 3 Sum / Avg
Output (million units) 308.1 212.8 374.9 895.8 0.105 0.118 0.112 0.111
Output ($ M) 3,911 2,747 4,756 11,414 0.020 0.026 0.021 0.022
Price ($ / unit) 12.69 12.91 12.68 12.74 -0.085 -0.092 -0.090 -0.089
Land Use Residential Industrial Public Sum / Avg Residential Industrial Public Sum / Avg
Land share (%) 63.35 15.58 21.07 100.00 -0.062 0.087 0.033 -
Land rent ($ /sq-yd) 20.20 25.82 14.19 19.94 0.604 -2.206 -0.131 -0.078
Floor space (M Sqyd) 123.75 66.67 190.42 -0.213 0.647 0.086
Floor rent ($ / sq-yd) 15.42 16.56 15.82 0.233 -0.457 -0.006
Trips Working Shopping Freight Sum / Avg Working Shopping Freight Sum / Avg
Trip (PCU / 4Hour) 2,285,880 649,262 109,947 3,045,089 0.000 0.107 0.114 0.027
VMT (Million mile) 208.66 8.49 217.14 0.008 0.014 0.008
VTT (1000 Hour) 4,960.8 200.2 5,161.1 0.004 0.000 0.004
Avg Distance (Miles) 71.089 77.177 71.309 -0.015 -0.100 -0.018
Avg Time (minutes) 101.4 109.3 101.7 -0.019 -0.113 -0.023
Avg Speed (MPH) 42.06 42.37 42.07 0.004 0.014 0.004
209
E.3 Land Use – No Land Use Plan Option
Model Estimation Ratio to Baseline (%)
Household Household 1 Household 2 Household 3 Sum / Avg Household 1 Household 2 Household 3 Sum / Avg
Utility / household 5.243 5.154 4.978 5.081 3.827 2.465 2.311 2.644
Wage ($/hour) 17.545 13.145 12.069 13.340 50.947 50.849 45.047 48.189
Income ($ /household) 3389.1 2124.0 1867.9 2206.8 38.611 37.778 38.359 38.236
Weekly work hour 42.18 38.23 31.64 35.59 -3.334 0.097 -1.452 -1.285
Production Industry 1 Industry 2 Industry 3 Sum / Avg Industry 1 Industry 2 Industry 3 Sum / Avg
Output (million units) 349.4 245.1 430.1 1024.5 13.502 15.304 14.853 14.496
Output ($ M) 5,817 4,100 7,120 17,036 48.751 49.265 49.744 49.288
Price ($ / unit) 16.65 16.73 16.55 16.63 31.056 29.453 30.378 30.388
Land Use Residential Industrial Public Sum / Avg Residential Industrial Public Sum / Avg
Land share (%) 53.68 33.90 12.42 100.00 -9.588 82.514 -43.677 -
Land rent ($ /sq-yd) 22.02 17.96 22.76 20.74 9.695 -31.957 60.213 3.919
Floor space (M Sqyd) 147.57 98.66 246.23 19.001 48.934 29.424
Floor rent ($ / sq-yd) 19.20 18.24 18.81 24.767 9.634 18.909
Trips Working Shopping Freight Sum / Avg Working Shopping Freight Sum / Avg
Trip (PCU / 4Hour) 2,285,880 733,042 127,470 3,146,392 0.000 13.025 16.069 3.355
VMT (Million mile) 214.59 10.20 224.79 2.852 20.281 3.533
VTT (1000 Hour) 5,089.9 241.3 5,331.3 2.607 20.517 3.302
Avg Distance (Miles) 71.081 80.058 71.445 -0.026 3.629 0.172
Avg Time (minutes) 101.2 113.6 101.7 -0.264 3.832 -0.051
Avg Speed (MPH) 42.16 42.29 42.17 0.239 -0.195 0.224
210
E.4 Unexpected Event – Coastal Zone
Model Estimation Ratio to Baseline (%)
Household Household 1 Household 2 Household 3 Sum / Avg Household 1 Household 2 Household 3 Sum / Avg
Utility / household 5.048 5.037 4.857 4.949 -0.037 -0.035 -0.022 -0.031
Wage ($/hour) 11.624 8.714 8.320 9.002 0.005 0.001 -0.003 0.000
Income ($ /household) 2459.1 1550.6 1357.8 1605.6 0.578 0.580 0.577 0.578
Weekly work hour 43.62 38.18 32.09 36.04 -0.039 -0.024 -0.034 -0.031
Production Industry 1 Industry 2 Industry 3 Sum / Avg Industry 1 Industry 2 Industry 3 Sum / Avg
Output (million units) 307.6 212.4 374.2 894.2 -0.060 -0.081 -0.063 -0.066
Output ($ M) 3910 2746 4754 11410 -0.007 -0.013 -0.010 -0.010
Price ($ / unit) 12.71 12.93 12.70 12.76 0.054 0.068 0.052 0.057
Land Use Residential Industrial Public Sum / Avg Residential Industrial Public Sum / Avg
Land share (%) 63.36 15.53 21.10 100.00 0.019 -0.118 0.241 -
Land rent ($ /sq-yd) 20.08 26.40 14.20 19.96 3.317 2.767 5.383 3.511
Floor space (M Sqyd) 124.01 66.24 190.25 -0.291 -0.396 -0.327
Floor rent ($ / sq-yd) 15.39 16.63 15.82 0.295 0.290 0.292
Trips Working Shopping Freight Sum / Avg Working Shopping Freight Sum / Avg
Trip (PCU / 4Hour) 2,285,880 648,265 109,730 3,043,876 0.000 -0.047 -0.084 -0.013
VMT (Million mile) 208.66 8.48 217.15 0.012 -0.034 0.010
VTT (1000 Hour) 4,962.8 200.3 5,163.0 0.043 0.007 0.042
Avg Distance (Miles) 71.116 77.293 71.339 0.023 0.050 0.023
Avg Time (minutes) 101.5 109.5 101.8 0.054 0.090 0.055
Avg Speed (MPH) 42.05 42.35 42.06 -0.031 -0.041 -0.032
211
E.5 Unexpected Event – Puente-Hill Earthquake Scenario
Model Estimation Ratio to Baseline (%)
Household Household 1 Household 2 Household 3 Sum / Avg Household 1 Household 2 Household 3 Sum / Avg
Utility / household 4.994 5.008 4.815 4.909 -1.096 -0.896 -0.599 -0.829
Wage ($/hour) 11.644 8.749 8.360 9.037 0.177 0.401 0.473 0.386
Income ($ /household) 2537.4 1601.0 1402.2 1657.6 3.778 3.850 3.860 3.836
Weekly work hour 43.58 38.05 31.98 35.93 -0.119 -0.382 -0.396 -0.335
Production Industry 1 Industry 2 Industry 3 Sum / Avg Industry 1 Industry 2 Industry 3 Sum / Avg
Output (million units) 302.2 207.8 367.8 877.8 -1.835 -2.240 -1.771 -1.904
Output ($ M) 3,940 2,762 4,787 11,489 0.751 0.570 0.689 0.682
Price ($ / unit) 13.04 13.29 13.01 13.09 2.634 2.874 2.504 2.636
Land Use Residential Industrial Public Sum / Avg Residential Industrial Public Sum / Avg
Land share (%) 63.53 15.29 21.18 100.00 0.621 -1.342 0.978 -
Land rent ($ /sq-yd) 24.53 31.44 17.70 24.20 22.171 19.087 24.599 21.287
Floor space (M Sqyd) 115.98 59.35 175.33 -6.474 -10.409 -7.844
Floor rent ($ / sq-yd) 16.58 18.07 17.08 7.749 8.628 7.979
Trips Working Shopping Freight Sum / Avg Working Shopping Freight Sum / Avg
Trip (PCU / 4Hour) 2,285,880 639,821 107,170 3,032,871 0.000 -1.349 -2.415 -0.374
VMT (Million mile) 209.07 8.28 217.34 0.207 -2.466 0.102
VTT (1000 Hour) 4,984.2 195.1 5,179.4 0.476 -2.559 0.358
Avg Distance (Miles) 71.460 77.215 71.663 0.506 -0.051 0.478
Avg Time (minutes) 102.2 109.2 102.5 0.776 -0.147 0.735
Avg Speed (MPH) 41.95 42.41 41.96 -0.268 0.096 -0.255
212
E.6 Transportation Investment - Speed 5% Increase Option
Model Estimation Ratio to Baseline (%)
Household Household 1 Household 2 Household 3 Sum / Avg Household 1 Household 2 Household 3 Sum / Avg
Utility / household 5.043 5.030 4.849 4.942 -0.125 -0.203 -0.158 -0.174
Wage ($/hour) 11.378 8.523 8.159 8.817 -2.109 -2.198 -1.938 -2.059
Income ($ /household) 2429.2 1536.0 1349.8 1591.7 -0.647 -0.364 -0.022 -0.292
Weekly work hour 44.32 38.84 32.81 36.74 1.570 1.684 2.209 1.894
Production Industry 1 Industry 2 Industry 3 Sum / Avg Industry 1 Industry 2 Industry 3 Sum / Avg
Output (million units) 305.4 210.6 371.8 887.8 -0.780 -0.901 -0.714 -0.781
Output ($ M) 3,830 2,688 4,661 11,180 -2.045 -2.118 -1.971 -2.032
Price ($ / unit) 12.54 12.76 12.54 12.59 -1.275 -1.228 -1.266 -1.260
Land Use Residential Industrial Public Sum / Avg Residential Industrial Public Sum / Avg
Land share (%) 63.38 15.56 21.06 100.00 - - - -
Land rent ($ /sq-yd) 19.96 26.30 14.14 19.84 -0.598 -0.385 -0.482 -0.561
Floor space (M Sqyd) 122.62 65.47 188.09 -1.117 -1.175 -1.137
Floor rent ($ / sq-yd) 15.22 16.45 15.65 -1.059 -1.123 -1.083
Trips Working Shopping Freight Sum / Avg Working Shopping Freight Sum / Avg
Trip (PCU / 4Hour) 2,285,880 644,512 108,862 3,039,254 0.000 -0.626 -0.874 -0.165
VMT (Million mile) 210.57 8.53 219.10 0.923 0.547 0.909
VTT (1000 Hour) 4,828.1 193.8 5,021.9 -2.671 -3.244 -2.693
Avg Distance (Miles) 71.856 78.362 72.089 1.063 1.434 1.075
Avg Time (minutes) 98.9 106.8 99.1 -2.536 -2.391 -2.533
Avg Speed (MPH) 43.61 44.03 43.63 3.693 3.918 3.702
213
E.7 Transportation Investment - Capacity 5% Increase Option
Model Estimation Ratio to Baseline (%)
Household Household 1 Household 2 Household 3 Sum / Avg Household 1 Household 2 Household 3 Sum / Avg
Utility / household 5.053 5.037 4.857 4.949 0.076 -0.042 -0.021 -0.014
Wage ($/hour) 11.916 8.950 8.556 9.248 2.519 2.712 2.836 2.728
Income ($ /household) 2494.0 1573.0 1381.6 1630.8 2.005 2.038 2.334 2.154
Weekly work hour 43.68 38.10 32.13 36.05 0.092 -0.249 0.100 -0.025
Production Industry 1 Industry 2 Industry 3 Sum / Avg Industry 1 Industry 2 Industry 3 Sum / Avg
Output (million units) 308.1 213.0 375.1 896.3 0.104 0.221 0.172 0.161
Output ($ M) 3,984 2,798 4,846 11,627 1.880 1.859 1.920 1.892
Price ($ / unit) 12.93 13.13 12.92 12.97 1.774 1.634 1.745 1.728
Land Use Residential Industrial Public Sum / Avg Residential Industrial Public Sum / Avg
Land share (%) 63.38 15.56 21.06 100.00 - - - -
Land rent ($ /sq-yd) 20.01 26.22 14.12 19.88 -0.349 -0.675 -0.586 -0.381
Floor space (M Sqyd) 125.00 67.18 192.18 0.801 1.413 1.014
Floor rent ($ / sq-yd) 15.53 16.78 15.97 0.953 0.903 0.946
Trips Working Shopping Freight Sum / Avg Working Shopping Freight Sum / Avg
Trip (PCU / 4Hour) 2,285,880 649,342 110,064 3,045,286 0.000 0.119 0.220 0.033
VMT (Million mile) 211.42 8.63 220.05 1.335 1.720 1.350
VTT (1000 Hour) 5,004.0 202.0 5,205.9 0.873 0.861 0.873
Avg Distance (Miles) 72.030 78.411 72.261 1.308 1.497 1.316
Avg Time (minutes) 102.3 110.1 102.6 0.847 0.640 0.839
Avg Speed (MPH) 42.25 42.73 42.27 0.458 0.852 0.473
214
E.8 Transportation Investment – Completion of I-710
Model Estimation Ratio to Baseline (%)
Household Household 1 Household 2 Household 3 Sum / Avg Household 1 Household 2 Household 3 Sum / Avg
Utility / household 5.050 5.038 4.859 4.951 0.013 0.015 0.014 0.014
Wage ($/hour) 11.628 8.718 8.324 9.006 0.040 0.048 0.044 0.045
Income ($ /household) 2445.7 1541.9 1350.1 1596.6 0.027 0.016 0.003 0.013
Weekly work hour 43.63 38.18 32.09 36.04 -0.011 -0.026 -0.044 -0.031
Production Industry 1 Industry 2 Industry 3 Sum / Avg Industry 1 Industry 2 Industry 3 Sum / Avg
Output (million units) 308.0 212.7 374.8 895.5 0.078 0.080 0.076 0.077
Output ($ M) 3,914 2,749 4,758 11,421 0.084 0.083 0.081 0.083
Price ($ / unit) 12.71 12.92 12.70 12.75 0.006 0.003 0.006 0.005
Land Use Residential Industrial Public Sum / Avg Residential Industrial Public Sum / Avg
Land share (%) 63.38 15.56 21.06 100.00 - - - -
Land rent ($ /sq-yd) 20.08 26.40 14.21 19.96 0.012 -0.009 0.004 0.007
Floor space (M Sqyd) 124.11 66.29 190.40 0.078 0.074 0.076
Floor rent ($ / sq-yd) 15.39 16.63 15.82 0.006 0.000 0.004
Trips Working Shopping Freight Sum / Avg Working Shopping Freight Sum / Avg
Trip (PCU / 4Hour) 2,285,880 649,077 109,905 3,044,862 0.000 0.078 0.075 0.019
VMT (Million mile) 208.67 8.49 217.16 0.015 0.049 0.016
VTT (1000 Hour) 4,961.6 200.4 5,161.9 0.019 0.054 0.021
Avg Distance (Miles) 71.098 77.234 71.320 -0.002 -0.027 -0.003
Avg Time (minutes) 101.4 109.4 101.7 0.002 -0.022 0.001
Avg Speed (MPH) 42.06 42.37 42.07 -0.004 -0.005 -0.004

215

APPENDIX F. NOTATIONS
Constants – Dimension
H Type of households in the system
I Industries in the system
R Zones, spatial unit in the system
M Transportation modes for households

Index – Right Superscripts
h Type of household, H ∈ h
i, j Industry or its commodity I ∈ j i,
u Housing developers
v Industrial space developers
g Local governments
m Transportation mode for working trip, M ∈ m
n Transportation mode for shopping trip, M ∈ n
w Working trip (Journey home to work; JHW)
s Shopping trip (Journey home to shop; JHS)
t Truck trip (Intermediate goods movement)

Index – Left Superscripts
n Iteration index n=0…N

Index – Right Subscript
p Residential location of household, R ∈ p
q Firms’ location or place of job, R ∈ q
r Location of supplying industry for final demand, R ∈ r
k Place of shopping, R ∈ k

216
Endogenous Variables
∈
h
pq
U U Utility of a household of type h, who lives in zone p, and works in
zone q
∈ Ω
h
pq
Ω Number of household of type h, who lives in zone p, and works in
zone q
∈
h
pq
V V Indirect utility of household of type h for selecting residential
location p and working place q
hmn
pq
V Indirect utility of household of type h for selecting residential
location p, working place q, and taking travel mode m for working
trip, n for shopping trip
∈
ih
pqk
Y Y Commodity i that produced at zone k, consumed by household h,
who lives in zone p and works in zone q
ihmn
pqk
Y Commodity i that produced at zone k, consumed by household h,
who lives in zone p, works in zone q, and takes travel mode m for
working trip, n for shopping trip
p
H Developed housing space in zone p
∈
h
pq
H H Housing consumption of household h, who lives in zone p, and
works in zone q
hmn
pq
H Housing consumption of household h, who lives in zone p, works in
zone q, and takes travel mode m for working trip, n for shopping trip
∈
h
pq
N N Endowment allocated for leisure by household h, who lives in zone
p, and works in zone q
hmn
pq
N Endowment allocated for leisure by household h, who lives in zone
p, works in zone q, and takes travel mode m for working trip, n for
shopping trip
∈
p
G G Amenity offered to households in zone p
g p
G
∈
Produced amenity by local government g, whose jurisdiction
includes zone p
j
q
ω Profit of industry j, operating in zone q

217
∈
j
q
X X Output quantity of commodity j, produced in zone q
∈
ij
rq
Z Z Intermediate input of commodity i produced in zone r for producing
commodity j in zone q
∈
hj
q
L L Labor input of type h to the industry to produce commodity j in
zone q
q
F Developed space for industrial use in zone q
∈
j
q
F F Developed space used by the industry to produce commodity j in
zone q
∈
j
q
Q Q Input of raw land to developers or local government j in zone q,
{ } g v u j , , ∈ ∀
∈
h
pq
D D Income of household h, who lives in zone p, works in zone q by
renting raw land
hmn
pq
D Income of household h, who lives in zone p, works in zone q, and
takes mode m for working trips, mode n for shopping trips by
renting raw land
∈
i
k
p p Price of commodity i in zone k
∈
h
q
w w Hourly wage for the labor from of commodity i in zone k
∈
u
p
r r
u
Housing rent in zone p
∈
v
p
r r
v
Rent of space for industrial use in zone p
∈
p
r r Land rent in zone p
w w
pq
f f ∈ Working trips from residential zone p to working zone q
wm
pq
f Working trips from residential zone p to working zone q by mode m
s s
pq
f f ∈ Shopping trips from residential zone p to shopping zone q
sm
pq
f Shopping trips from residential zone p to shopping zone q by mode
m
t t
pq
f f ∈ Truck trips from zone p to zone q

218
∈
pq
c c Travel cost in dollar between zone p and q when only one travel
mode is used
∈
pq
t t Travel time in hour between zone p and q when only one travel
mode is used
m
pq
c Travel cost in dollar by travel mode m between zone p and q
m
pq
t Travel time in hour by travel mode m between zone p and q
ih
pk
α Coefficient of utility function of household h who lives at zone p for
consumption of commodity i that was produced at zone k
ih
η Factor of scale of economy in shopping behavior of household, type
h
ih
µ Decay coefficient to total shopping price of household, type h
ij
rq
ϕ Input share coefficient for intermediate input of commodity i that is
produced in zone r to production of j in zone q
ij
µ Decay coefficient to total input price of industry j for intermediate
input i

Constants – Exogenous Variables
T Number of working days
E Endowment
h
Θ Number of household of which type is h
p
A Available land in zone p

Constants – Parameters on production technology and preference.
ih
α Coefficient of utility function of household h for consumption of
commodity i
h
β Coefficient of utility function of household h for consumption of
housing
h
δ Coefficient of utility function of household h for leisure

219
h
φ Coefficient of utility function of household h for amenity
h
pq
ε Random part of utility in selection of residential location p, and
working place q by household h
ij
ϕ Input share coefficient for intermediate input of commodity i in
production of j, { } g v u j , , ∪ ∈ ∀ I
hj
ρ Input share coefficient for labor from household type h in
production of j, { } g v u j , , ∪ ∈ ∀ I
j
σ Input share coefficient for developed space in production of j,
{ } g v u j , , ∪ ∈ ∀ I
j
θ Elasticity of substitution in production of j, {} g v u j , , ∪ ∈ ∀ I
j
q
τ Tax rate of industry j that is operating in zone q, { } v u j , ∪ ∈ ∀ I
i
ζ Number of trips for shopping per dollar value of commodity i
purchased
i
ξ Number of trips for delivery of intermediate goods i per commodity
value in dollar
j
γ Return trip rate of trip type i within analysis time horizon,
{ } t s w j , , ∈ ∀
h
ψ Dispersion factor of utility function for household type h

Abstract (if available)

Abstract The objective of this study is to establish an alternative modeling scheme for comprehensive urban transportation studies. The model developed in this study is based on Anas-Kim's computable general equilibrium model and expanded in applicability to address real world problems existent in complex urban systems.

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Asset Metadata

Creator Cho, Sungbin (author)

Core Title A multi-regional computable general equilibrium model in the Haritage of A. Anas and I. Kim

School School of Policy, Planning, and Development

Degree Doctor of Philosophy

Degree Program Planning

Publication Date 12/01/2006

Defense Date 08/24/2006

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Land use,multiregional computable general equilibrium,OAI-PMH Harvest,regional economic analysis,transportation plan

Language English

Advisor Moore, James E. (committee chair), Dessouky, Maged M. (committee member), Gordon, Peter (committee member)

Creator Email sc@imagecatinc.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m207

Unique identifier UC1451651

Identifier etd-Cho-20061201 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-165670 (legacy record id),usctheses-m207 (legacy record id)

Legacy Identifier etd-Cho-20061201.pdf

Dmrecord 165670

Document Type Dissertation

Rights Cho, Sungbin

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu