Traffic Congestion and Land Use Regulations: Theory and Policy Analysis 0128170204, 9780128170205

Table of contents :
Cover
TRAFFIC CONGESTION
AND LAND USE
REGULATIONS
Theory and Policy Analysis
Copyright
1
Introduction
References
2
Necessity of a minimum floor area regulation
Introduction
The model
The city
Household behavior
Developers behavior
Market clearing conditions
Optimal FAR regulations
Harberger's welfare function and optimal conditions for FAR regulation
Optimal FAR regulations and market equilibrium FAR
A case of a monocentric city: Numerical simulation
The model city
Setup for numerical simulation
Numerical results
Conclusion
Technical Appendix 2
Supplementary note: Harberger welfare function
References
3
Differences in optimal land use regulation between a closed city and an open city
Introduction
The model
The city
Household behavior
Developers behavior with no FAR regulation
Developers behavior under FAR regulation
Commuting cost: The external factor
Market equilibrium conditions
Optimal FAR regulation and urban boundary
Definition
Closed city
Open city
Comparison of results between closed- and open-city models
Conclusion
Technical Appendix 3
Mathematical note: Lagrangian method and optimal control theory
References
4
Optimal land use regulation in a growing city
Introduction
The model
The city
Household behavior
Developers behavior
Market clearing conditions
Maximizing social welfare using FAR regulation
Social welfare
Optimal dynamic FAR regulation
Differences from a static city
Exceptional cases
FAR regulation in a growing monocentric city under traffic congestion externality
Conclusion
Technical Appendix 4
References
5
Optimal land use regulations in a city with business areas
Introduction
The model
The city
Firms behavior
Developers behavior
Commuting cost-An external factor
Household behavior
Market clearing conditions and definitions
Optimal land use regulations
Maximizing social welfare
Optimal FAR and LS regulations
Optimal zonal boundaries
Numerical simulation
The setup
Zones C and H
Zone B
Numerical results
Conclusion
Technical Appendix 5
References
6
Introducing cordon pricing in a regulated city
Introduction
The model
The city
Household behavior
Developers behavior under FAR regulation
Traffic flow and commuting cost
The model under cordon pricing
Market equilibrium conditions
Optimal cordon pricing and location
Optimal land use regulation under cordon pricing
Characterization of optimal FAR and UGB regulation
Numerical example of optimal cordon pricing and FAR regulation
Conclusion
Technical Appendix 6
References
Further reading
7
Road investment evaluation under land use regulation
Introduction
The model
The city
Household behavior
Landowners benefit
Developers behavior
Commuting cost: The external factor
Market equilibrium
Welfare functions of FAR and UGB regulations and city regimes
Optimal FAR
Change in FAR according to road investment in each regime
Cost-benefit analysis of road investment in a closed city
Social welfare
Closed city with optimal FAR regulation
Closed city with fixed FAR
Closed city with no FAR regulation
Comparing the three regimes of the closed city
Cost-benefit analysis of road investment in an open city
Social welfare
Open city with optimal FAR regulation
Open city with fixed FAR
Open city with no FAR regulation
Conclusion
Technical Appendix 7
References
8
Changes with future ICT technologies and future studies
Acknowledgments
References
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
W
Z
Back Cover

Citation preview

TRAFFIC CONGESTION AND LAND USE REGULATIONS

TRAFFIC CONGESTION AND LAND USE REGULATIONS Theory and Policy Analysis

TATSUHITO KONO KIRTI KUSUM JOSHI

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States © 2019 Elsevier Inc. All rights reserved.

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CHAPTER ONE

Introduction Land use regulations are common urban policies in most cities all over the world. Common regulations include (1) zoning by which land use is restricted zone by zone; (2) lot size (LS) regulation, which restricts the size of each housing lot; (3) urban growth boundary (UGB) control, which separates urban development areas from urbanization control areas; and (4) floor area ratio (FAR) regulation,a which restricts building sizes. The adoption and implementation of these regulations vary according to the country or the city. In some cases, multiple regulations may be applied to a single building; likewise, each regulation could be implemented in slightly different ways.b Why do cities impose land use regulations? In practice, cities impose land use regulations for various reasons such as to mitigate traffic congestion and noise, improve urban aesthetics, control air pollution, recover public service cost, or reduce frictions between agents (e.g., landowners and residents) and conflicts in land use.c Similar to other public policies, the targets of practical land use regulations are not necessarily economically reasonable. Nevertheless, sufficient accountability is required for regulations because the regulations restrict residents and landowners from freely using their property as they wish, and in most cases, regulations result in costs for them. Hence, land use regulations should be justifiable. Moreover, buildings are probably one of the most durable goods ever produced. Accordingly, if a regulation at a certain time leads to inefficient urban land use, it remains inefficient for many years. An inefficient result of land use regulation can be seen in Moscow. As shown in Fig. 1.1 borrowed from Bertaud and Renaud (1997), the Moscow bureaucratic density control led to a perverse, inverted population density pattern in which suburban areas have more residents than the central areas, in contrast to Paris where the reverse is true. The density pattern in Moscow generates heavy traffic burdens. Another example is seen in suburbs in Tokyo. Fig. 1.2 shows a

FAR is the ratio of the total floor area of a building to the size of the plot on which the building is built. For example, an urban growth boundary can be implemented by using a greenbelt. In many countries, for regulating a building, a LS restriction is set along with a restriction on the FAR of the building. c Some land use regulations might be used for less benign purposes such as to serve landowners’ benefits (see Brueckner and Lai, 1996). b

Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00001-7

© 2019 Elsevier Inc. All rights reserved.

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Traffic congestion and land use regulations

Fig. 1.1 Comparison of density gradients of Moscow and Paris. (Source: Bertaud, A., Renaud, B., 1997. Socialist cities without land markets. J. Urban Econ. 41 (1), 137–151.)

Fig. 1.2 Comparison of density gradients among towns in Tokyo. (Source: Sagamihara City, 2011. Outlook of Sagamihara and Machida Using Maps, p. 52.)

Introduction

3

change in the population density in three suburb towns according to the distance from the nearest station in 2005 (Sagamihara City, 2011). These three towns are new towns, informally called “bed towns,” which were built as residential places for employees working in the center of Tokyo. Comparing the density patterns among the three towns, while the population density decreases with distance in Hachiouji and Hashimoto, the density is almost constant in Machida. The constant density of Machida generates greater congestion of commuting trips than in the other two towns. This difference in population density patterns is probably caused by past land use planning. As these different density patterns show, it is important to set appropriate land use regulations or policies to achieve efficient population density patterns. However, as this book will show, the urban mechanism behind land use regulations is not straightforward. Accordingly, careful consideration in city planning is required at any time. In economics, policies can be evaluated and accounted for from two viewpoints: efficiency and income distribution. In this book, we explore land use regulations from these two perspectives. However, we focus more on efficient land use regulations because the effect of land use regulations on income distribution is so indirect and complex that policy makers do not adopt land use regulations from the viewpoint of income distribution in most cases. Nevertheless, because land use regulations do not normally involve income redistribution, it is important to know the effects of land use regulation on the income distribution between agents. This book explores the different effects on landowners and residents. Urban activities in the market mechanisms lose efficiency in various manners. Land use regulations can deal with several types of market failures such as agglomeration economies in business areas, congestion externality, pollution (e.g., noise and air pollution), blocked sunlight or air circulation between buildings, aesthetic degradation of landscape, and nonoptimal investment costs of public facilities such as roads and water and sewage systems. These market failures incur huge social costs. One important market failure is traffic congestion externalities. Traffic congestion in the United States in 2007 caused an additional 4.2 billion hours of travel and an extra 2.8 billion gallons of fuel consumption, costing a loss of $87.2 billion in travel time and fuel alone (Schrank and Lomax, 2009). In Japan, about 8 billion hours per year are lost due to traffic congestion, and this amount corresponds to about 40% of the travel time (Ministry of Land, Infrastructure, Transportation and Tourism, 2015). Traffic congestion externality spatially extends from the center of a city to its boundary,

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Traffic congestion and land use regulations

and its magnitude depends on the location. So, land use and densities at locations in a city with traffic congestion are obviously inefficient. Another important market failure is agglomeration economies, which arise from high employment density because of easy access to intermediate goods and labor, facilitating job matching, and knowledge spillovers, among others (Fujita and Thisse, 2013; Rosenthal and Strange, 2004; Puga, 2010). The employment elasticity of city productivity, which is a typical measure of agglomeration economies, is estimated to be 0.05 in the EU region (Ciccone, 2002) and Japan (Nakamura, 1985) and 0.06 in the United States (Ciccone and Hall, 1996).d In other words, doubling the employment density would increase the city output by 5% in the EU region and Japan and 6% in the United States. Agglomeration economies generate spatial concentration of workers, although the concentration level is insufficient. The geographical concentration of workers simultaneously produces commuting trips from residential areas to the business areas. As agglomeration economies increase, the number of concentrated workers increases, and simultaneously the total length of trips in a city increases. We have to deal with such spatial land use patterns to increase the welfare of city residents. These externalities can be completely internalized by spatially differentiated Pigouvian tax (or subsidies), which are differences between the social marginal cost (benefit) and the private marginal cost (benefit). However, for political reasons in particular, it is hard to implement such space-dependent Pigouvian taxes and subsidies. Indeed, Pigouvian taxes, or even diluted versions of Pigouvian taxes, have never been the common measures to address urban spatial externalities such as congestion and agglomeration economies. For example, although most cities in the world suffer from severe traffic congestion, when a few advanced cities (e.g., London, Milan, Oslo, Singapore, and Stockholm) introduced congestion pricing, it had been more than 50 years since John d

Elasticity of productivity can also be measured in terms of industry size (employment) and city size (population). For example, the employment elasticity of productivity in Japanese cities is estimated at 0.05 by Nakamura (1985), that in Brazilian cities at 0.11 by Henderson (1986), and that in the US metropolitan statistical areas at 0.19 by Henderson (1986). The population elasticity of productivity in Japanese cities is estimated at 0.03 by Nakamura (1985) and 0.04 by Tabuchi (1986) and that in Greek regions at 0.05 by Louri (1988). Recently, such quantitative analyses have been conducted at microlevels (individual firms and plants) owing to the increase in data availability. This body of research includes Henderson (2003), Rosenthal and Strange (2003), Moretti (2004), and Jofre-Monseny et al. (2014). Henderson (2003), using panel data, estimates plant level production functions that allow for scale externalities from other local plants in the same industry and from the diversity of local economic activities outside the industry.

Introduction

5

Kain and William Vickrey proposed practical versions of congestion pricing in the 1950s following Pigou’s (1920) initial proposal (see Harsman and Quigley, 2011). Furthermore, current practical applications of congestion pricing are in the form of cordon or area pricing and are far from the first-best congestion pricing. For agglomeration economies, labor subsidies have been proposed by many studies (e.g., Kanemoto, 1990; Fujita and Thisse, 2013; Lucas and Rossi–Hansberg, 2002). Nevertheless, labor subsidies have never been introduced in any cities and probably never been discussed either. In contrast to such first-best policies, land use regulations have been imposed for a long time in many cities around the world.e This common use of land use regulations is partly because governments tend to prefer quantity regulations to price regulations. For example, in the United States, 92% of the jurisdictions in the 50 largest metropolitan areas have zoning ordinances of one kind or another in place, and only 5% of the metropolitan population lives in jurisdictions without zoning (Pendall et al., 2006). In Japan, most cities have their own local city planning councils to make city plans. However, it is not an easy task for the governments to rationally design optimal land use regulations because of the need to consider change in price distortions caused by the regulations and change in spatial externalities.f Indeed the mechanisms involving a spatial equilibrium are complex. Against this background, it is very important to find optimal land use regulations, by clarifying how the distortions and externalities change according to land use regulations and by clarifying the mechanisms, which depend on the urban situation (e.g., whether or not population changes in response to land use regulation) or the externality characteristics.g For this purpose, we need a theoretical model in which the outcomes of all the agents’ behaviors are in equilibrium and the equilibrium depends on land use regulations. In this chapter, we review theoretical studies on land use regulations. The purpose of this review is to capture the overall flow of development of models for analyzing land use regulations and not to show a comprehensive review of the studies. To address the spatial mechanisms of land use e

The prevalence of land use regulations all over the world could be attributed to the social preference for policies that involve no direct money payment, making it easy for policy makers to implement such policies. f Sometimes, some members in the local city planning councils in Japan argue that they follow market equilibrium to set regulations. However, such comment is illogical because if the market equilibrium is followed, there is no need to set the regulations. g Note that land use regulations can be replaced by equivalent property tax policies (see Pines and Kono, 2012).

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regulations, we basically review only general equilibrium models and ignore empirical research on land use regulations.h In addition, we do not review growth control papers that consider only the population distribution across cities and ignore heterogeneous spaces with transportation and amenities within a city. Basically, we focus on the ingredients, which play an important role in determining land use regulations in a city. The following discussion classifies the theoretical studies into five categories. The first four categories of the studies explore efficiency of land use regulations rather than income distribution. The first category includes studies up to the year 2000. Many studies in this period use Alonso-type models and regard the central business district (CBD) as a point in the center of the city (a point CBD). The second category of studies, published after 2000, also features Alonso-type models but with some modification (e.g., a city with high-rise residential buildings but a point CBD). The third category extends the Alonso model from a point CBD by adding nonzero business areas or considering duocentric city. The fourth category leaps from the Alonso model to demonstrate dynamics and a system of cities and is published after 2000. The last category explores mainly income distribution of land use regulations, rather than efficiency. The studies in each category are summarized in the tables on the following pages. Theoretical studies on the efficiency of land use regulations first appear in the 1970s, following the Muth (1961) model, the Mills and De Ferranti (1971) model, and the Solow (1973) model, which incorporate road congestion into the Alonso (1964) model. Table 1.1 summarizes such studies on land use regulations from the 1970s up to 2000. Studies in this period, except for Arnott and MacKinnon (1978) that numerically calculate the welfare cost of building size regulation, take account of cities composed of only detached houses and roads. Most studies use an Alonso-type model, that is, a static monocentric city. For example, Kanemoto (1977), Arnott (1979), Pines and Sadka (1985), and Wheaton (1998) use Alonso-type models to explore deviation of shadow prices from market prices of housing at different locations under unpriced congestion. A graphical representation of this type of cities is shown in Fig. 1.3, where each cylinder on the city circle indicates a house, and the base of the cylinder represents the lot size. As a result of rent competition among residents, the h

Land use interventions and their effects on the welfare level of urban residents have been discussed in many previous studies (see Brueckner, 2009, for a survey of theoretical analyses, or Evans, 1999; Brueckner, 2009, for a survey of empirical investigations). In addition, huge empirical research is being produced (e.g., Brueckner et al., 2017; Albouy et al., 2017).

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Introduction

Table 1.1 Studies on efficiency of land use regulations (1970–2000). Land use Targeted Study regulation externalities Model characteristics

Stull (1974)

Zoning

Helpman and Pines (1977)

Zoning

Neighborhood externalities Neighborhood externalities

Kanemoto (1977) UGB control Road congestion Arnott and MacKinnon (1978) Arnott (1979)

FAR regulation

No specified externalities

UGB control Road congestion

LS regulation Road congestion and UGB control Sullivan (1983) Zoning Agglomeration economies, road congestion Brueckner (1990) UGB control Population congestion Engle et al. (1992) UGB control Road congestion and pollution Sakashita (1995) UGB control Road congestion Sasaki (1998) UGB control Congestion, public goods, production Wheaton (1998) LS regulation Road congestion Ding et al. (1999) UGB control Congestible public good Pines and Sadka (1985)

Nonzero business area, detached housing Nonzero business area, detached housing, multiple cities Road space, detached housing Condominiums

Road space, detached housing Detached housing

Nonzero business area

Dynamic modelling, an open city Detached housing, twocity model Two-city model Different landownership systems Detached housing Dynamic modelling, an open city

Abbreviations: UGB, urban growth boundary; FAR, floor area ratio; LS, lot size.

CBD Road

Detached houses

Fig. 1.3 Alonso-type model incorporating congested roads.

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Traffic congestion and land use regulations

lot size is larger in the suburbs than in the central area. This Alonso-type model has only detached houses and no high-rise buildings so that the inverse of the lot size expresses population density. Another important feature is a point CBD. Among other studies using similar models, Stull (1974) and Helpman and Pines (1977) explore zoning, taking account of nonzero business areas in addition to the residential areas. Sullivan (1983) considers external economies of scale in production in nonzero business areas under traffic congestion, using numerical simulations. Helpman and Pines (1977), Engle et al. (1992), and Sakashita (1995) extend the Alonso-type models to include multiple cities. In contrast to the previously mentioned static models in this period, Brueckner (1990) and Ding et al. (1999) derive the efficient dynamic path of the UGB. As an optimal regulation, Kanemoto (1977) shows that the UGB should be smaller than the market equilibrium urban boundary. Pines and Sadka (1985)i and Wheaton (1998) show that the excess burden of unpriced traffic congestion can be completely eliminated by appropriate LS regulations. Accordingly, under optimal LS regulation, the UGB can be determined by the market in their model. This is hardly surprising because implementing LS regulations is equivalent to determining the population’s distribution in a city model composed of only detached houses. Comparing Kanemoto (1977) and Pines and Sadka (1985), we find that simultaneous consideration of multiple land use regulations (LS regulation and UGB) gives a different optimal solution (useless UGB in Pines and Sadka) when addressing a single regulation (useful UGB in Kanemoto). Accordingly, multiple land use regulations should be explored simultaneously. In this period, most papers explore zoning, LS regulations, or UGB control, ignoring floor area size regulations because they treat only detached houses. Since 2000, variations of the model exploring land use regulations have expanded. We have classified this variety into three types. Each type is summarized in Tables 1.2–1.4. As shown in Table 1.2, most papers take account of high-rise buildings including condominiums in a city with point CBD and explore FAR regulations. This point is different from the studies before 2000. A graphical representation of this type of cities is shown in Fig. 1.4, where high-rise buildings are added to Fig. 1.2. Unlike the Alonso-type i

Pines and Sadka (1985) use housing tax to control lot size. However, this is equivalent to LS regulation in terms of social welfare. Land use regulations can be replaced by equivalent property tax policies (see Pines and Kono, 2012).

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Introduction

Table 1.2 City with a point CBD (2000 onward). Study

Land use regulation

Targeted externalities

Model characteristics

Bertaud and Brueckner (2005) Brueckner (2007) Pines and Kono (2012)

Maximum FAR regulation

No externalities

Numerical simulation

Maximum FAR regulation, UGB control Maximum and minimum FAR regulations, UGB control Maximum and minimum Kono et al. (2012) FAR regulations, UGB control Kono and Joshi FAR regulation, UGB (2012) control Borck (2016) Maximum FAR regulation Tikoudis et al. FAR regulation, UGB (2018) control FAR regulation, UGB Kono and control Kawaguchi (2017)

Road congestion Numerical simulation Road congestion

Road congestion

Road congestion Closed and open cities GHG emissions Road congestion Road congestion

Abbreviations: UGB, urban growth boundary; FAR, floor area ratio; LS, lot size; GHG, greenhouse gas.

Table 1.3 Extensions of the Alonso model from a point CBD (2000 onward). Land use Study regulation Targeted externalities Model characteristics

Rossi-Hansberg (2004)

Zoning

Anas and Rhee (2006) Rhee et al. (2014)

UGB

Buyukeren and Hiramatsu (2016) Zhang and Kockelman (2016) Kono and Joshi (2018)

UGB

Zoning

Zoning, UGB Zoning, FAR, UGB

Agglomeration economies, no road congestion Road congestion Agglomeration economies, road congestion Road congestion

Agglomeration economies, road congestion Agglomeration economies, road congestion

Nonzero business area

Mixed land use, nonzero business area Mixed land use, nonzero business area An uncongested public transit mode and a car mode Nonzero business area

Nonzero business area

Abbreviations: UGB, urban growth boundary; FAR, floor area ratio; LS, lot size.

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Table 1.4 Leaps from the Alonso model (2000 onward). Land use Study regulation Targeted externalities

Lin et al. (2004) FAR regulation Anas and Rhee (2007) Anas and Pines (2008) Joshi and Kono (2009) Kono et al. (2010) Jou (2012)

UGB control UGB control FAR regulation FAR regulation UGB control Anas and Pines UGB (2012) control

Neighborhood externalities, road congestion Road congestion Road congestion Population congestion

Model characteristics

Dynamic modelling

Existence of a suburban business district Two-city model

Population congestion

Two-zone city with growing population Two-zone city

Population congestion

Stochastic rents

Road congestion

Multiple-city model

Abbreviations: UGB, urban growth boundary; FAR, floor area ratio.

CBD Road High-rise building Detached houses

Fig. 1.4 Alonso-type model with high-rise buildings and roads.

models, the inverse of the lot size does not represent population density anymore because high-rise buildings include many households. Still, this type of model assumes a point CBD. As these post-2000 studies have clarified, we should treat FAR regulation and LS regulation separately to explore optimal regulations because FAR regulation necessarily generates deadweight loss caused by the regulation itself (see Chapter 2 for details), whereas LS regulation has no deadweight losses (see Pines and Sadka, 1985; Wheaton, 1998). Under FAR regulation, households can choose their optimal floor size within the regulated buildings. In other words, FAR regulation controls population density indirectly, whereas LS regulation does this directly.

Introduction

11

Considering high-rise buildings, Bertaud and Brueckner (2005), Brueckner (2007), and Brueckner and Sridhar (2012) quantitatively calculate in a general equilibrium framework how much the welfare cost of maximum building size (or FAR) regulation increases with an increase in the commuting costs. Kono et al. (2012) and Pines and Kono (2012), using a closed city model, show that minimum FAR regulation should be simultaneously imposed along with maximum FAR regulation to achieve optimal regulation. Next, Kono and Joshi (2012) show how optimal land use regulations differ between closed and open cities. Tikoudis et al. (2018) and Kono and Kawaguchi (2017) consider road tolls and FAR regulations simultaneously. Indeed, real-world cities implementing congestion pricing impose land use regulations simultaneously. Land use regulations can also contribute toward making cities environment friendly by changing population distribution. In a related study, Borck (2016) estimates how greenhouse gas emissions as CO2-equivalent change with FAR regulation. The studies listed in Table 1.2 have taken account of only road congestion or environmental damage in the residential areas, assuming a point CBD. In contrast to such negative externalities, concentration of workers in business areas in cities enhances communication and thus facilitates exchange of innovative ideas (see Rauch, 1993; Ciccone and Hall, 1996; Duranton and Puga, 2001; Moretti, 2004). Such positive agglomeration economies in business areas can be explored by taking account of nonzero business area as shown by post-2000 papers listed in Table 1.3. In the 1970s, Stull (1974) and Helpman and Pines (1977) already considered nonzero business areas to explore optimal zoning to tackle externalities between manufacturing and residential land use. However, agglomeration economies arising from employment density are not considered. Rossi-Hansberg (2004) takes account of the existence of agglomeration spillovers of firms to explore zoning. Rhee et al. (2014), Zhang and Kockelman (2016), and Kono and Joshi (2018) consider the existence of agglomeration economies and traffic congestion to explore land use regulations. Rhee et al. (2014) focus on mixed land use with residences and businesses. One land use pattern of these models is shown in Fig. 1.5. An essential feature of this type of cities is no use of a point CBD. So, how land is allocated for different land use purposes matters. All previous studies so far have shown that in a monocentric city, residential locations should be centralized by optimal land use regulations when there is only car commuting. In contrast, Buyukeren and Hiramatsu (2016),

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Road

High-rise commercial building High-rise residential building Detached houses

Fig. 1.5 A city with business zones.

assuming a congested car mode and an uncongested public transit mode, analytically demonstrate that an expansionary UGB would be optimal under certain conditions in a simple model composed of two discrete zones. Interestingly, by setting mixed-use zones in a monocentric and nonmonocentric city having residents with idiosyncratic tastes, Anas and Rhee (2006) also show that an expansive UGB may be necessary, which contradicts the traditional conclusions based on a monocentric city model. The extensions of Alonso models presented in Tables 1.2 and 1.3 are all static. Land use regulations influence population density for a long time, and the regulations cannot be changed easily because of the durability of buildings. In a city with low population growth, a one-time decision on FAR regulation might be useful over a long period. However, many cities, particularly those in developing countries, have a high population growth rate. To explore a dynamic situation, Lin et al. (2004) explore the dynamic effects of exogenous change in FAR regulations on the equilibrium in a monocentric city. Joshi and Kono (2009) demonstrate a socially optimal path of FAR regulation under the presence of negative population externalities that change dynamically with a growing population. Jou (2012) explores when and how UGB control should be implemented with a stochastically increasing population. These models are summarized in Table 1.4. Land use regulation can affect the welfare of agents (e.g., landowners, developers, and renters) differently at different times; so, some of the studies analyzing income distribution effects adopt a dynamic framework. An interesting characteristic of the papers listed in Tables 1.2 and 1.3 is that all of them target only one city. As Anas and Pines (2008, 2012) show, if we consider more than one city, optimal land use regulations can differ from those when only one city is supposed. These studies clarify that the geographical setting plays an important role in determining the properties of optimal land use regulations (e.g., see Anas and Rhee, 2007). These varieties of models are still developing, and the current model settings are limited in many aspects. This leaves room for future works.

Introduction

13

We finally review the studies that focus on the income distribution effects of land use regulations in Table 1.5. Our division of the studies into the “efficiency” studies, reviewed earlier, and the “income distribution” studies is arbitrary in the sense that some papers explore both. Existing residents in a city have an incentive to control future development of the community to serve their own interests. Hence the income distribution between the current residents and potential future residents matters. To this end a dynamic model of community formation deserves attention. This type of studies includes Epple et al. (1988) and Helsley and Strange (1995), the latter of which assumes endogenous growth control. In a city, residents with different incomes tend to reside separately because of income effects on lot sizes and different values of time. Land use regulations affect residents spatially, so the residents with different incomes are affected differently. This effect is explored by Pasha (1996). Land use regulations straightforwardly affect land prices, so the income distribution between landowners and nonlandowning residents also changes. As a result, these two kinds of residents, who often bid different rents, reside separately from one another in the city. This situation is explored by Brueckner (1995) and Brueckner and Lai (1996). Such income effects depend, among others, on the system of landownership such as absentee landownership or resident landownership. Only a few studies have addressed income distribution effects, probably because setting heterogeneity would complicate the model. Future studies could focus on these effects. This chapter has so far reviewed how studies on land use regulations have developed over time. As those papers demonstrate, the geographical setting (e.g., open or closed setting, one city or a system of cities, and monocentric or duocentric) and the targeted externalities (e.g., agglomeration economies and traffic congestion) play an important role in designing optimal land use regulations. This book addresses land use regulations, disentangling such complex mechanisms of spatial equilibrium theoretically, and derives practical ways for optimizing them. The contents of the remaining chapters are as follows: Chapter 2 shows a necessity for minimum FAR regulation and maximum FAR regulation to maximize social welfare in a closed city. In other words, considering only maximum FAR regulation, a common practice in actual city planning,j is j

Although not as popular as maximum FAR regulation, cities like Oregon City, Buffalo, and Colorado Springs have practiced minimum FAR regulation in designated areas to prevent underdevelopment. In Japan, too, minimum FAR regulation can be imposed according to urban planning law.

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Table 1.5 Studies on income effects across agents or endogenously determined regulations. Land use Targeted Study regulation externalities Model characteristics

Epple et al. (1988) Helsley and Strange (1995)

Brueckner (1995)

Pasha (1996) Brueckner and Lai (1996)

UGB

Public good

Population controls (e.g., FAR and UGB) UGB

Community-scale externalities, conflicts among communities No externalities

LS UGB

No externalities No externalities

Two-period model, endogenous regulations Endogenous growth control

Landowner within or without UGB, residents, trade-offs between agents Two income groups Resident landowners and residents with no land

Abbreviations: UGB, urban growth boundary; FAR, floor area ratio.

insufficient from a welfare perspective. For obtaining a second-best policy, the deadweight losses in the floor space market play an important role, and this chapter shows that, to minimize the total deadweight losses, a minimum FAR regulation should also be imposed. Chapter 3 shows that optimality of land use regulation—in this case, FAR regulation—differs between a closed city and an open city. This chapter demonstrates that a closed city requires not only downward adjustment to the market density at boundary locations using maximum FAR regulation but also upward adjustment at central locations using minimum FAR regulation. On the other hand an open city requires only the former. Therefore city planners should identify whether the city is open or closed before determining policies. We will discuss what factors determine the openness of a city. Chapter 4 explores optimal land use regulation in a growing city with congestion. Many cities, particularly those in developing countries, have a high population growth rate. In such cities the population distribution and thereby the level of externality at a location can change remarkably over time. For optimality, FAR regulation should also change dynamically and concomitantly with population growth. This chapter explores the optimal timing and scale of FAR regulations in a dynamic setting.

Introduction

15

Chapter 5 addresses optimal land use regulations in a city with agglomeration economies and traffic congestion, closely modelling on a real-world city. A typical city is generally composed of distinct land use zones consisting of business firms, condominiums, and detached houses. This chapter shows how regulations on building size, lot size, and the zonal boundaries should be imposed in the presence of agglomeration economies and traffic congestion. Results show that optimal regulations depend on the trade-off between agglomeration economies and traffic congestion costs. As an important result the Appendix of this chapter shows the formulae demonstrating by how much the building size, the lot size, and the zone sizes should differ from those determined at the market equilibrium. These formulae readily evaluate whether the current land use regulations are optimal or not. Chapter 6 introduces cordon pricing in a city with regulated land use. Several cities have indeed imposed cordon pricing to mitigate traffic congestion and impose land use regulations as well. The chapter explores the optimal level of a single cordon toll and its location and the optimal FAR regulation. Results show existence of price distortions in the presence of cordon pricing and FAR regulation. Chapter 7 focuses on road investment. Up to Chapter 6, we discuss how to design optimal density and zonal regulations in congested cities. In such cities, road investment can also reduce congestion, but this requires a large area of land, which could otherwise be used for residential and business uses. Chapter 7 explores how much of the land should be allocated for roads in a city without congestion tolls. Chapter 8 summarizes results and their implications from the perspective of urban policy making and discusses future studies as well. Spatially variable land use regulations can change the spatial distribution of population. But it is not easy to forecast the effect of land use regulation on the spatial equilibrium and capture all the changes in price distortions, which are important factors for optimizing the regulations. These complex mechanisms can be captured by rigorous derivation using optimal control theory. We also provide intuitive explanation in each chapter to supplement mathematical derivation. Optimal control theory has often been used to analyze the first-best allocation in urban economics. However, land use regulations are second-best policies. So, we have to impose market equilibrium conditions when applying optimal control theory. Furthermore, land use regulations are not necessarily spatially continuous. For example, as shown in Chapters 5 and 6, land use regulations should be simultaneously optimized over distinct zones

16

Traffic congestion and land use regulations

(e.g., business zones, condominium zones, and detached housing zones). We provide detailed explanation of how optimal control theory is applied to such discontinuous cases to analyze the optimal regulations. Understanding the mechanisms rigorously provides not only theoretical insights but also knowledge of the practical usefulness of land use regulations for the maximization of social welfare. Each chapter provides concise tips for land use regulations for policy makers. Regarding some of the optimal regulations proposed in this book, it is hard to compare our results with empirical evidence due to the lack of actual implementation of such regulations. In such case, we show simulation-based results of the effects of optimal regulations.

References Albouy, D., Ehrlich, G., Shin, M., 2017. Metropolitan land values. Rev. Econ. Stat. 100 (3), 454–466. Alonso, W., 1964. Location and Land Use. Harvard University Press, Cambridge, MA. Anas, A., Pines, D., 2008. Anti-sprawl policies in a system of congested cities. Reg. Sci. Urban Econ. 38 (5), 408–423. Anas, A., Pines, D., 2012. Public goods and congestion in a system of cities: how do fiscal and zoning policies improve efficiency? J. Econ. Geogr. 13 (4), 649–676. Anas, A., Rhee, H.-J., 2006. Curbing excess sprawl with congestion tolls and urban boundaries. Reg. Sci. Urban Econ. 36 (4), 510–541. Anas, A., Rhee, H.-J., 2007. When are urban growth boundaries not second-best policies to congestion tolls? J. Urban Econ. 61 (2), 263–286. Arnott, R.J., 1979. Unpriced transport congestion. J. Econ. Theory 21 (2), 294–316. Arnott, R.J., MacKinnon, J.G., 1978. Market and shadow land rents with congestion. Am. Econ. Rev. 68 (4), 588–600. Bertaud, A., Brueckner, J.K., 2005. Analyzing building-height restrictions: predicted impacts and welfare costs. Reg. Sci. Urban Econ. 35 (2), 109–125. Bertaud, A., Renaud, B., 1997. Socialist cities without land markets. J. Urban Econ. 41 (1), 137–151. Borck, R., 2016. Will skyscrapers save the planet? Building height limits and urban greenhouse gas emissions. Reg. Sci. Urban Econ. 58, 13–25. Brueckner, J.K., 1990. Growth controls and land values in an open city. Land Econ. 66 (3), 237–248. Brueckner, J.K., 1995. Strategic control of growth in a system of cities. J. Public Econ. 57 (3), 393–416. Brueckner, J.K., 2007. Urban growth boundaries: an effective second-best remedy for unpriced traffic congestion? J. Hous. Econ. 16 (3–4), 263–273. Brueckner, J.K., 2009. Government land use interventions: an economic analysis. In: Lall, S.V., Freire, M., Yuen, B., Rajack, R., Helluin, J.-J. (Eds.), Urban Land Markets: Improving Land Management for Successful Urbanization. Springer, Dordrecht, pp. 3–23. Brueckner, J.K., Lai, F.C., 1996. Urban growth controls with resident landowners. Reg. Sci. Urban Econ. 26 (2), 125–143.

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Brueckner, J.K., Sridhar, K.S., 2012. Measuring welfare gains from relaxation of land-use restrictions: the case of India’s building-height limits. Reg. Sci. Urban Econ. 42 (6), 1061–1067. Brueckner, J.K., Fu, S., Gu, Y., Zhang, J., 2017. Measuring the stringency of land use regulation: the case of China’s building height limits. Rev. Econ. Stat. 99 (4), 663–677. Buyukeren, A.C., Hiramatsu, T., 2016. Anti-congestion policies in cities with public transportation. J. Econ. Geogr. 16 (2), 395–421. Ciccone, A., 2002. Agglomeration effects in Europe. Eur. Econ. Rev. 46 (2), 213–227. Ciccone, A., Hall, R., 1996. Productivity and the density of economic activity. Am. Econ. Rev. 86 (1), 54–70. Ding, C., Knaap, G.J., Hopkins, L.D., 1999. Managing urban growth with urban growth boundaries: a theoretical analysis. J. Urban Econ. 46 (1), 53–68. Duranton, G., Puga, D., 2001. Nursery cities: urban diversity, process innovation, and the life cycle of products. Am. Econ. Rev. 91 (5), 1454–1477. Engle, R., Navarro, P., Carson, R., 1992. On the theory of growth controls. J. Urban Econ. 32 (3), 269–284. Epple, D., Romer, T., Filimon, R., 1988. Community development with endogenous land use controls. J. Public Econ. 35 (2), 133–162. Evans, A.W., 1999. The land market and government intervention. In: Cheshire, P., Mills, E.S. (Eds.), Handbook of Regional and Urban Economics. In: vol. 3. Elsevier, pp. 1637–1669. Fujita, M., Thisse, J.F., 2013. Economics of Agglomeration: Cities, Industrial Location, and Globalization. Cambridge University Press. Harsman, B., Quigley, J.M., 2011. Political and Public Acceptability of Congestion Pricing: Ideology and Self-Interest in Sweden. University of California Transportation Center. Helpman, E., Pines, D., 1977. Land and zoning in an urban economy: further results. Am. Econ. Rev. 67 (5), 982–986. Helsley, R.W., Strange, W.C., 1995. Strategic growth controls. Reg. Sci. Urban Econ. 25 (4), 435–460. Henderson, J.V., 1986. Efficiency of resource usage and city size. J. Urban Econ. 19 (1), 47–70. Henderson, J.V., 2003. Marshall’s scale economies. J. Urban Econ. 53 (1), 1–28. Jofre-Monseny, J., Marı´n-Lo´pez, R., Viladecans-Marsal, E., 2014. The determinants of localization and urbanization economies: evidence from the location of new firms in Spain. J. Reg. Sci. 54 (2), 313–337. Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511. Jou, J.B., 2012. Efficient growth boundaries in the presence of population externalities and stochastic rents. Q. Rev. Econ. Finance 52 (4), 349–357. Kanemoto, Y., 1977. Cost-benefit analysis and the second best land use for transportation. J. Urban Econ. 4 (4), 483–503. Kanemoto, Y., 1990. Optimal cities with indivisibility in production and interactions between firms. J. Urban Econ. 27 (1), 46–59. Kono, T., Joshi, K.K., 2012. A new interpretation on the optimal density regulations: closed and open city. J. Hous. Econ. 21 (3), 223–234. Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598. Kono, T., Kawaguchi, H., 2017. Cordon pricing and land-use regulation. Scand. J. Econ. 119 (2), 405–434. Kono, T., Kaneko, T., Morisugi, H., 2010. Necessity of minimum floor area ratio regulation: a second-best policy. Ann. Reg. Sci. 44 (3), 523–539. Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630.

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Lin, C.-C., Mai, C.-C., Wang, P., 2004. Urban land policy and housing in an endogenously growing monocentric city. Reg. Sci. Urban Econ. 34 (3), 241–261. Louri, H., 1988. Urban growth and productivity: the case of Greece. Urban Stud. 25 (5), 433–438. Lucas, R.E., Rossi–Hansberg, E., 2002. On the internal structure of cities. Econometrica 70 (4), 1445–1476. Mills, E.S., De Ferranti, D.M., 1971. Market choices and optimum city size. Am. Econ. Rev. 61 (2), 340–345. Ministry of Land, Infrastructure, Transportation and Tourism, 2015. Retrieved from:http:// www.mlit.go.jp/road/sisaku/tdm/Top03-01-01.html. (in Japanese). Moretti, E., 2004. Workers’ education, spillovers, and productivity: evidence from plantlevel production functions. Am. Econ. Rev. 94 (3), 656–690. Muth, R.F., 1961. Spatial structure of the housing market. In: Papers and Proceedings of the Regional Science Association . vol. 7, pp. 207–220. Nakamura, R., 1985. Agglomeration economies in urban manufacturing industries: a case of Japanese cities. J. Urban Econ. 17 (1), 108–124. Pasha, H.A., 1996. Suburban minimum lot zoning and spatial equilibrium. J. Urban Econ. 40 (1), 1–12. Pendall, R., Puentes, R., Martin, J., 2006. From Traditional to Reformed: A Review of the Land Use Regulations in the Nation’s 50 Largest Metropolitan Areas. The Brookings Institution, Washington, DC. Pigou, A.C., 1920. The Economics of Welfare, fourth ed. Macmillan, London. Pines, D., Kono, T., 2012. FAR regulations and unpriced transport congestion. Reg. Sci. Urban Econ. 42 (6), 931–937. Pines, D., Sadka, E., 1985. Zoning, first-best, second-best, and third-best criteria for allocation land for roads. J. Urban Econ. 17 (2), 167–183. Puga, D., 2010. The magnitude and causes of agglomeration economies. J. Reg. Sci. 50 (1), 203–219. Rauch, J.E., 1993. Productivity gains from geographic concentration of human capital: evidence from the cities. J. Urban Econ. 34 (3), 380–400. Rhee, H.-J., Yu, S., Hirte, G., 2014. Zoning in cities with traffic congestion and agglomeration economies. Reg. Sci. Urban Econ. 44 (C), 82–93. Rosenthal, S.S., Strange, W.C., 2003. Geography, industrial organization, and agglomeration. Rev. Econ. Stat. 85 (2), 377–393. Rosenthal, S.S., Strange, W.C., 2004. Evidence on the nature and sources of agglomeration economies. In: Handbook of Regional and Urban Economics. vol. 4. Elsevier, pp. 2119–2171. Rossi-Hansberg, E., 2004. Optimal urban land use and zoning. Rev. Econ. Dyn. 7 (1), 69–106. Sagamihara City, 2011. Outlook of Sagamihara and Machida Using Maps, p. 52. Sakashita, N., 1995. An economic theory of urban growth control. Reg. Sci. Urban Econ. 25 (4), 427–434. Sasaki, K., 1998. Optimal urban growth controls. Reg. Sci. Urban Econ. 28 (4), 475–496. Schrank, D.L., Lomax, T.J., 2009. 2009 Urban Mobility Report. Texas Transportation Institute, Texas A & M University. Solow, R.M., 1973. Congestion cost and the use of land for streets. Bell J. Econ. Manag. Sci. 4 (2), 602–618. Stull, W.J., 1974. Land use and zoning in an urban economy. Am. Econ. Rev. 64 (3), 337–347. Sullivan, A.M., 1983. The general equilibrium effects of congestion externalities. J. Urban Econ. 14 (1), 80–104.

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Tabuchi, T., 1986. Urban agglomeration, capital augmenting technology, and labor market equilibrium. J. Urban Econ. 20 (2), 211–228. Tikoudis, I., Verhoef, E.T., van Ommeren, J.N., 2018. Second-best urban tolls in a monocentric city with housing market regulations. Transp. Res. B Methodol. 117 (Part A), 342–359. Wheaton, W.C., 1998. Land use and density in cities with congestion. J. Urban Econ. 43 (2), 258–272. Zhang, W., Kockelman, K.M., 2016. Optimal policies in cities with congestion and agglomeration externalities: congestion tolls, labor subsidies, and place-based strategies. J. Urban Econ. 95 (C), 64–86.

CHAPTER TWO

Necessity of a minimum floor area regulation

2.1 Introduction Urban areas in most countries are confronted by various externalities. Agglomeration externalities arise in business areas, based on communication among people and firms. Negative externalities, which typically include traffic congestion, congestion in public facilities, noise, and insufficient sunlight, are prevalent in business areas and residential areas. This chapter argues that, in order to manage these externalities with floor area ratio (FAR) regulations, a minimum FAR regulation, though not common in current practices, is necessary nevertheless, along with a maximum FAR regulation. Regardless of whether the target externalities are positive or negative, the overall level of externalities in a zone depends on population densities. For example, since communication between agents is promoted by easy accessibility, agglomeration economies depend on population densities. Likewise, negative externalities, such as congestion, noise, and insufficient sunlight, also depend on population densities overall. If a certain zone in an urban area has both positive and negative externalities, the net effect is also dependent on population densities. To manage population density externalities in an urban area with highrise buildings, FAR regulations are effectivea in the sense that such regulations control building sizes, which are generally large in densely populated areas. Maximum FAR regulations control population density by restricting the size of buildings accommodating many households, which in turn helps reduce negative externalities. a

FAR regulations can deal with both building-height diseconomies and population density diseconomies. However, for the former, shape regulations are more appropriate than FAR regulations. This is because FAR regulations impose a common FAR on a large area, whereas building-height diseconomies arise within a specific site (e.g., a local area around a high building).

Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00002-9

© 2019 Elsevier Inc. All rights reserved.

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Lot size (LS) regulation is also used to control population density, particularly in detached housing zones. Although LS regulation and FAR regulation are both intended to control population density externalities, they must be examined separately because of their distinctly different characteristics. In particular, FAR regulations are second-best policies, whereas the optimal LS regulation can be a first-best policy. How does that difference arise? In the zones regulated by FAR regulations, households can choose their optimal floor space in the regulated buildings, whereas in the zones regulated by LS regulation, households cannot choose their optimal LSs. If FAR regulation reduced the total floor space of a building, floor rents would rise. In response, residents might simply choose to consume less floor space than before, causing population density to increase. Therefore, to reduce population density by a certain amount, a sufficiently small FAR would have to be imposed so that such a response to the regulations by residents could be overcome. In other words, FAR regulation can only control the population density indirectly, whereas LS regulation can do so directly. Correspondingly, FAR regulations are second-best policies because a difference pertains between the rent and the marginal supply cost in the “total floor space of a building” market. The optimal minimum LS regulation, on the other hand, achieves the first-best outcome, as described in Wheaton (1998). Designing secondbest policies is more difficult than designing first-best policies because the former deals with the residual deadweight losses.b This chapter examines where and how to impose FAR regulations in the presence of negative externality arising from population density using a simple two-zone city model in which population density externality is larger in one zone than in the other zone. Applying the model to obtain optimal FAR regulations following Kono et al. (2010), we show that a maximum FAR regulation should be imposed in the zone with larger externality and a minimum FAR regulation in the other zone. In other words the imposition of only maximum FAR regulations, a practice common in actual city planning, is not optimal. This necessity of minimum FAR regulations has also been argued by Joshi and Kono (2009). For designing a second-best policy, the deadweight losses in the related markets play an important role. This chapter

b

Lipsey and Lancaster (1956) define a general definition of the second-best policies. Accordingly, if there is some constraint within the general equilibrium system that prevents attainment of at least one of the Pareto optimal conditions, then attainment of the other Pareto optimal conditions is no longer necessarily welfare improving.

Necessity of a minimum floor area regulation

23

shows how a minimum FAR regulation minimizes the total deadweight loss, hereby justifying its necessity. We also verify, through numerical simulation, the necessity of simultaneous imposition of minimum and maximum FAR regulations in a monocentric city with traffic congestion externality. We also compare welfare gains under market equilibrium and different policy regimes including congestion toll (first-best policy), FAR regulation, urban growth boundary (UGB) regulation, and FAR with UGB regulations. The remaining part of this chapter is laid out as follows. Section 2.2 develops a two-zone city model with population density externality, and Section 2.3 examines FAR regulations for the same. Section 2.4 presents numerical simulation to explore different policy regimes including FAR regulation in a monocentric city with traffic congestion externality. Section 2.4 presents conclusions. The final subsection provides Technical Appendix.

2.2 The model 2.2.1 The city This chapter uses a simple city model composed of two residential zones, Zone h and Zone l with fixed areas, denoted Ah and Al , respectively.c Fig. 2.1 shows an image of the target city. The land is divided into an urbanization promotion area, which is composed of the two aforementioned residential zones, and an urbanization restriction area. This division is fixed. Without any loss of generality, we assume that the externality level in Zone h (high-externality zone) is more severe than in Zone l (low-externality zone) because of the difference in zone-specific environmental factor q
i , which represents all location-relevant utility-enhancing factors such as weather, existence of a shopping center, or transportation facilities. The variable q
i engenders different equilibrium population densities in Zones h and l. Although each zone in the model city could have both positive and negative externalities, we assume, for simplicity, that the net externality in each zone is negative.d The economy has two types of agents: homogeneous households and developers. Each household member works outside the residential areas. External diseconomies arise only from the population density. No resident c

The setting of two zones is for simplicity; multiple zones can also be grouped into two zones. The concluding section of the chapter explains how the results are applicable to cases in which positive externality is greater than negative externality.

d

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Traffic congestion and land use regulations

Fig. 2.1 City configuration.

migrates to another city (i.e., the city is closed),e reflecting that workers are unable to move to another city to work.

2.2.2 Household behavior The household utility in Zone i (i 2 {h,l}), denoted Vi, comprises floor space fi; composite goods zi; zone population Ni; and zone-specific environmental variable qi , which is exogenously given. The zone population Ni (i 2 {h,l}) in the utility function expresses the population density externality. Note that population density is proportional to zone population because the zone area is fixed by assumption. For that reason the externality in the model is related directly to both population density and zone population. Typical examples of such externalities are familiar urban problems such as traffic congestion and noise caused by the residents. For suburban residents also, such externalities might exist in the form of traffic congestion on specific roads leading to downtown or congestion in large-scale stores and public facilities. We adopt a quasilinear utility function for simplicity, and use consumer surplus for welfare analysis. The income is the sum of the exogenous wage w e

An open city is explored in the next chapter.

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Necessity of a minimum floor area regulation

and income from the land and developers’ profit. We assume public ownership of the land and developers (i.e., all households share the land and developers’ profit equally). The household model is formulated as " # X 1 Vi ¼ max uð fi , Ni , qi Þ + zi , s:t: zi + ri fi ¼ w + Ri Ai , i 2 fh, lg (2.1) Π+ z i , fi N i

where u() is subutility function, ri is the rent of floor space, Ri is land rent, N is total population (constant), Ai is the supply of land (constant), and Π is developers’ profit. Regarding the preference, we make the following plausible assumptions. First the subutility function u() decreases as Ni increases, that is, ∂u/∂Ni < 0, which expresses negative population externality. Moreover, uð fi , Ni , qi Þ increases as fi increases, that is, ∂u/∂fi > 0; also, ∂2 u=∂Ni 2 < 0 and ∂2 u=∂fi 2 < 0. Second the cross derivative of u() with regard to Ni and fi is zero, that is, ∂2u/∂fi ∂Ni ¼ 0. This assumption implies that the preference for floor space is independent of the external population density externality.

2.2.3 Developers’ behavior Landowner-cum-developers produce floor space Fi in Zone i (i 2 {h, l}). Floor space and land are supplied under perfect competition. The inputs for producing floor space are composite goods zdi and land Adi . Developer’s profit is formulated as X X Π¼ ri Fi  Si ðRi Fi Þ, i 2 fh, lg (2.2) i



i

   where Si ðRi Fi Þ ¼ min zdi , Adi zdi + Ri Adi , s:t: Fi ¼ Fi zdi , Adi (cost function of developers). The production function is assumed to be homogeneous of one degree for expressing perfect competition—the developers’ profit Π is zero. Floor space Fi is exogenous for developers because it is controlled under FAR regulations by the government. FAR is defined as floor space Fi divided by the respective fixed zone area Ai , that is, Fi =Ai .

2.2.4 Market clearing conditions The market equilibrium conditions are given in Eqs. (2.3)–(2.6) and are explained as follows. First the sum of all residential lots demanded equals the total supply of the floor area in each zone:

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Traffic congestion and land use regulations

Floor space: Ni fi ¼ Fi , i 2 fh, lg

(2.3)

The land consumed by developers equals the supply of land: Land area: Adi ¼ Ai , i 2 fh,l g

(2.4)

Residents migrate between zones without incurring any cost, so the utility levels must be equal between Zones 1 and 2: Utility: V1 ¼ V2 Finally the population constraint holds: Population: X Ni ¼ N , i 2 fh, lg

(2.5)

(2.6)

i

2.3 Optimal FAR regulations 2.3.1 Harberger’s welfare function and optimal conditions for FAR regulation Maximizing the utility of each household is equivalent to maximizing aggregate household utility because all households are homogeneous.  P  The aggregate household utility function is written as W ¼ i Ni Vi (i 2 {h, l}), which is hereinafter called social welfare. Optimal FAR regulations are defined as maximizing W by controlling both F1 and F2 subject to Eqs. (2.1)–(2.6). Through calculations provided in Technical Appendix 2(1), the total differential form of social welfare, dW, is obtained as       ∂Sh ∂Sl ∂u ∂u dW ¼ rh  dF h + rl  dF l + Nh dN h + Nl ½dN h  ∂Fh ∂Fl ∂Nh ∂Nl (2.7) Eq. (2.7) shows the total change in deadweight losses in the following three markets: FARs in Zones h and l and the population density market. The first term and second term in Eq. (2.7), respectively, express the changes in deadweight losses in the floor space markets in Zones h and l. The third term

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Necessity of a minimum floor area regulation

expresses the change in population density externality in Zones h and l. Note that dNh ¼  dNl because the city is closed by assumption. Eq. (2.7) is consistent with Harberger’s (1971) welfare change measurement formula, which represents the change in welfare with the existence of deadweight loss in an economy.f Social welfare W is written as W(Fh, Fl), that is, a function of floor spaces, which are policy controllable variables. To maximize W(Fh, Fl) with respect to floor spaces, we require ∂W/∂Fh ¼ 0 and ∂W/∂Fl ¼ 0. This implies that a change in the floor space does not change the social welfare at optimal condition. The optimization conditions for Fi (i 2 {h, l}) are obtained as follows, using Eq. (2.7):     ∂W ∂Sh ∂u ∂u dN h + Nh ¼ rh   Nl and ∂Fh ∂Fh ∂Nh ∂Nl dF h     ∂W ∂Sl ∂u ∂u dN h + Nh ¼ rl   Nl ∂Fl ∂Fl ∂Nh ∂Nl dF l

(2.8a)

(2.8b)

The first parenthesis in Eq. (2.8a), that is, floor rent minus the marginal cost in Zone h, implies the marginal change in the deadweight loss in the floor space market in Zone h according to the change in Fh. The second term in Eq. (2.8a) implies the marginal change in total population density externality in Zones h and l, resulting from the migration from Zone h to Zone l according to the change in Fh. Eq. (2.8b) can be similarly explained. An important point to note is that, to maximize the social welfare with respect to the supply of Zone i floor space (i 2 {h, l}), the marginal change in the sum of the deadweight loss in the floor space market in Zone i and the total population density externality in Zones h and l is expected to be zero.

2.3.2 Optimal FAR regulations and market equilibrium FAR Optimal FAR regulations can be compared with the market FAR: whether the regulated floor space is expected to be smaller or larger than the market equilibrium floor space. To that end, we rewrite the optimization conditions (2.8a) and (2.8b) in terms of two new functions, DFi and DN, and two new e variables, FM i and F i , with definitions following next. f

P Harberger’s welfare formula is expressed as dW/dQ ¼ iΞi ∂Xi/∂Q where Ξi is the distortion (e.g., price minus marginal cost) in market i, X is the output in market i, and Q is the policy variable. See Supplementary Note at the end of this Chapter.

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Function DFi (i 2 {h, l}) expresses marginal change in the deadweight loss in the floor space market in Zone i, given by the first parentheses in Eqs. (2.8a), (2.8b): ∂Sh ∂Sl and DFl ðFh , Fl Þ  rl  ∂F DFh ðFh , Fl Þ  rh  ∂F h l

The floor rent ri and the marginal cost ∂Si/∂Fi in these expressions are determined endogenously, given Fh and Fl. Therefore DFi is expressed as DFi(Fh, Fl) (i 2 {h, l}). M Next, we define variables FM h and Fl , which are market equilibrium floor spaces. These variables satisfy the following conditions:     ∂Sh   M  M  Rh Fh ,Fl , Fh ¼ 0 8Fl and DFh FhM ,Fl  rh FhM ,Fl  ∂Fh     ∂Sl     DFl Fh ,FlM  rl Fh , FlM  Rl Fh ,FlM , FlM ¼ 0 8Fh ∂Fl The earlier equations imply that the floor rent in Zone i (i 2 {h, l}) is equal to the marginal cost of floor area FM h with no regulation in Zone i. A similar argument applies to FM . l Likewise, we define FAR regulation mode by Fei (i 2 {h, l}) as FhM ðFl Þ + Feh ¼ Fh and FlM ðFh Þ + Fel ¼ Fl Therefore Fei represents Fi (the real or regulated FAR in Zone i) minus FM i (the floor space in Zone i at market equilibrium with no regulation in Zone i). In other words, Fei represents how much regulated FAR in Zone i is smaller or larger than the market equilibrium FAR. Fig. 2.2 portrays the mutual relation among DFi, Fei , FM i , and Fi (i 2 {h, l}), e where F i can be negative or positive. Fig. 2.1 presents an example in which Fei is negative. Finally, we define function DN(Fh, Fl), which represents marginal change in total population density externality in the two zones due to migration from Zone l to Zone h. The function DN(Fh, Fl) expresses the difference in population density externality between the two zones, that is, between the second parentheses in Eqs. (2.8a), (2.8b). Mathematically the definition is expressed as DN ðFh , Fl Þ  DNh ðFh , Fl Þ  DNl ðFh , Fl Þ where DNh(Fh, Fl)  Nh(Fh, Fl)∂u/∂Nh and DNl(Fh, Fl)  Nl(Fh, Fl)∂u/∂Nl.

Necessity of a minimum floor area regulation

29

Fig. 2.2 Relation among DFi,e F i , FM i , and Fi. (Source: Kono, T., Kaneko, T., Morisugi, H., 2010. Necessity of minimum floor area ratio regulation: a second-best policy. Ann. Reg. Sci. 44 (3), 523–539.)

Applying the four definitions described earlier to Eqs. (2.8a), (2.8b), the optimal conditions can be expressed as     dN h ∂W ¼ DFh FhM + Feh , FlM + Fel + DN FhM + Feh , FlM + Fel ¼ 0 and dF h ∂Fh (2.9a)     ∂W dN h ¼ DFl FhM + Feh , FlM + Fel + DN FhM + Feh , FlM + Fel ¼ 0 (2.9b) dF l ∂Fl Eqs. (2.9a), (2.9b) can be interpreted in a way similar to Eqs. (2.8a), (2.8b). The important difference is the division of Fi (i 2 {h, l}) into two new variables: Fei and FM i . While Fi represents regulated floor area in Zone i in Eqs. (2.8a), (2.8b), Eqs. (2.9a), (2.9b) represent FAR regulations in terms of Fei by how much does the regulated FAR exceed or get exceeded by the market equilibrium FAR, FM i . Technical Appendix 2(2) presents a summary of the characteristics of the terms in Eqs. (2.9a), (2.9b). A second-best policy is obtained according to the total deadweight loss in all markets affected by the regulation. Multiplying dFh to both sides of Eq. (2.9a) shows that the regulation-induced change in the Zone h floor area generates (1) change in the deadweight loss in the floor area market in Zone   h, that is, DFh FhM + Feh , FlM + Fel dF h , and (2) change in the population den  sity externality, that is, DN FhM + Feh , FlM + Fel dN h =dF h  dF h , which is the

30

Traffic congestion and land use regulations

target externality. Eq. (2.9a) shows that to achieve the optimal regulation, the two factors must be balanced such that the net sum is zero. A similar argument applies to Eq. (2.9b). This section, without loss of generality, is based on the assumption that   under no regulation, Feh , Fel ¼ ð0, 0Þ. Also by assumption, the population density externality in Zone h is greater than in Zone l, that is, j DNh(FM h , M M FM )j > j D (F , F )j because of the difference in zone-specific environl Nl h l mental factor. Under this setting, we can achieve the following proposition (see Technical Appendix 2(3) for the formal proof ). Proposition 2.1 (Optimal FAR regulation) Assuming, without loss of generality, that the population density externality in Zone M h is more severe than that in Zone l without regulations, that is, jDNh(FM h , Fl )j > M j DNl(FM h , Fl )j, the optimal FAR regulation must meet the following conditions: Feh < 0 and Fel > 0, that is, a smaller FAR in Zone h and a larger FAR in Zone l than the respective unregulated market equilibrium FAR. According to Proposition 2.1, to achieve a smaller FAR in Zone h and a larger FAR in Zone l (see Fig. 2.3), not only maximum FAR regulation must be implemented in Zone h, but also minimum FAR regulation must be implemented in Zone l. Fig. 2.3A shows the floor market in Zone h, and Fig. 2.3B shows the floor market in Zone l. The maximum FAR regulation in Zone h implies that the developers are regulated to build the floor space within the interval [0, Fh ∗ ), whereas the minimum FAR regulation in Zone l

M M M Fig. 2.3 Optimal regulation (when j DNh(FM h , Fl )j > j DNl(Fh , Fl )j). (Source: Kono, T., Kaneko, T., Morisugi, H., 2010. Necessity of minimum floor area ratio regulation: a second-best policy. Ann. Reg. Sci. 44 (3), 523–539.)

Necessity of a minimum floor area regulation

31

implies the same within the interval ðFl ∗ , ∞ where Fh ∗ and Fl ∗ are the optimal floor spaces. An intuitive interpretation of the necessity of such FAR regulations follows. Maximum FAR in Zone h in addition to minimum FAR in Zone l induces migration from Zone h to Zone l, given the city is closed by assumption. Therefore it reduces the aggregate population density externality, revealing the benefit of FAR regulations in both zones. The cost of such FAR regulations is the aggregate deadweight loss in floor space markets in Zones h and l. The marginal deadweight loss of inducing migration through effective FAR increases from zero in each zone. Therefore the combination of maximum FAR in Zone h with minimum FAR in Zone l is more efficient than FAR regulation imposed in only one zone to minimize the cost of reducing aggregate population density externality. The earlier interpretation can be explained by using Fig. 2.4. Presume first that FAR regulation is adopted only in Zone h to mitigate a certain amount of externality through migration to Zone l. The dotted lines in Fig. 2.3 show demand curves that are shifted because of FAR regulation. Such regulation generates deadweight loss in floor area market in Zone h. The area bound by a thick line in the left graph in Fig. 2.3 (i.e., Area X) is the deadweight loss due to such regulation. On the other hand, assuming that FAR regulation is adopted in both zones in which FAR in Zone h is decreased and FAR in Zone l is increased from the respective market equilibrium FAR, this regulation would generate deadweight loss in both zones as given by the shaded areas in the left and right graphs in Fig. 2.3 (i.e., Area

Fig. 2.4 Interpretation of optimal FAR regulations. Note: (1) Area X: Deadweight loss when only maximum FAR is imposed in Zone h. (2) Area Y + Z (total shaded area): Total deadweight loss when a maximum FAR in Zone h is combined with a minimum FAR in Zone l. (Source: Kono, T., Kaneko, T., Morisugi, H., 2010. Necessity of minimum floor area ratio regulation: a second-best policy. Ann. Reg. Sci. 44 (3), 523–539.)

32

Traffic congestion and land use regulations

Y + Z). Still the total amount of deadweight loss (i.e., Area Y + Z) is smaller than that related to the first regulation (i.e., Area X) if the mitigated externality amount is the same as in the first regulation. In other words, FAR regulation imposed in both zones is more efficient than FAR regulation imposed in only one zone. Another explanation might further elucidate the interpretation presented earlier. FAR regulations are measures to increase the social welfare. If only maximum FAR regulation can be used in the economy with no minimum FAR regulation, the maximum FAR will be imposed only in Zone h, where the population density externality is more severe than in Zone l, in order to reduce the total externality in the economy. However, a minimum FAR in Zone l, administered along with a maximum FAR in Zone h, can reduce the total congestion externality more efficiently because such FAR regulation induces migration from Zone h to Zone l, given the city is closed. Summing up, the sole imposition of maximum FAR regulation in one zone can yield the same population distribution attained by the combination of maximum and minimum FAR regulations, but building sizes in the zones would differ under these two cases, resulting into different deadweight losses. The combination of maximum FAR in Zone h with minimum FAR in Zone l turn outs to be more efficient than FAR regulation imposed in only one zone. This mechanism resembles the generalized Le Chatelier’s principle, which states that, for a maximum condition of economic equilibrium subject to some constraints, excluding one of the constraints increases the response to a parameter change. That is, a large flexibility of measures, that is, the use of a minimum FAR regulation in addition to maximum FAR regulation, can increase social welfare. Next, we confirm the feasibility of optimal FAR regulations under a decentralized economy. More specifically, we will check whether the regulations are compatible with the nonnegative profit constraint. There is no problem in the supply of floor space under both Feh < 0 and Fel > 0 because the developer always makes zero profit under perfect competition. Problems might be encountered with the supply of land in Zone l where minimum FAR is imposed. When Fe2 > 0, under the optimal regulation, the land rent can be negative. Fig. 2.5 shows a relation between demand and supply for floor space in Zone l, the land area of which is fixed. The land rent is negative if triangle CDE has an area smaller than triangle ABC. This is because rectangle AGDE expresses the total revenue from floor rents, and the building cost is triangle DBG. Therefore the land rent is expressed by rectangle AGDE minus triangle DBG.

Necessity of a minimum floor area regulation

33

Fig. 2.5 Feasibility of minimum FAR regulations in Zone l. (Source: Kono, T., Kaneko, T., Morisugi, H., 2010. Necessity of minimum floor area ratio regulation: a second-best policy. Ann. Reg. Sci. 44 (3), 523–539.)

In the case of a negative land rent, the landowner does not sell or rent land to the developers, and as such, buildings cannot be built because of no land supply. However, when land rent is positive, land can be supplied. In other words, optimal FAR regulations can be realized to the extent that land rent is positive.g If the land rent is negative, some policy intervention such as transfers to landowners is necessary to impose the optimal regulation.

2.4 A case of a monocentric city: Numerical simulation 2.4.1 The model city The preceding sections explain the necessity of minimum FAR regulation in a two-zone city. In fact the necessity of minimum FAR regulation holds in a monocentric city as well. We present a numerical example in this section, which assumes traffic congestion externality in a monocentric closed city. The example is from Kono et al. (2012). We adopt the standard model of a congested monocentric city a` la Wheaton (1998) and Brueckner (2007). The model is described in detail in Chapter 3 along with theoretical analysis; here, we present a summary. g

The land in the present model has no opportunity cost for residential use, so the land is supplied if the land rent is positive. If the land has a positive opportunity cost (e.g., agricultural rent), the land is supplied for residential use if the rent is greater than the opportunity cost.

34

Traffic congestion and land use regulations

The city is circular and is symmetric along any radial axis. The residential area expands from x ¼ 1 at the CBD edge to x ¼ xb at the city boundary. The city is divided into several concentric bands of equal width. The total floor supply per unit area of land at x is denoted F(x), which represents FAR at x.h At each x a fraction ρ(x) of the land is used for road. The share of land available for housing at x from the CBD edge is then given by 2πx[1  ρ(x)]. In our model the city land is owned by absentee landowners from whom land is rented by developers to build dwelling units to be rented out to households. The city government imposes FAR regulation on all buildings and UGB regulation to ease traffic congestion. For simplicity, we define population in terms of the number of households. The city is inhabited by N identical households. Every household chooses only one residence although it could be anywhere within the city. Residents have a quasilinear utility function v ¼ z + υ(f ), where subutility υ(f ) is a function of household floor space consumption f, and z denotes numeraire composite goods that include all consumer goods except floor space. Each household earns income y per period, which is spent on floor rent and commuting. The income constraint is expressed as z + rf ¼ y  T(x), where r is floor rent and T(x) denotes commuting cost to and from the CBD for the household residing at x. Because many households can reside in a building, floor rent r equals maximum floor rent bid by a household as a result of competition among residents. Such behavior is expressed by max r ¼ f ,z

y  T ðxÞ  z s:t: v ¼ z + υð f Þ f

(2.10)

Developers combine housing capital (building materials) and land to produce dwellings under prevailing FAR regulation. Developers are assumed to be perfectly competitive and are, therefore, price takers. Floor space supply per unit of land is given by housing production function F(S), where S is capital-to-land ratio. Using the reverse production function S(F), the sum of developers’ net profit from the total floor space supply, denoted Π, is given by ð xb (2.11) Π ¼ 2πx½1  ρðxÞ½F ðxÞr ðxÞ  SðF ðxÞÞ  RðxÞdx 1 h

For sufficiently large city radius compared with the lot size of a building, the building size (or FAR) can be treated as a continuous function of distance.

35

Necessity of a minimum floor area regulation

where R(x) is the land rent paid to the absentee landowners. The price of capital is normalized at unity. Under no FAR regulation, developers maximize profit per unit of land with respect to F(x). Solving the relevant first-order condition for F(x) yields r ðxÞ 

∂SðF ðxÞÞ ¼0 ∂F ðxÞ

(2.12)

solving which gives F(x) ¼ F(y  T(x), v(x)) and S(F(x)) ¼ S(y  T(x), v(x)). Plugging these into the profit function (2.11) and noting that developers’ profit at each location is zero because of perfect competition, the land rent is expressed as R(x) ¼ R(y  T(x), v(x)). Finally, population density, denoted P(x), is defined as housing square feet per unit of land divided by square feet per dwelling, as P ðxÞ ¼

F ðy  T ðxÞ, vðxÞÞ f ðy  T ðxÞ, vðxÞÞ

(2.13)

Under the FAR regulation, the floor space supply F is set exogenously, which implies that the developers cannot maximize profit with respect to F. Using zero-profit condition at each location, the land rent is given by RðxÞ ¼ F ðxÞr ðxÞ  SðF ðxÞÞ

(2.14)

For simplicity, we assume that automobiles are the only mode of commuting and that only one radial route extends across the residential zone between the CBD and city boundary. Moreover the commuting cost is incurred only when commuting to and from the CBD edge. Because vehicles from more distant locations increasingly join traffic while moving toward the CBD, traffic volume is not uniform across all locations but increases toward the central locations. Under congestion the unit cost of commuting for a commuter at x, denoted Tunit(x), depends on the number of commuters passing through the ring at x, denoted n(x). Assuming that one member Ð of each household commutes to the CBD to work, n(x) is given by n(x)  xxb2πm[1  ρ(m)]P(m)dm. Consistent with most of the prior literature including Brueckner (2007), Tunit(x) is given by   nðxÞ γ Tunit ðxÞ ¼ η + δ (2.15) 2πxρ where η, δ, γ are positive parameters. The term δ[n(x)/2πxρ]γ in Eq. (2.15) denotes congestion. Note that n(1) ¼ N and n(xb) ¼ 0. When an additional

36

Traffic congestion and land use regulations

commuter joins traffic at x, the resultant change in congestion cost is given by ∂Tunit(x)/∂n(x), which when multiplied by n(x) gives the total externality caused by unpriced congestion, expressed as   ∂Tunit ðxÞ nðxÞ γ  ptoll ðxÞ (2.16) nðxÞ ¼ γδ ∂nðxÞ 2πxρ where ptoll(x) equals congestion toll at x that fully internalizes congestion externality. A commuter at x pays the total commuting cost, denoted T(x), which is given by ðx T ðxÞ ¼ ½Tunit ðmÞ + ptoll ðmÞdm (2.17) 1

When no congestion toll is levied in the city, ptoll in Eq. (2.17) is set to zero. Finally, to help set the stage for what will come later, we differentiate n(x) and T(x) with respect to x, which yields dnðxÞ ¼ 2πx½1  ρP ðxÞ  n0 ðxÞ and dx dT ðxÞ  T 0 ðxÞ ¼ Tunit ðxÞ + ptoll ðxÞ dx

(2.18)

2.4.2 Setup for numerical simulation Similar to the setup adopted by Brueckner (2007), we divide the city into narrow, discrete rings with an equal width denoted ε. Fig. 2.6 shows the city

Fig. 2.6 Structure of our model city.

Necessity of a minimum floor area regulation

37

structure. The rings are indexed by j, where j ¼ 1 at the CBD edge. The inner radius of ring j is given by xj ¼ 1 + ε[ j  1], where xj now represents a distance variable for the corresponding ring. The utility function and the floor production function are defined as v(z, f ) ¼ z + α ln f and F(S) ¼ θSβ, respectively, where α and θ are positive multiplicative factors and 0 < β < 1. Using Eqs. (2.10)–(2.14), we obtain Unit floor rent:   (2.19) rj ¼ α exp κ j Floor space per household:   fj ¼ exp κj

(2.20)

  Sj ¼ Λ exp ^κj

(2.21)

  Rj ¼ Ωexp ^κ j

(2.22)

  Pj ¼ Ψexp ^κ j

(2.23)

Capital-to-land ratio:

Unit land rent:

Population density:

where κ j  [V  y + Tj + α]/α; Λ  [βθα]1/[1β]; Ω  Λ[1  β]/β; Ψ  θΛβ; ^κ j  κj =½1  β; and V is the parametric utility level, which is exogenously set. Note that household utility across all locations should be equal to the equilibrium utility level, which in our case is V, that is, v(x) ¼ V. Note that income y is also set exogenously. The total commuting cost from ring j, Tj, is derived asi   (2.24) T1 ¼ 0; Tj + 1 ¼ Tj + εT 0 ðxÞ ¼ Tj + ε Tunit j + ptoll j where Tunitj and ptollj are set per kilometer, using Eqs. (2.15), (2.16), as     nj γ nj γ and ptoll j ¼ γδ (2.25) Tunit j ¼ η + δ 2πxj ρ 2πxj ρ P where nj ¼ jk¼i*[1  ρ]π[x2k+1  x2k]Pk and j* denotes the outermost ring such that nj* ¼ 0. Finally the social welfare is given as i

Note that T1 ¼ 0 because of zero commuting cost from the CBD edge, by assumption. Likewise, Ti+1 is expressed using first-order approximation.

38

Traffic congestion and land use regulations

W¼

j* X

h i  ½1  ρπ x2j + 1  x2j Pj V + Rj  Ra + nj ptoll j

(2.26)

j¼1

where ptollj is set to zero when no congestion toll is levied. The iterative process starts at j ¼ 1 with Tunit1 ¼ 0 and n1 ¼ N and is carried out conditional on the value of V that should satisfy the equilibrium conditions. In the laissez-faire and toll-regime cases, the iteration stops when j reaches a value j* such that nj*0 but nj*+1 < 0, indicating that the population N is just accommodated within the radius of xb ¼ xj*; the increment in nj is expressed, using Eq. (2.18), as   nj + 1 ¼ nj + εn0 xj ¼ nj  ε2πxj ½1  ρPj

(2.27)

We then check the equilibrium condition whether the land rent at the city boundary should be equal to the agricultural rent, that is, Rj* ¼ Ra. Until we achieve this within a reasonable degree of accuracy, the iteration process is repeated by adjusting V. In the UGB case, the city boundary, indexed by, say j**, is exogenously set—that is, xb ¼ xj** is predetermined. The UGB equilibria are computed for a range of xb values that lie below the laissez-faire urban radius by reducing xb in steps of ε. The optimal xb value is that which : maximizes W. The iteration, as explained earlier, stops when nj** ¼ 0 within a reasonable degree of accuracy; otherwise the iteration process is repeated by adjusting V. Note that the condition Rj* ¼ Ra does not hold in the UGB case. In the FAR regulation case, the floor space supply on each ring j, denoted Fj, is set exogenously. Eqs. (2.13), (2.14) are rewritten as Dj ¼ Fj/fj and Rj ¼ Fjrj  Sj, where rj and fj are defined in Eqs. (2.19), (2.20), respectively, and Sj ¼ [Fj/θ]1/β. We set parameters as follows. The total number of households N is set at 100,000. The income per household is set at $40,000 as in Brueckner (2007). The housing parameter α in the quasilinear utility function is set at 8000, implying 20% of the income of $40,000. Next, setting ρ ¼ 0.2 as in Brueckner (2007), 20% of the land in each ring is allocated for roads. Agricultural land rent Ra is set at $150,000 per square km. The parameter β in the housing production function F(S) ¼ θSβ is set at 0.85, and the multiplicative factor θ is set at 0.0001 as in Brueckner (2007). The ring width ε is

Necessity of a minimum floor area regulation

39

set at 0.5 km.j The intercept parameter η in the commuting-cost function expresses the commuting cost incurred while driving through 1 km in the case of no congestion. Setting the number of trips to the CBD as 225 round trips per year, average speed as 30 km/hour, travel cost including travel time as US$15/hour, and one worker per household, η is set at US$225.k For sensitivity analysis the values of congestion exponent γ and multiplicative factor δ differ across the examples.

2.4.3 Numerical results This section presents five numerical examples to simulate different levels of congestion externality. We set the following five sets of the combination of γ and δ,l which are referred to as Examples 1–5: Example 1: γ ¼ 1.25, δ ¼ 0.001, Example 2: γ ¼ 1.20, δ ¼ 0.001, Example 3: γ ¼ 1.25, δ ¼ 0.0002, Example 4: γ ¼ 1.35, δ ¼ 0.0002, Example 5: γ ¼ 1.40, δ ¼ 0.00015, where ‘< >’ denotes each example’s ranking with regard to the resultant total congestion externality level in decreasing order. Each of the five examples involves comparison of five equilibria: (i) the laissez-faire equilibrium, (ii) the equilibrium under the congestion-toll regime, (iii) the equilibrium with an optimally chosen UGB regulation, (iv) the equilibrium with an optimally chosen FAR regulation, and (v) the equilibrium with an optimally chosen FAR and UGB regulation. The goal is to gauge the efficacy of an optimal UGB and FAR regulation by comparing the resultant welfare gain to that achieved under the firstbest congestion-toll regime. The numerical results are presented in Table 2.1. Although the congestion externality level depends on the distance away from the CBD, one way of measuring the total city congestion externality is to consider the welfare gain from “laissez-faire” to “toll regime,” which can be calculated using the results shown in the second column of Table 2.1. The ranking order of the examples, therefore, is as follows: Example 1 This ring width ε is larger than that adopted by Brueckner (2007). Because our case necessities setting FAR in each ring, a moderate value of ε is adopted for faster results. k This is because 225 (times)  2 (round trip)  [1.0 (km)/30 (km/hour)]  15 ($/hour) ¼ $225, where the value within the square parenthesis expresses the average time to pass through 1 km. l Note that the effects of a marginal change in δ and in γ on congestion externality are different, expressed respectively, by ϕ(x)γ and δγϕ(x)γ1, where ϕ(x)  n(x)/[2πxρ]. j

xb (km)

40

Table 2.1 Numerical results W (108 $)

Welfare gain (%)

V

F(1)

F(xb)

P(1) (per square km)

R(1) (106 $/sq. km)

9.0739 9.1721 9.0822 9.1354 9.1445

100.0 8.4 62.6 71.9

8332 6621 8265 8389 8318

123.51 229.58 129.41 220.31 228.17

11.19 11.71 14.21 10.78 11.62

2157 4476 2281 3824 3994

2.5 5.3 2.7 2.7 1.7

9.7303 9.7601 9.7305 9.7543 9.7554

100.0 0.7 80.5 84.2

8987 7885 8904 9007 8967

95.22 150.45 100.95 146.49 149.57

11.92 11.81 15.14 10.48 11.81

1590 2726 1702 2439 2503

1.9 3.2 2.0 1.5 1.6

10.3834 10.4086 10.3857 10.4010 10.4010

100.0 9.1 69.8 69.8

9700 9246 9603 9694 9694

68.71 83.41 73.28 83.25 83.25

11.43 10.96 14.40 10.96 10.96

1082 1360 1168 1311 1311

1.3 1.6 1.4 1.2 1.2

Example 1 (γ 5 1.25, δ 5 0.001)

Laissez-faire Toll regime Optimal UGB Optimal FAR Optimal FAR and UGB

10.0 9.0 9.0 10.0 9.0

Example 2 (γ 5 1.20, δ 5 0.001)

10.0 9.5 9.0 10.0 9.5

Example 3 (γ 5 1.25, δ 5 0.0002)

Laissez-faire Toll regime Optimal UGB Optimal FAR Optimal FAR and UGB

10.5 10.5 10.0 10.5 10.5

Traffic congestion and land use regulations

Laissez-faire Toll regime Optimal UGB Optimal FAR Optimal FAR and UGB

Laissez-faire Toll regime Optimal UGB Optimal FAR Optimal FAR and UGB

10.5 9.5 9.5 10.0 10.0

9.7275 9.7667 9.7307 9.7601 9.7601

100.0 8.1 83.2 83.2

9029 7749 8952 9012 9012

90.29 150.60 95.35 146.30 146.30

10.84 12.17 13.66 10.79 10.79

1493 2726 1592 2425 2425

1.8 3.3 1.9 1.5 1.5

9.4609 9.5129 9.4655 9.5028 9.5047

100.0 8.8 80.6 84.2

8722 7158 8642 8755 8716

102.23 180.36 108.21 175.72 178.51

11.93 11.63 15.14 10.36 11.59

1728 3370 1847 2958 3019

2.1 4.1 2.2 1.4 1.5

Example 5 (γ 5 1.4, δ 5 0.00015)

Laissez-faire Toll regime Optimal UGB Optimal FAR Optimal FAR and UGB

10.0 9.5 9.0 10.0 9.5

Necessity of a minimum floor area regulation

Example 4 (γ 5 1.35, δ 5 0.0002)

Source: Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630.

41

42

Traffic congestion and land use regulations

Fig. 2.7 City size under Examples 1–5. (Source: Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630.)

($0.0982  108), Example 5 ($0.0520  108), Example 4 ($0.0392  108), Example 2 ($0.0298  108), and Example 3 ($0.0252  108), where the value within the parenthesis represents the welfare gain. The results on the city radius (i.e., xb) under each policy are shown in the first column of Table 2.1 and depicted in Fig. 2.7. In Example 1 the city radius under the toll regime and the optimal UGB policy with and without FAR restriction is smaller—at 9 km in each case—than that in laissez-faire equilibrium at 10 km.m,n However, under the optimal FAR policy without UGB, there is no significant change in the city size. When congestion externality is reduced from that in Example 1 to that in Example 2, the city expands to 9.5 km under the toll regime and the optimal FAR policy with UGB. When congestion externality is further decreased as in Example 3, there is an expansion of city size under all five equilibria. Considering two more examples—Examples 4 and 5—we find that the city shrinks from the laissez-faire equilibrium the most under the optimal UGB policy. The optimal FAR policy with no UGB does not reduce the city size in most cases, an exception being Example 4. As expected the city size is largest in the laissez-faire equilibrium, whereas the toll regime and optimal UGB

Note that because ε is set at 0.5 km, the city size is expressed as a multiple of 0.5 km. F(xb) slightly differs between the laissez-faire and the toll regime cases as shown in Table 2.1. Because land rent at the city boundary equals the agricultural rent in both cases and F is unregulated, the value of F(xb) should be the same in both cases. The difference arises from the assumption of discrete rings for the numerical analysis as the city is divided into discrete rings with an equal width (0.5 km). Accordingly, land rent at the urban boundary ring does not equal the agricultural rent exactly. However, at the outside ring next to the boundary ring, land rent is below the agricultural rent.

m n

Necessity of a minimum floor area regulation

43

policy with and without FAR restriction result in a more compact city only in the case of high congestion externality as evident from Examples 1 and 5. The results on the land rent at the CBD edge (i.e., R(1)) under each policy are shown in the last column of Table 2.1 and depicted in Fig. 2.8. The land rent reflects the household density. The household density (resp. building size) at the CBD edge is highest (resp. largest) under the toll regime in all five examples, being closely followed by optimal FAR policy with and without UGB. When congestion externality is high (Examples 1 and 5), the household density under the toll regime significantly exceeds that in the laissez-faire equilibrium, whereas when congestion externality is low (Example 3), no policy increases household density at the CBD edge significantly, which is an expected result. In all cases the land rent at the CBD edge is highest under the toll regime and is significantly so when the congestion externality is high as in Examples 1 and 5. We next compare spatial variation of household (population) density under different equilibria, using Example 1. As shown in Fig. 2.9, the household density under the toll regime significantly exceeds market density and that under the optimal FAR policy at the more central locations. At the peripheral locations the household density under the toll regime is less than that in the laissez-faire or optimal FAR equilibria. The household density under the optimum UGB policy exceeds market density across all locations. On the other hand the household density profiles under the optimal FAR policy, with and without UGB, closely follow that under the toll regime. But does that mean the optimal FAR policy is a useful substitute for the first-best congestion toll policy? To answer this, we next gauge the efficacy

Fig. 2.8 Land rent at the CBD edge under Examples 1–5. (Source: Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630.)

44

Traffic congestion and land use regulations

Fig. 2.9 Household density profiles under Example 1. Note: Because simulation results are obtained for each zone, our results are stepped curves. However, we use smooth curves to better facilitate comparison between results. (Source: Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630.)

of an optimal FAR policy by comparing the resultant welfare gain to that achieved under the first-best toll regime. The welfare gain, which is our main focus, is highest under the toll regimeo among all policies under consideration.p This is not surprising given that the toll regime is a first-best policy. In contrast the optimal UGB policy produces only a small fraction—at best 9% (Example 3)—of the welfare gain achieved under the toll regime. This justifies Brueckner (2007)’s conclusion that the UGB policy is a poor substitute for the toll regime. However, the welfare gain under the optimal FAR policy is impressive—and even more so if UGB is also in place. The welfare gain under the optimal FAR policy with UGB accounts for about 70% to 84% of that achieved under the toll regime in our five examples, whereas even without UGB regulation the optimal FAR policy still yields from 63% to over 80% of the welfare gain under the toll regime (Fig. 2.10).

o

The dollar-equivalent welfare gain under the toll regime over the laissez-faire equilibrium in Example 1 is about $98 per household (given by dividing difference in W by N ), but the same is only $25 in the less congested case of Example 3. Brueckner (2007) also achieves a low dollar-equivalent welfare gain under the toll regime. p On the contrary, the toll regime produces the lowest household utility among all five equilibria (see Table 2.1); the difference in utility under the toll regime and in other equilibria is less pronounced in the case of high congestion externality (Examples 1 and 5). Except the toll regime, all other policies yield almost similar utility levels as achieved in the laissez-faire equilibrium although the optimal UGB policy tends to lower utility, and especially so when the congestion externality is low (Example 3).

Necessity of a minimum floor area regulation

45

Fig. 2.10 Welfare gains under Examples 1–5. (Source: Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630.)

Such large improvement in the welfare by FAR and UGB regulation can be explained as follows. As shown in Fig. 2.9, FAR regulation can achieve almost the same household density profile as the toll regime. This is because FAR regulation can control the building size at each point although it engenders deadweight loss in the floor market. But under FAR regulation only, the city size is larger than that under the toll regime, as shown in Fig. 2.9. So, if the UGB regulation is combined with FAR regulation, the density profile looks similar to the toll regime, and accordingly a large gain in the welfare is achieved. The result that the optimal FAR regulation involves enforcement of minimum FAR regulation at certain locations (which is central locations in the current model) and maximum FAR regulation at other locations (which is in suburban locations in the current model) is verified by the numerical results. Results of Example 1 are depicted in Fig. 2.11, which shows that the population density under the optimal FAR policy exceeds (resp. is exceeded by) the market density near the CBD (resp. city boundary).q Note that under the maximum (resp. minimum) FAR regulation, the population density is lower (resp. higher) than the market density. The combination of minimum FAR policy at the central locations with maximum FAR policy at the farthest locations is more efficient than the enforcement of only maximum FAR regulation to minimize total q

Although Fig. 2.6 shows the case of optimal FAR policy with UGB, the same result holds even if the UGB policy is absent.

46

Traffic congestion and land use regulations

Fig. 2.11 Optimal FAR and UGB regulations under Example 1. Note: Because simulation results are obtained for each zone, our results are stepped curves. However, we use smooth curves to better facilitate comparison between results. (Source: Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630.)

deadweight loss, which is the cost of reducing negative externality through the FAR policy.r Moreover the qualitative result related to FAR regulation holds irrespective of whether UGB regulation is imposed or not. Regarding UGB regulation, under simultaneous imposition of FAR and UGB regulation, the optimal city is more compact than the market city. In summary the optimal regulation on the building size requires not only downward adjustment to the market population density at the boundary locations but also upward adjustment at the central locations. Accordingly, minimum FAR regulation that raises the population density cannot achieve a socially optimal solution on its own: maximum FAR regulation at city boundary locations, which decreases population density, is also necessary. One important numerical result is that optimal FAR regulation is effective all over the city in the simulation. In other words the land rent is not less than zero in the simulation even if the minimum FAR regulation is imposed. Land rent profiles in Fig. 2.12 shows the land rent curves under market equilibrium (i.e., no policies) and the following policy regimes: congestion toll, UGB, and FAR with UGB. The land rents under combined FAR and UGB r

In other words, the combination of “maximum FAR in one part of the city” with “minimum FAR in another part of the city” is more efficient than “FAR regulation imposed in only one part of the city” to minimize total deadweight loss, which is the cost of reducing negative externality. See related discussion in Kono et al. (2010) for an intuitive explanation.

Necessity of a minimum floor area regulation

47

Fig. 2.12 Land rent profile depending on policies.

regulation are lower than those under other cases at the central locations, but the land rents are still positive. The minimum FAR regulation should be imposed only at locations close to the city center. In the center the floor rent is normally quite high. So, even if the building size is regulated to be greater than the market FAR and accordingly even if the unit floor rent falls below the market rent, the land rent can be greater than zero. Another important point shown in Fig. 2.12 is that FAR regulation makes land rents around the CBD lower than under other policy regimes. This implies that these policies yield different income distributions. In addition to the efficiency of policies, we should also consider such income distribution effects.

2.5 Conclusion In this chapter, we have examined optimal conditions for FAR regulations in the presence of negative externality associated with population density. The model considers two zones—Zone h and Zone l—in a closed city containing multistory buildings. Real-world examples of the model city include island cities (e.g., Singapore and Hong Kong), urbanized valleys (e.g., Kathmandu), and any arbitrarily defined urban areas such as the Japanese-style urbanization promotion areas. In the model, Zone h has greater negative externality associated with population density, without loss of generality. To maximize social welfare

48

Traffic congestion and land use regulations

in the closed city, not only must maximum FAR regulations be adopted in Zone h, but also minimum FAR regulations must be implemented in Zone l. Regulations imposing a maximum FAR in Zone h, but no regulation in Zone l, cannot achieve social welfare maximization. Maximum FAR regulation alone, a general practice that is common in actual city planning, is thus insufficient from a welfare perspective. Although this model uses only two zones in a closed city, the results can be extended to many zones in a closed city, by categorizing multiple zones into two zones.s This model considers optimal FAR regulations only for residential areas, but one can apply the conclusions to business areas by replacing the utility functions with profit functions. Business offices tend to locate only in central areas with many buildings, which is consistent with the model. However, positive externalities from proximity economies must also be considered for business areas. In the case of proximity economies rather than population density diseconomies, one can easily reinterpret the optimal regulations in the opposite way: the imposition of a minimum FAR in Zone h and a maximum FAR in Zone l is needed if the proximity economies are greater in Zone h than in Zone l. Under such regulation the FAR in Zone h must be greater than the market equilibrium. Consequently the urban area enjoys the proximity economies, while the deadweight loss generated by the regulation is minimized. The mechanism is fundamentally identical to that outlined in this chapter. In Chapter 5, we explore optimal FAR regulations in business zone in a continuous monocentric city. In addition to theoretical treatment of a two-zone city with population density externality, we have also provided numerical examples using a monocentric closed city with traffic congestion externality to demonstrate that the necessity of a minimum FAR regulation also holds in such a setup. Besides FAR regulation the numerical simulation compares welfare gain among market equilibrium and different policy regimes including congestion pricing (first-best policy), UGB regulation, and FAR with UGB regulation. This demonstrates the necessity for simultaneous imposition of multiple land use regulations (FAR and UGB in our case) to obtain efficient second-best policy alternatives to congestion pricing against traffic congestion externality. We consider multiple land use regulations in detail in Chapter 5.

s

For example, if the urban area has three zones, at least one zone needs a minimum FAR regulation.

49

Necessity of a minimum floor area regulation

Technical Appendix 2 (1) The total differential form of social welfare (Proof of Eq. 2.7) Social welfare changes can be described as follows. From Eq. (2.1), omitting constant variable qi and substituting the optimality condition with regard to fi, we have " # X 1 Ri Ai  ri fi∗ (A2.1) Vi ¼ uðfi ∗ , Ni Þ + w + π+ N i where superscript “∗” denotes the optimal solution. Maximizing social welfare W is equivalent to maximizing aggregate household utility because all households are homogeneous. Hence W¼

X

Ni V i

(A2.2)

i

From Eqs. (A2.1), (A2.2) and noting that dN1 ¼ dN2, we have dW ¼ Nh dV h + Nl dV l + ðVh  Vl ÞdN h     ∂u ∂u ¼ Nh dN h  dr h fh + Nl dN l  dr l fl + dΠ + Ah dRh + Al dRl ∂Nh ∂Nl (A2.3) From Eq. (2.2) the total differential form of profit dΠ is obtained as dΠ ¼ Fh dr h + Fl dr l + rh dF h + rl dF l  Adh dRh  Adl dRl  

∂SðFl Þ dF l ∂Fl

∂SðFh Þ dF h ∂Fh (A2.4)

Substituting Eq. (A2.4) into Eq. (A2.3) and using market equilibrium conditions Eqs. (2.3)–(2.6), we obtain expression Eq. (2.7). (2) Characteristics of the terms in Eqs. (2.9a), (2.9b)     DFh FhM + Feh , FlM + Fel in Eq. (2.9a) and DFl FhM + Feh , FlM + Fel in Eq. (2.9b) have the following relations:

50

Traffic congestion and land use regulations

8  M < < 0 for Feh > 0, 8Fl e DFh Fh ðFl Þ + F h , Fl ¼ 0 for Feh ¼ 0, 8Fl : > 0 for Feh < 0, 8Fl 8 < < 0 for Fel > 0, 8Fh   DFl Fh , FlM ðFh Þ + Fel ¼ 0 for Fel ¼ 0, 8Fh : > 0 for Fel < 0, 8Fh

and

(A2.5)

Eq. (A2.5) indicates whether the floor rent is greater or smaller than the marginal cost given Fei (i 2 {h, l}). Noting the definition of Fei (recall Fig. 2.1), it is clear that if Fei is positive (resp. negative), then the floor rent is lower (resp. greater) than the marginal cost. Second the signs of dNh/dFh in Eq. (2.9a) and dNh/dFl in Eq. (2.9b) are given as  dN h dN h dN l 0, ¼ dF h dF l dF l Eq. (A2.6) states that an increase in the floor space in a zone increases population in the zone. Eq. (A2.6) can be derived rigorously, but the meaning is apparent without any derivation.   The term DN FhM + Feh , FlM + Fel in Eqs. (2.9a), (2.9b) satisfies ∂DN ∂DN < 0, >0 ∂Feh ∂Fel

(A2.7)

The left-hand sides of the inequalities in Eq. (A2.7) show the effect of change in Feh and Fel on DN, which is “population density externality in Zone h minus that in Zone l” through the market mechanism. An increase in Feh increases the absolute value of the population density externality in Zone h and decreases that in Zone l because it would engender a population increase in Zone h and a decrease in Zone l. Consequently an increase in Fe1 decreases DN; that is, the first inequality of (A2.7) holds. A similar discussion applies for an increase in Fe2 . (3) Proof of Proposition 2.1 Optimal regulation on floor space is derived by using Fig. A2.1 in the following manner. The horizontal axis in Fig. A2.1 expresses Feh , and the vertical axis expresses Fel . Actually, Fei (i 2 {h, l}) represents the regulation mode of FAR in Zone i: how much regulated FAR is smaller or larger than the market equilibrium FAR, FM i . Among Quadrants I, II, III, and IV presented   in Fig. A2.1, we explore a quadrant that has an optimal Feh , Fel solution,

51

Necessity of a minimum floor area regulation

Fig. A2.1 Quadrants of regulation modes. (Source: Kono, T., Kaneko, T., Morisugi, H., 2010. Necessity of minimum floor area ratio regulation: a second-best policy. Ann. Reg. Sci. 44 (3), 523–539.) M satisfying Eqs. (2.9a), (2.9b) simultaneously. Notably the origin (FM h , Fl ) is   M M endogenous. So the origin (Fh , Fl ) changes as the regulation policy Feh , Fel changes. We will next show that excluding the axes, only Quadrant II meets the requirements of Eqs. (2.9a), (2.9b) simultaneously. First, we show that an optimal solution cannot exist in Quadrant   I (including the axes) where Feh  0 and Fel  0, with Feh , Fel 6¼ ð0, 0Þ. In Quadrant I, because of Eq. (A2.5), we have   DFh FhM + Feh , FlM + Fel  0,   DFl FhM + Feh , FlM + Fel  0 (A2.8)

both of which cannot be equal to zero simultaneously. Now, for this case to meet optimal conditions (2.9a), (2.9b), it must satisfy both:   DFh FhM + Feh , FlM + Fel   ð0 orÞ DN FhM + Feh , FlM + Fel ¼   0 and (A2.9) dN h =dF h 

  DN FhM + Feh , FlM + Fel ¼ 

ð +Þ

DFl FhM

+ Feh , FlM + Fel ð0 orÞ

dN h =dF l

 0

(A2.10)

ðÞ

where the signs for the terms are obtained from Eqs. (A2.6), (A2.8). For optimality, both Eqs. (A2.9), (A2.10) must be satisfied simultaneously. However, DN() 0 in Eq. (A2.9) and DN()  0 in Eq. (A2.10) are contradictory because both expressions cannot be set equal to zero simultaneously because of Eq. (A2.8). Consequently an optimal solution cannot exist in Quadrant I.

52

Traffic congestion and land use regulations

Next, we will demonstrate that an optimal solution can also not exist in Quadrant III (again, including the axes), that is, Feh  0 and Fel  0, whereas   Feh , Fel 6¼ ð0, 0Þ. In Quadrant III, Eq. (A2.5) implies   DFh FhM + Feh , FlM + Fel  0,   DFl FhM + Feh , FlM + Fel  0 (A2.11) both of which cannot be set equal to zero simultaneously. To meet optimal conditions (2.9a), (2.9b), this case must satisfy both: 



DFh ðFhM + Feh FlM + Fel Þ ð0 or +Þ

DN FhM + Feh FlM + Fel ¼ 

dN h =dF h

 0 and

(A2.12)

ð +Þ

  DN F1M + Fe1 F2M + Fe2 ¼ 

DF2 ðF1M + Fe1 F2M + Fe2 Þ ð0 or +Þ

dN 1 =dF 2

0

(A2.13)

ðÞ

Similar to the impossibility of an optimal solution in Quadrant I, DN()  0 in Eq. (A2.12) and DN() 0 in Eq. (A2.13) cannot both be set equal to zero simultaneously because of Eq. (A2.11). Hence an optimal solution does not exist in Quadrant III. Finally, we will show that a solution cannot exist in Quadrant IV, that is, Feh > 0, Fel < 0, because Eq. (A2.5) yields the relations     and DFl FhM + Feh , FlM + Fel > 0. DFh FhM + Feh , FlM + Fel < 0 Rearrangement of Eqs. (2.9a), (2.9b) gives   DFh FhM + Feh , FlM + Fel   ðÞ DN FhM + Feh , FlM + Fel ¼  > 0 and (A2.14) dN h =dF h ð +Þ

  DN FhM + Feh , FlM + Fel ¼ 

  DFl FhM + Feh , FlM + Fel ð +Þ

dN h =dF l

>0

(A2.15)

ðÞ

Both equations imply that DN() > 0, which contradicts the presupposition with regard to the relative population density externality between Zone h and Zone l, that is, between DNh(Fh, Fl) and DNl(Fh, Fl), which implies   DN FhM + Feh , FlM + Fel < 0. Hence an optimal solution does not exist in Quadrant IV.

53

Necessity of a minimum floor area regulation

Careful inspection shows that sign conditions offer no contradiction in Quadrant II. Therefore, we conclude that an optimal solution exists only in Quadrant II. Put simply, conditions Feh < 0 and Fel > 0 must be met to have optimal FAR regulations. Proposition 2.1 thus holds. Note that the horizontal axis Fel ¼ 0 and vertical axis Feh ¼ 0 of the quadrant are excluded from the solution region.

Supplementary note: Harberger welfare function Throughout this book the Harberger welfare function is used because it is the welfare evaluation formula in an economy with price distortions. The price distortions addressed in this book are caused not only by congestion externalities but also by land use regulations. This section explains the Harberger welfare function. The Harberger welfare function is based on Harberger (1971), who originally explored project evaluation methods in the presence of distortionary taxation. The Harberger welfare function can be expressed as X ∂Xi dW Di ¼ direct effect + ∂Ξ dΞ i

(S.1)

where Di  (ri  ∂Si/∂Xi) is price distortion, that is, price minus marginal cost, in market i; Xi is the equilibrium output in market i; and Ξ is the policy variable. P i The part i Di ∂X ∂Ξ is the key feature of the Harberger welfare function. We explain this part in detail. Fig. S.1 shows market equilibrium and price distortions in a certain market. When there are no factors generating price distortions, the equilibrium output is determined by the intersection of the demand function and the marginal cost function. In this point, there is no distortion. So, when a Price, cost Deadweight losses

Shifted supply function (shifts caused by imperfect competition, taxation, regulations, and so on) Marginal cost

Price distortions This point has no distortion. Demand function Output

Fig. S.1 Price distortions.

54

Traffic congestion and land use regulations

Price, cost

Change in equilibrium output Xi

Change in shifted supply function Marginal cost

Price distortion Di Change in demand function Output

Fig. S.2 Change in price distortions.

certain project is carried out, the cost-benefit analysis does not take account P i of i Di ∂X ∂Ξ , the part in Eq. (S.1). If supply functions shift from the marginal cost function, we need to conP i sider i Di ∂X ∂Ξ , the part in Eq. (S.1). Factors shifting the supply function from the marginal cost function are imperfect competitive markets, distortionary taxation, and regulations. In this case, deadweight losses, which are shown by the triangle in Fig. S.1, exist in the market. Price distortion is shown in the vertical length of the deadweight loss triangle. So the price distortion can be expressed by price minus marginal cost. If a project is carried out in the presence of price distortions, in the distorted market, the equilibrium output changes according to the project. This is shown in Fig. S.2. The project generally changes the supply function and the demand function. So the intersection of the two functions, that is, the P i equilibrium output, changes. The part i Di ∂X ∂Ξ in Eq. (S7.1) is expressed by the rectangle in Fig. S.2. In the earlier discussion, we have focused on only one market. Actually, many real markets have distortions due to various reasons. If we strictly follow the Harberger benefit function, we have to consider all the markets, but this is almost impossible. Then, how can we use the Harberger welfare function practically? To see this, first, we can deconstruct the Harberger welfare function into several terms as follows: X ∂Xi X ∂Xi X ∂Xi dW Di + Di + Di ¼ direct effect + ∂Ξ ∂Ξ ∂Ξ dΞ i2a i2c i2b |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} group a

X

∂Xi X ∂Xi + Di + Di ∂Ξ i2e2 ∂Ξ i2e1 |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} group e1

group e2

group b

group c

(S.2)

Necessity of a minimum floor area regulation

55

where In markets in group a, price distortion Di and the change in the equilibrium output Xi are both large. (In other words the market is far from perfect competition, and the effect of the target policy on this market is very large.) In markets in group b, price distortion Di is small, but the change in the equilibrium output Xi is large. (In other words the market is close to perfect competition, and the effect of the target policy on this market is very large.) In markets in group c, price distortion Di is large, but the change in the equilibrium output Xi is small. (In other words the market is distant from perfect competition, and the effect of the target policy on this market is small.) In markets in group e1, price distortion Di is zero, but the change in the equilibrium output Xi is arbitrary. (In other words the market outcome is the same as that of perfect competition, and the effect of the target policy on this market is arbitrary.) In markets in group e2, price distortion Di is arbitrary, but the change in the equilibrium output Xi is zero. (In other words the market can be far from perfect competition, but the effect of the target policy on this market is zero). Eq. (S.2) is useful in practical situations. Whether the market is close to perfect competition or not can be checked by referring to the number of suppliers in the market. That is, if there are many suppliers, basically the market is close to perfect competition. However, if there are regulations in the market, the output is not determined by the marginal cost. An important point of Eq. (S.2) is that we can ignore the terms related to group e1 and group e2. For example, the market output is regulated by a policymaker; the change in the equilibrium output is zero. So, in this case, this market can be classified in e2 group.

References Brueckner, J.K., 2007. Urban growth boundaries: an effective second-best remedy for unpriced traffic congestion? J. Hous. Econ. 16 (3–4), 263–273. Harberger, A.C., 1971. Three basic postulates for applied welfare economics: an interpretive essay. J. Econ. Lit. 9 (3), 785–797. Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511. Kono, T., Kaneko, T., Morisugi, H., 2010. Necessity of minimum floor area ratio regulation: a second-best policy. Ann. Reg. Sci. 44 (3), 523–539.

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Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630. Lipsey, R.G., Lancaster, K., 1956. The general theory of second best. Rev. Econ. Stud. 24 (1), 11–32. Wheaton, W.C., 1998. Land use and density in cities with congestion. J. Urban Econ. 43 (2), 258–272.

CHAPTER THREE

Differences in optimal land use regulation between a closed city and an open city

3.1 Introduction Land use regulations affect the urban housing market (in particular, housing supply and price), and the impacts are reflected in the urban spatial pattern as well. To understand how regulations affect urban spatial pattern and social welfare, urban economic literature often defines cities as closed or open with important implications. Introduced by Wheaton (1974), in a closed city, population size is exogenous, while the level of consumer welfare or household utility is determined endogenously. In contrast, in an open city, the level of consumer welfare is predetermined and is equal to “that of the rest of the economy” (Fujita, 1989), whereas population is determined endogenously. In an opencity model, it is assumed that households can move costlessly across the city boundary. Real-world cities exhibit, at least partly, characteristics of either an open city or a closed city (Brueckner, 1987). For instance, a closed-city situation is “typical of industrially advanced societies, where there is essentially no alternative to living in urban areas” (Wheaton, 1974). Likewise, low-density desert towns also exhibit more closed-city than open-city properties (Brueckner, 1987). An open-city situation, on the other hand, is typical of less developed countries where migration to cities occurs as long as the urban utility levels exceed the base utility level of the economy, which is often established by rural life. In summary the choice of the open-city or closed-city model is dictated by the features of the problem being considered. The closed-city model is more suitable to study land use in large or average cities of developed countries, whereas the open-city model better addresses urban conditions in developing countries with surplus labor in rural areas (Fujita, 1989). Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00003-0

© 2019 Elsevier Inc. All rights reserved.

57

58

Traffic congestion and land use regulations

In this chapter, following Kono and Joshi (2012), we determine and compare optimal land use regulations in monocentric closed and open cities with traffic congestion externality. We define optimization as the maximization of social welfare. In our model, floor area ratio (FAR) and urban growth boundary (UGB) are regulated. Our definitions of closed and open cities slightly modify the standard definitions. In our closed city, population does not change in response to the land use regulation.a An example is Silicon Valley that mostly attracts skilled ICT workers, and, as such, the total population is not affected much by the regulations. In contrast, in our open city, population changes in response to the regulation. Examples include towns that primarily consist of residences for workers employed elsewhere such as the “bed towns” in Japan.b The remainder of this chapter is organized as follows. Section 3.2 develops a monocentric city model with traffic congestion. Section 3.3 explores optimality of the FAR and UGB regulations in a closed city and an open city. Section 3.4 concludes the chapter. The final subsection provides a technical appendix.

3.2 The model 3.2.1 The city We adopt the standard model of a congested monocentric city a` la Wheaton (1998) and Brueckner (2007). The city is circular and is symmetric along any radial axis. The residential area expands from x ¼ 1 at the central business district (CBD) edge to x ¼ xb at the city boundary. The city is divided into several concentric bands of equal width. The total floor supply per unit area of land at x is denoted F(x), which represents FAR at x.c At each distance x a fraction ρ(x) of the land is used for roads,d and so the share of land available for housing at x from the CBD edge is given by 2πx[1  ρ(x)]. a

In other words, population might change due to other policies, such as subsidies for firms’ location, but the impact of population density regulations on the population size is assumed to be neutral in our closed city. b In Japan, in late 1980s, new towns, informally called “bed towns,” were built exclusively as residential places for workers employed in the nearby large cities to ease overconcentration of people and firms in the latter. For example, towns in Western Tokyo (or Tama area) act as “bed towns” for the workers in Central Tokyo. c For sufficiently large city radius compared with the lot size of a building, the building size (or FAR) can be treated as a continuous function of distance. d Wheaton (1998) and Brueckner (2007) assume a constant fraction.

Differences in optimal regulation between closed and open cities

59

In our model the city land is owned by absentee landowners from whom land is rented by developers to build dwelling units to be rented out to households. Like in the earlier chapters, the city government imposes FAR regulation on all buildings and UGB regulation to ease traffic congestion. The city government sets the FAR per building as an equality constraint such that if the optimum FAR is less than the market FAR, then the government should impose maximum < resp. minimum> FAR regulation.

3.2.2 Household behavior The city is inhabited by N identical households. For simplicity, we define population in terms of the number of households. Every household chooses only one residence although it could be anywhere within the city. Residents have a quasi-linear utility function v ¼ z + υð f Þ

(3.1)

where subutility υ( f ) is a function of housing square footage f, and z denotes numeraire composite goods that include all consumer goods except floor space. Each household earns income y per period. The household rents floor space from developers at the price of r per square foot of floor space per period. The commuting cost to and from the CBD for the household residing at x is denoted T(x). The income constraint is, therefore, expressed as z + rf ¼ y  T ðxÞ

(3.2)

Because many households can reside in a building, floor rent r equals the maximum floor rent bid by a household as a result of competition among residents. Mathematically, using Eqs. (3.1) and (3.2), such behavior is expressed by max r ¼ f ,z

y  T ðxÞ  z subject to v ¼ z + υð f Þ f

(3.3)

Solving Eq. (3.3) for r(x) and f(x), given a parametric utility level v(x), yields r ðxÞ ¼ r ðy  T ðxÞ, vðxÞÞ and f ðxÞ ¼ f ðy  T ðxÞ, vðxÞÞ

(3.4)

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Traffic congestion and land use regulations

3.2.3 Developers’ behavior with no FAR regulation Developers combine housing capital (building materials) and land to build dwellings under FAR regulation. Developers are assumed to be perfectly competitive and are, therefore, price-takers.e Floor space supply per unit of land is given by housing production function F(S) where S is capitalto-land ratio. Using the reverse production function S(F), the sum of developers’ net profit from the total floor space supply in the city, denoted Π, is given by ð xb (3.5) Π ¼ 2πx½1  ρðxÞ½F ðxÞr ðxÞ  SðF ðxÞÞ  RðxÞdx 1

where R(x) is the land rent paid to the absentee landowners. The price of capital is normalized at unity. Under no FAR regulation, developers maximize profit per unit of land with respect to F. Solving the relevant firstorder condition for F(x) yields r ðxÞ 

∂SðF ðxÞÞ ¼0 ∂F ðxÞ

(3.6)

Solving this for F(x) and S(F(x)) gives F ðxÞ ¼ F ðy  T ðxÞ, vðxÞÞ and SðF ðxÞÞ ¼ Sðy  T ðxÞ, vðxÞÞ

(3.7)

Plugging these solutions into profit function (3.5) and noting that developers’ profit at each location is zero because of perfect competition, the land rent is expressed as RðxÞ ¼ Rðy  T ðxÞ, vðxÞÞ

(3.8)

Finally, population density, denoted P(x), is defined as housing square feet per unit of land divided by square feet per dwelling: P ðxÞ ¼

F ðy  T ðxÞ, vðxÞÞ f ðy  T ðxÞ, vðxÞÞ

(3.9)

3.2.4 Developers’ behavior under FAR regulation Under the FAR regulation the floor space supply F is set exogenously, which implies that the developers cannot maximize profit with respect to F. e

As a result, they can be treated as an aggregate developer who maximizes net profit per unit of land.

Differences in optimal regulation between closed and open cities

61

The developers’ total net profit is given by Eq. (3.5). Using zero-profit condition at each location, the land rent is given by RðxÞ ¼ F ðxÞr ðxÞ  SðF ðxÞÞ

(3.10)

3.2.5 Commuting cost: The external factor For simplicity, we assume that automobiles are the only mode of commuting and that only one radial route extends across the residential zone between the CBD and city boundary. Commuting cost is assumed to be incurred only when commuting to and from the CBD boundary—that is, commuting cost within the CBD is unchanged by regulations in the residential zone. Because vehicles from more distant locations increasingly join traffic while moving toward the CBD, traffic volume is not uniform across all locations but increases toward the central locations. With congestion the unit cost of commuting for a commuter at x depends on the traffic flow across the ring at x.f Let n(x) denote the of households residing beyond the ring at x, which is given Ð xnumber b as n(x) ¼ x 2πm[1  ρ(m)]P(m)dm. Assuming that one member of each household commutes to the CBD to work, n(x) denotes the number of commuters passing through the ring at x. We adopt the following commonly used functional form of the commuting cost to cross the ring at x (e.g., Kanemoto, 1977):   nðxÞ Tunit ðxÞ ¼ c (3.11) 2πxρðxÞ where Tunit(x) denotes commuting cost per unit distance, and c() is the unitdistance commuting cost function with both the first derivative c0 () and the second derivative c00 () being positive. The term n(x)/[2πxρ(x)] is the traffic volume-to-road capacity ratio at x. A commuter at x pays Ðthe total commuting cost T(x), which is the demand-side cost, whereas x1Tunit(m)dm is the supply-side cost. The commuter cannot commute by paying less than the supply-side cost, but she could be paying more than the supply-side cost (e.g., by driving more slowly or by consuming more fuel through needless acceleration). Accordingly, T(x) is equal to or greater than the supply-side costg:  ðx ðx  nðmÞ T ðxÞ  Tunit ðmÞdm ¼ c dm (3.12) 2πmρðmÞ 1 1 f

This setting on commuting costs is identical to that adopted by Wheaton (1998) or Brueckner (2007). A different setting is possible but without fundamental differences in the results. g The inequality condition in Eq. (3.12) implies that a commuter at x pays at least the supply-side cost. This inequality condition is important for determining the sign of shadow price for this constraint as we show later.

62

Traffic congestion and land use regulations

Fig. 3.1 Transportation cost and population distribution.

The latter inequality implies that the commuter pays at least Tunit(x) per unit distance.h In this way the commuting cost is determined by the population density distribution in the city. The relation between population density and commuting cost is illustrated in Fig. 3.1.

3.3 Market equilibrium conditions The market equilibrium conditions are summed up in Eqs. (3.13)– (3.17), which are explained next. First, because households can choose a residential location costlessly, the household utility is the same across all locations, which is denoted V in the closed-city case and V
in the open-city case as expressed in Eq. (3.13). Note that V is endogenously determined, whereas V
is exogenous. Next, Eq. (3.14) expresses population constraint. The commuting cost constraint is expressed by Eq. (3.15),i which follows Eqs. (3.11)
in and (3.12). Next, Eq. (3.16) implies that the total population, denoted N the closed-city case and N in the open-city case, is accommodated within
is exogenous and fixed, whereas N is deterthe city boundary. Note that N mined endogenously. Finally, Eq. (3.17) implies that the land rent at the city boundary, denoted xb, is equal to the agricultural land rent, denoted Ra. When the city boundary is regulated, Eq. (3.17) does not hold because xb is set exogenously:  vðxÞ ¼ h

V ðclosed cityÞ V
ðopen cityÞ 8x 2 ½1, xb 

(3.13)

This inequality condition directly implies that the Lagrange multiplier corresponding to Eq. (3.13) is positive as shown later in Eq. (3.26). This property is instrumental in proving propositions. i The inequality condition implies that the commuter pays at least Tunit(x) per unit distance. This inequality condition directly implies that the Lagrange multiplier corresponding to Eq. (3.15) is positive as shown later in Eq. (3.26). This property is instrumental in proving propositions.

Differences in optimal regulation between closed and open cities

dnðxÞ ¼ 2πx½1  ρðxÞP ðxÞ dx   dT ðxÞ nðmÞ 0 T ðxÞ  c dx 2πmρðmÞ 
N ðclosed cityÞ nð1Þ ¼ N ðopen cityÞ

n0 ðxÞ 

F ðxb Þr ðxb Þ  SðF ðxb ÞÞ  Ra ¼ 0

63

(3.14) (3.15) (3.16) (3.17)

3.4 Optimal FAR regulation and urban boundary 3.4.1 Definition The optimality of regulations is defined in terms of the maximization of social welfare denoted as W. Noting that developers’ profit is zero, W is given by 8 ð xb > >
2πx½1  ρðxÞ½RðxÞ  Ra dx ðclosed cityÞ N V + < 1 ð W¼ (3.18) xb > > : 2πx½1  ρðxÞ½RðxÞ  Ra dx ðopen cityÞ 1

Note that in a closed city, W is the sum of total household utility (in monetary terms) and differential land rents. In an open city, W consists of differential land rents only because household utility is exogenous. The welfare level W depends on the land use regulation that controls the FAR F ¼ F ðxÞ 8x 2 ½1, xb  and the city boundary xbj and is thereby expressed as W ¼ W ðF, xb Þ. The optimal regulations at x 2 [1,xb] require F∗(x) and   x∗b such that F ∗ ðxÞ,x∗b ¼ arg max F, xb W ðF, xb Þ subject to Eq. (3.10) and Eqs. (3.13)–(3.17).k

3.4.2 Closed city The Lagrangian corresponding to the definition of optimal FAR in a closed city is given asl j

Pines and Kono (2012) point out that FAR regulation renders UGB regulation redundant because a zero level of FAR imposed on a certain lot implies the setting of UGB regulation. In our study the city boundary implies that FAR is set to zero beyond it. k We have six equations as constraints—Eqs. (3.10), (3.13)–(3.17)—using which we determine six endogenous variables: n(x), T(x), V, v(x), R(x), and R(y  T(xb), V)jNo UGB, given F(x) exogenously; W is optimized with regard to F(x). l See mathematical note at the end of this chapter.

64

Traffic congestion and land use regulations

ð xb
L ¼ NV + 2πx½1  ρðxÞ½F ðxÞr ðxÞ  SðF ðxÞÞ  Ra dx   ð xb 1  nðxÞ 0 λðxÞ T ðxÞ  c dx + 2πxρðxÞ ð1xb μðxÞ½2πx½1  ρðxÞP ðxÞ + n0 ðxÞdx + 1 ð xb + ϕðxÞ½vðxÞ  V dx

(3.19)

1

where λ(x), μ(x), and ϕ(x) are the shadow prices of the corresponding constraints. The boundary conditions are as follows: n(1) ¼ N, nðxb Þ ¼ 0, and T(1) ¼ 0. Because T(xb) is unconstrained, λ(xb) ¼ 0. Ð Ð Integrating x1bλ(x)T 0 (x)dx and x1bμ(x)n0 (x)dx by partsm and applying the following three relations to the first-order conditions, (1) ∂ r/∂T ¼  1/f (Muth’s condition), (2) ∂ r/∂ v ¼  1/f (quasi-linearity of the utility function), and (3) Eq. (3.14), we obtain the following arranged forms of the first-order conditionsn: ∂L ∂SðF ðxÞÞ μðxÞ ¼ 0 : r ðxÞ  + ¼0 ∂F ðxÞ ∂F ðxÞ f ðxÞ ∂L P ðxÞ ∂f ðxÞ ¼ 0 : n0 ðxÞ  2πx½1  ρðxÞμðxÞ  ϕðxÞ ¼ 0 ∂vðxÞ f ðxÞ ∂vðxÞ ∂L P ðxÞ ∂f ðxÞ ¼ 0 : n0 ðxÞ  2πx½1  ρðxÞμðxÞ  λ0 ðxÞ ¼ 0 ∂T ðxÞ f ðxÞ ∂T ðxÞ   ∂L nðxÞ 1 0 ¼ 0 : λðxÞc  μ0 ðxÞ ¼ 0 ∂nðxÞ 2πxρðxÞ 2πxρðxÞ ð xb ∂L
 ϕðxÞdx ¼ 0 ¼0:N ∂V 1 ∂L ¼ 0 : ½F ðxb Þr ðxb Þ  C ðF ðxb ÞÞ  Ra  + μðxb ÞP ðxb Þ ¼ 0 ∂xb    ∂L nðxÞ ∂L 0 λðxÞ ¼ λðxÞ T ðxÞ  c ¼ 0, λðxÞ  0, 0 ∂λðxÞ 2πxρðxÞ ∂λðxÞ ∂L ¼ 2πx½1  ρðxÞP ðxÞ + n0 ðxÞ ¼ 0 ∂μðxÞ ∂L ¼ 0 : vðxÞ  V ¼ 0 ∂ϕðxÞ

(3.20) (3.21) (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28)

Ð xb 0 Ð xb Ð xb 0 Ð xb xb xb 0 0 1 λ(x)T (x)dx ¼ [λ(x)T(x)]1  1 λ (x)T(x)dx and 1 μ(x)n (x)dx ¼ [μ(x)n(x)]1  1 μ (x)n(x)dx. Ð Ð n
¼ xb 2πx½1  ρðxÞDðxÞdx ¼  xb n0 ðxÞdx. To derive Eq. (3.22), we used N m

1

1

Differences in optimal regulation between closed and open cities

65

where r(x) ¼ r(y  T(x), V) and f(x) ¼ f(y  T(x), V). The optimal FAR regulation in a closed-city model should satisfy the first-order conditions given by Eqs. (3.20)–(3.24), (3.26)–(3.29). Eq. (3.25) expresses the optimal condition for the city boundary. The following discussion interprets the firstorder conditions. First, plugging Eq. (3.20) into Eq. (3.22) while also using constraint (3.28) and integrating the rearranged form of Eq. (3.22) with regard to x, we obtain ð xb

λ0 ðmÞdm ¼

x

ð xb

n0 ðmÞdm +

x

ð xb 

r ðmÞ 

x

 ∂SðF ðmÞÞ ∂f ðmÞ 2πm½1  ρðmÞP ðmÞ dm ∂F ðmÞ ∂T ðmÞ

leading to λðxÞ ¼ nðxÞ +

ð xb  x

r ðmÞ 

 ∂SðF ðmÞÞ ∂f ðmÞ 2πm½1  ρðmÞP ðmÞ dm ∂F ðmÞ ∂T ðmÞ (3.29)

given λ(xb) ¼ 0 and n(xb) ¼ 0. Eq. (3.29) is interpreted as follows. Note that [λ(x)] denotes the shadow price of Tunit(x); in other words, it expresses ∂W/∂ Tunit(x). The first term in the right side of Eq. (3.29) is the direct effect of the total increase in commuting time for the commuters passing through x. The second term is the effect of change in the per-capita floor area beyond x induced by an increase in T(m)(m 2 [x, xb]). Note that ∂ f(m)/∂ T(m) multiplied by P(m) is the change in the total floor area F at m. The term r ðxÞ  ∂S∂FðFððxxÞÞÞ is the marginal change in the deadweight loss (or distortion) in the floor space market at x caused by the FAR regulation (see Fig. 3.2). If the developers are perfectly competitive, r ðxÞ ¼ ∂S∂FðFððxxÞÞÞ as shown in Eq. (3.6), and correspondingly the second term in the right side of Eq. (3.29) vanishes. However, when the floor area is regulated, r ðxÞ 6¼ ∂S∂FðFððxxÞÞÞ. A similar equation to Eq. (3.29) appears in Kanemoto (1977), Arnott (1979), and Pines and Sadka (1985). In our model, r ðxÞ  ∂S∂FðFððxxÞÞÞ expresses the distortion arising from the FAR regulation. But in those previous papers,

66

Traffic congestion and land use regulations

Fig. 3.2 Floor rent and marginal cost curve.

the distortion arises from the allocation of land between road and residential area, ρ(x), which is exogenously fixed in our model. Next, we interpret the optimality condition of FAR regulation by combining the first-order conditions. First, plugging Eq. (3.29) into Eq. (3.23) and rearranging yields A  μ0 ðxÞdx      ð xb nðxÞ 1 ∂SðF ðmÞÞ ∂f ðmÞ 0 dx nðxÞ + r ðmÞ  2πm½1  ρðmÞP ðmÞ dm ¼c 2πxρðxÞ 2πxρðxÞ ∂F ðmÞ ∂T ðmÞ x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ∂T ðxÞ=∂nðxÞ

λðxÞ

(3.30)

Recalling that μ(x) denotes the shadow price of 2πx[1  ρ(x)]P(x), μ(x) ¼ ∂ W/∂(2πx[1  ρ(x)]P(x)). Moreover, μ0 (x) ¼ [μ(x + dx)  μ(x)]/dx. In a closed city, if the population at a location increases by one person, the population must decrease by one person somewhere else in the city; μ0 (x) expresses the welfare change when population increase at x + dx results in population decrease at x (see Fig. 3.3 for a graphical representation of the relocation of a household under consideration). x

x + dx UGB

CBD

One household moves

Fig. 3.3 Relocation of a household.

Differences in optimal regulation between closed and open cities

67

The term A in Eq. (3.30) denotes the welfare change with regard to an increase in the commuting cost induced by the relocation of one person from x to x + dx. In this situation, recalling Eq. (3.11), the commuting cost at x increases by [c 0 ()/[2πxρ(x)]]dx(¼[∂Tunit(x)/∂n(x)]dx) for all commuters beyond x because of one added person (commuter) at x + dx. The term [ c 0 ()/[2πxρ(x)]dx  n(x)] in Eq. (3.30) denotes social marginal commuting cost, whereas the remaining term denotes the change in social cost arising from the FAR regulation associated with the change in commuting cost. Thus the term A is composed of the direct effect of traffic congestion externality and the indirect effect on the deadweight loss in the floor area caused by the change in per-capita floor area consumption beyond x. This explains that spatial efficiency holds when reallocation of households across locations cannot increase the social welfare function. This condition has been discussed in detail by previous papers.o Likewise, differentiating Eq. (3.20) with regard to x yields B  μ0 ðxÞdx     ∂SðF ðxÞÞ ∂SðF ðx + dxÞÞ ¼ f ðxÞ r ðxÞ   f ðx + dxÞ r ðx + dxÞ  ∂F ðxÞ ∂F ðx + dxÞ (3.31) which can be interpreted as follows. First, if the population at x decreases by one person, F(x) decreases by f(x). Accordingly the deadweight loss in the h i floor market increases by f ðxÞ r ðxÞ  ∂S∂FðFððxxÞÞÞ (see Fig. 3.1). Second, if the population at x + dx increases by one person, the deadweight loss in h i the floor market decreases by f ðx + dxÞ r ðx + dxÞ  ∂S∂FðFððxx++dxdxÞÞÞ . Summing up the welfare changes by [B]. Because both A and B are equal to μ0 (x)dx, the right side of Eq. (3.30) should be equal to that of Eq. (3.31) for the optimality of FAR regulation. In other words, A  B ¼ 0, which implies that as a person relocates from x to x + dx, the welfare change associated with FAR regulation-induced o

A similar interpretation for the first-best condition is given in Kanemoto (1977), Arnott and MacKinnon (1978), Arnott (1979), Pines and Sadka (1985), and Pines and Kono (2012) although these papers, except for the latter, consider only lot housings but not condominiums and accordingly do not explore FAR regulations. Pines and Kono (2012) interpret the condition corresponding to “A  B ¼ 0” in the case of a resident’s relocation from x to a distant location (not x + dx as in the current study). However, interpretation in any of these papers is not fully linked to Harberger welfare measurement formula as described in Kono and Joshi (2012) or in this chapter although a suggestive reference to the Harberger formula is found in Pines and Kono (2012).

68

Traffic congestion and land use regulations

deadweight loss,p that is, [B], should be offset by the welfare change associated with the increased commuting cost, that is, A. An important economic interpretation of the relation “A  B ¼ 0” is possible vis-a`-vis Harberger’s welfare formula, which measures the welfare change in a distorted economy (see Harberger, 1971). As also stated in Chapter P2, Harberger’s welfare formula is expressed as dW/dΞ ¼ iΦi∂Xi/∂Ξ where Φi is the distortion in the market i, X is the output in the market i, and Ξ is the policy variable. In the earlier “A  B” context, the distortions Φi are the deadweight losses in the floor h i h i area market, that is, r ðxÞ  ∂S∂FðFððxxÞÞÞ and r ðx + dxÞ  ∂S∂FðFððxx++dxdxÞÞÞ and the congestion externality, that is, c 0 ()/[2πxρ(x)]  n(x). The policy Ξ implies relocation of one person from x to x + dx. The changes in the output associated with policy change, ∂Xi/∂Ξ, consist of the change in the number of commuters at x (one person) and the change in the floor area consumption. Minimizing these distortions maximizes the welfare, as demonstrated in Joshi and Kono (2009) and Kono et al. (2010). We next obtain the property of the optimal FAR regulation. The combination of Eq. (3.20) and Lemma 3.1 presented in Technical Appendix 3(1) yields Proposition 3.1. Note r ðxÞ  ∂S∂FðFððxxÞÞÞ ¼ 0 if the FAR is determined in the market (i.e., is unregulated). Therefore, to satisfy Lemma 3.1, FAR should be greater than the market FAR (i.e., r ðxÞ  ∂S∂FðFððxxÞÞÞ < 0) for any x 2 ½1, x^Þ and smaller than the market FAR (i.e., r ðxÞ  ∂S∂FðFððxxÞÞÞ > 0) for any x 2 ðx^, xb  where 1 < x^ < xb (see Fig. 3.1). Furthermore, this property holds regardless of Eq. (3.25). Proposition 3.1 (Optimal FAR regulation in a closed monocentric city with traffic congestion) (1) The optimal FAR regulation requires minimum (resp. maximum) FAR regulation at locations closer to the CBD (resp. the city boundary). h i (2) The level of FAR regulation, denoted by f ðxÞ r ðxÞ  ∂S∂FðFððxxÞÞÞ , increases from a negative value at the CBD edge (x ¼ 1) up to a positive value at the city boundary (x ¼ xb). p

The existence of deadweight loss even in the case of optimal FAR regulation implies that the FAR regulation is second best because of the indirect adjustment of population density. Understandably the FAR regulation can only control the total floor space of a building without controlling the percapita floor space consumption directly.

Differences in optimal regulation between closed and open cities

69

Proposition 3.1, which is illustrated in the left hand side of Fig. 3.2, implies that an increase in floor supply at the more central locations (resp. farther locations) decreases (resp. increases) the total congestion externality in the city. This is partly because more households (or commuters) living closer to the CBD (resp. the city boundary) would mean shorter (resp. longer) commuting distances to and from the CBD. However, the necessity of minimum FAR policy has a more important meaning. As also discussed in earlier chapters, minimum FAR policy regulates total floor space to be larger than the market equilibrium floor space, whereas maximum FAR policy regulates total floor space to be smaller than the market equilibrium floor space.q Proposition 3.1 implies that the combination of maximum FAR at peripheral locations with minimum FARr at central locations is more efficient than imposing only maximum or minimum FAR regulation, to minimize total cost composed of deadweight loss in the floor area market and congestion externality.s Next, we determine optimal city boundary. The first-order condition with regard to xb is Eq. (3.25). From Lemma 3.1 presented in Technical Appendix 3(1), μ(xb) < 0. This leads to Proposition 3.2. Proposition 3.2 (Optimal city boundary in a closed city) At the optimal city boundary, denoted x∗b , F(x∗b )r(x∗b )  S(F(x∗b ))  Ra > 0. Proposition 3.2 implies that at the optimal city boundary, the land rent is greater than the agricultural rent (i.e., opportunity cost). This result is unsurprising because when the spatial size of the city expands, the total congestion externality also increases, reducing social welfare. However, this point is important for understanding the difference between FAR regulation and lot size (LS) regulation. The LS regulation can attain first-best solution because it directly regulates population density and hence the distribution of commuters. Such direct regulation against population density externality generates no deadweight loss (see Pines and Sadka, 1985 and Wheaton, 1998) unlike the FAR regulation. Hence, under optimal LS regulation, the land rent at the city boundary is equal to the agricultural rent. The minimum FAR F∗ implies that the developer is regulated to build floor space within the interval [F∗, ∞], whereas the maximum FAR F∗ is within the interval [0, F∗]. r See Chapter 2 for explanation on the feasibility of minimum FAR regulation. s Bertaud and Brueckner (2005), considering only maximum FAR regulation and ignoring benefit of decrease in congestion externality, show that the welfare cost is about 2% of household income. Our results suggest that judicious imposition of both maximum and minimum FAR regulations might further reduce welfare cost. q

70

Traffic congestion and land use regulations

3.4.3 Open city The Lagrangian corresponding to the definition of optimal FAR in an open city is given as L¼

ð xb

2πx½1  ρðxÞ½F ðxÞr ðxÞ  SðF ðxÞÞ  Ra dx    ð xb nðxÞ 0 + λðxÞ T ðxÞ  c dx 2πxρðxÞ ð1xb μðxÞ½2πx½1  ρðxÞP ðxÞ + n0 ðxÞdx + ð1xb + ϕðxÞ½vðxÞ  V
dx 1

(3.32)

1

The last term expresses that vðxÞ  V
. Different from the closed-city model, in the open-city case, V
is given exogenously, and N is endogenous. Accordingly the boundary condition n(1) ¼ N is not fixed, but is unconstrained. With such free endpoint, μ(1) ¼ 0 in the open-city model. The other boundary conditions remain valid: n(xb) ¼ 0, T(1) ¼ 0, and λ(xb) ¼ 0. The first-order conditions are as follows:   ∂L ∂SðF ðxÞÞ μðxÞ ¼ 0 : r ðxÞ  + ¼0 ∂F ðxÞ ∂F ðxÞ f ðxÞ ∂L P ðxÞ ∂f ðxÞ ¼ 0 : n0 ðxÞ  2πx½1  ρðxÞμðxÞ + ϕðxÞ ¼ 0 ∂vðxÞ f ðxÞ ∂vðxÞ ∂L P ðxÞ ∂f ðxÞ ¼ 0 : n0 ðxÞ  2πx½1  ρðxÞμðxÞ  λ0 ðxÞ ¼ 0 ∂T ðxÞ f ðxÞ ∂T ðxÞ   ∂L nðxÞ 1 0 ¼ 0 : λðxÞc  μ0 ðxÞ ¼ 0 ∂nðxÞ 2πxρðxÞ 2πxρðxÞ ∂L ¼ 0 : ½F ðxb Þr ðxb Þ  SðF ðxb ÞÞ  Ra  + μðxb ÞP ðxb Þ ¼ 0 ∂xb    ∂L nðxÞ ∂L 0 λðxÞ ¼ λðxÞ T ðxÞ  c ¼ 0, λðxÞ  0, 0 ∂λðxÞ 2πxρðxÞ ∂λðxÞ ∂L ¼ 2πx½1  ρðxÞP ðxÞ + n0 ðxÞ ¼ 0 ∂μðxÞ ∂L ϕðxÞ ¼ ϕðxÞ½vðxÞ  V
 ¼ 0, ϕðxÞ  0, vðxÞ  V
 0 ∂ϕðxÞ

(3.33) (3.34) (3.35) (3.36) (3.37) (3.38) (3.39) (3.40)

Differences in optimal regulation between closed and open cities

71

where r ðxÞ ¼ r ðy  T ðxÞ, V
Þ and f ðxÞ ¼ f ðy  T ðxÞ, V
Þ. Here, Eq. (3.37) expresses the condition for optimal city boundary. If the city boundary is determined in the market, Eq. (3.17) holds instead of Eq. (3.37). The combination of Eq. (3.33) and Lemma 3.2 presented in Technical Appendix 3(2) yields Proposition 3.3. Note that r ðxÞ  ∂S∂FðFððxxÞÞÞ ¼ 0 if the FAR is determined in the market. Therefore, to satisfy Lemma 3.2, FAR should be smaller than the market FAR (i.e., r ðxÞ  ∂S∂FðFððxxÞÞÞ > 0) for any x 2 (1, xb] (see Fig. 3.1). At x ¼ 1, no FAR regulation is optimal. Moreover, this property holds regardless of Eq. (3.37). Proposition 3.3 (Optimal FAR regulation in an open monocentric city with traffic congestion) (1) The optimal FAR regulation requires maximum FAR regulation at all locations beyond the CBD, regardless of UGB regulation. h i (2) The level of FAR regulation, denoted by f ðxÞ r ðxÞ  ∂S∂FðFððxxÞÞÞ , increases from zero at the CBD edge (x ¼ 1) up to a positive value at the city boundary (x ¼ xb). To further understand Proposition 3.3 in an open city, we can look at the change in the social welfare if the FAR at x is reduced from the market equilibrium FAR. In an open city, where the equilibrium utility level V
and the income y are exogenous, the price of floor area r and its demand f by a representative household are both uniquely determined by the commuting cost T. Now, suppose that the FAR at x is marginally reduced below its market equilibrium FAR by ε, keeping the FAR elsewhere unchanged. Since V
is constant the effect on welfare is fully captured by the change in aggregate land rent. The latter is the sum of (1) the change in the profit from housing production at x and (2) the increase in the price of floor area, which results from the change in the commuting cost. The first effect can be ignored because, at the margin, the price of floor area equals its marginal cost, and the change in FAR is ε. Regarding the second effect, since r and f at x remain unchanged, the population density at x decreases by ε/f. This decrease changes the commuting cost of each commuter in the interval [1, x] by 0 (n(x))ε/f(x). MultiplyingÐ this by ∂ r/∂ T ¼  1 implies that the change Tunit 0 (m)ε/f(x)dm. This value is positive in the total price of floor area is x1n(m)Tunit at any x 2 (1, xb], implying that excluding x ¼ 1, an open city needs maximum FAR regulation that reduces the FAR from the market equilibrium FAR. A more elaborative discussion is presented in Technical Appendix 3(3).

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Regarding the city boundary, because Eq. (3.37) is analogous to Eq. (3.25) upon replacing V by V
, Proposition 3.2 holds in the case of an open-city model as well, which is reproduced as Proposition 3.4. Proposition 3.4 (Optimal city boundary in an open city) At the optimal city boundary, denoted x∗b , F(x∗b )r(x∗b )  S(F(x∗b ))  Ra > 0. Proposition 3.4 expresses that at the optimal city boundary, the land rent is greater than the agricultural rent in an open city like in a closed city. On the other hand, Proposition 3.3 is different from Proposition 3.1, implying a difference in FAR regulation between a closed city and an open city.

3.4.4 Comparison of results between closed- and open-city models The difference in optimal regulations between a closed-city and an opencity model arises from the treatment of household utility and population: in an open city (resp. a closed city), utility is exogenously given (resp. endogenously determined), whereas population is endogenously determined (resp. exogenously given). Such fundamental difference between the closed- and open-city models produces dissimilar characteristics of optimal FAR regulation against traffic congestion externality in a monocentric city setup. Results show that a congested, monocentric closed city requires minimum FAR regulation near the CBD and maximum FAR regulation near the city boundary to produce the upward adjustment to the market population density near the CBD and the downward adjustment near the city boundary, respectively. However, in an open city, only the downward adjustment of the market population density is required with no adjustment necessary at the CBD edge. Regarding the city size the optimal UGB regulation results into shrinkage of the FAR-regulated city regardless of whether the city model is closed or open. The results are shown in Table 3.1 (see also Fig. 3.4). Table 3.1 Optimal regulation in closed and open city. Regulation Regulation toward Model toward CBD city boundary Optimal city size

Closed city Minimum FAR regulation Open city No regulation

Maximum FAR regulation Maximum FAR regulation

At the optimal city boundary, the land rent is greater than the agricultural rent

Source: Kono, T., Joshi, K.K., 2012. A new interpretation on the optimal density regulations: closed and open city. J. Hous. Econ. 21 (3), 223–234.

Differences in optimal regulation between closed and open cities

73

Fig. 3.4 Optimal FAR and UGB regulations in a closed city and an open city. (Source: Kono, T., Joshi, K.K., 2012. A new interpretation on the optimal density regulations: closed and open city. J. Hous. Econ. 21 (3), 223–234.)

The difference between the closed- and open-city models in the case of FAR regulation stems from the behavior of μ(x), that is, the shadow price of population at x. In a closed city, marginal population increase at a location is counterbalanced by an equal decrease in population elsewhere in the city because the total population is exogenously fixed. The optimal FAR regulation in a closed city requires upward (resp. downward) adjustment to the pop: ulation density near the central (resp. peripheral) locations, that is, μ(x ¼ 1)> 0 : (resp. μ(x ¼ xb) < 0) to minimize the total deadweight losses. But in an open city, marginal population increase at a location does not affect the population distribution elsewhere in the city because the total population is endogenous. The optimal FAR regulation in an open city requires upward (resp. no) adjustment to the population density at locations beyond the CBD boundary (resp. at the CBD boundary), that is, μ(x 2 (1,xb]) < 0 (resp. μ(1) ¼ 0).

3.5 Conclusion In this chapter, we have investigated the effects of FAR and UGB regulations on the social welfare in a closed and an open monocentric city and achieved dissimilar results on the optimal FAR regulation against traffic congestion externality—a closed monocentric city requires enforcement of minimum FAR regulation near the CBD and maximum FAR regulation near the city boundary, whereas an open monocentric city requires only the maximum FAR regulation outside the CBD. Because the optimal FAR regulation differs qualitatively between a closed city and an open city, we conclude that whether the city is open or closed is critical to determine how FAR regulation should be imposed.

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Regarding the optimal city boundary, regardless of whether the city is closed or open, at the boundary, the land rent should be greater than the agricultural land rent. This result is in contrast to the optimal lot size zoning in which the land rent at the boundary is equal to the agricultural land rent. This is because a FAR regulation—even if it is optimal—is not a first-best solution to the traffic congestion externality.

Technical Appendix 3 (1) Lemma 3.1 The combination of Eqs. (3.21) and (3.24) yields ð xb P ðxÞ ∂f ðxÞ  2πx½1  ρðxÞμðxÞ dx ¼ 0 f ðxÞ ∂V 1

(A3.1)

We have P(x)/f(x) > 0 and ∂ f(x)/∂V > 0. The first inequality is trivial, and the second one can be derived from utility maximization.t To satisfy Eq. (A3.1), μ(x) has only two possible solution patterns: pattern (1) μ(x) should be positive for some x 2 ½1, xb  and negative for other x 2 [1, xb], and pattern (2) μ(x) is zero all over the city (i.e., μ(x) ¼ 0 at any x 2 [1, xb]). Now, from Eq. (3.23), μ0 (x) ¼  λ(x)c 0 ()/[2πxρ(x)]. Here, λ(x) > 0 holds as shown in Eq. (3.26) because of the binding constraint. Intuitively also noting that [λ(x)] ¼ ∂W/∂Tunit(x), λ(x) > 0 because an increase in the commuting cost (e.g., adding a bottleneck or consuming fuel for needless acceleration)u decreases welfare.v Accordingly, μ0 (x) < 0. Therefore, we can exclude the pattern 2 as a solution. As a result, μ(x) as a continuous function yields Lemma 3.1: Lemma 3.1 μ(x) > 0 for any x 2 ½1, x^Þ and μ(x) < 0 for any x 2 ðx^, xb , where 1 < x^ < xb and μðx^Þ ¼ 0. In addition, μ0 (x) < 0. The first-order condition of utility maximization with respect to f is ∂υ/∂ f ¼ r(y  T(x), V). Differentiating this with respect to u, [∂2υ/∂f 2]df ¼ [∂ r/∂ V]dV. From this, df/dV ¼  [1/f]/[∂2υ/∂ f 2] > 0 because ∂2υ/∂ f 2 < 0. u This increase in the commuting cost is different from congestion pricing. In the latter case, the commuting cost increases, but the toll revenue simultaneously increases. v That is, positive λ(x) directly implies that an increase in commuting time reduces welfare. If this property does not hold, it would imply a paradoxical situation that adding a bottleneck would increase welfare. In fact, in a road network with multiple roads, such a paradox (e.g., well-known Brasse paradox) can occur but a single road has no paradoxical situation. Moreover, our model has no externality other than congestion. Hence, λ(x) should be positive. This economic reason also supports the use of inequality constraints in Eqs. (3.12), (3.13). t

Differences in optimal regulation between closed and open cities

75

(2) Lemma 3.2 The optimal FAR regulation in an open-city model should satisfy the firstorder conditions given by Eqs. (3.33)–(3.36), (3.38)–(3.40). The interpretation of Eqs. (3.29)–(3.31) as in the closed-city model can be extended to the open-city model as well. Similar to the closed-city case, λ(x) 0 from c 0 ðÞ Eq. (3.38), which yields μ0 ðxÞ ¼ λðxÞ 2πxρ ðxÞ  0 using Eq. (3.36). However, a distinct feature of an open-city model is the boundary condition: μ(1) ¼ 0. Then, μ(x) as a continuous function yields Lemma 3.2: Lemma 3.2 μ(x) < 0 for any x 2 (1, xb], and μ(1) ¼ 0. In addition, μ0 (x)  0. (3) Elaborative discussion on Proposition 3.3 We look at the change in the social welfare if some FAR is added to the market equilibrium FAR. For such purpose the Lagrangian (3.32) is modified to represent the market equilibrium FAR condition as follows: L ¼ M

ð xb

  

  2πx½1ρðxÞ F M ðxÞ+F G ðxÞ r ðxÞS F M ðxÞ+F G ðxÞ Ra dx 1   ð xb ð xb nðxÞ xb 0  λ ðxÞT ðxÞdx + ½λðxÞT ðxÞ1  λðxÞc dx 2πxρðxÞ 1 1 ð xb ½F M ðxÞ + F G ðxÞ + μðxÞ2πx½1  ρðxÞ dx f ðxÞ 1 ð xb  μ0 ðxÞnðxÞdx + ½μðxÞnðxÞx1b 1   ð xb ∂SðF M ðxÞÞ + κ ðxÞ2πx½1  ρðxÞ r ðxÞ  dx ∂F M ðxÞ 1 (A3.2)

where r ðxÞ ¼ r ðy  T ðxÞ, V
Þ and f ðxÞ ¼ f ðy  T ðxÞ, V
Þ. Likewise, FM(x) is the market equilibrium FAR, and the last term of Eq. (A3.2) expresses that the floor rent is equal to the marginal cost of the floor area at any x 2 [1, xb], implying that the FAR is determined at market equilibrium; κ(x) is the shadow price for that constraint. On the other hand, FG(x) is the additional FAR to the market equilibrium FAR, which is initially set at zero (or at the market equilibrium). The boundary conditions are same as in the original Lagrangian Eq. (3.32). The first-order conditions of Eq. (A3.2) are shown as follows:

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  ∂L ∂SðF M Þ μðxÞ ∂2 SðF M Þ ¼0 ¼ 0 : r ð x Þ  +  κ ð x Þ ∂F M ðxÞ ∂F M ðxÞ f ðxÞ ∂F M ðxÞ2 ∂L P ðxÞ ∂f ðxÞ ¼ 0 : n0 ðxÞ  2πx½1  ρðxÞμðxÞ ∂T ðxÞ f ðxÞ ∂T ðxÞ κðxÞ ¼0  λ0 ðxÞ  2πx½1  ρðxÞ f ðxÞ   ∂L nðxÞ 1 0 ¼ 0 : λðxÞc  μ0 ðxÞ ¼ 0 ∂nðxÞ 2πxρðxÞ 2πxρðxÞ

(A3.3)

(A3.4)

(A3.5)

 

 ∂L F M ðxb Þ ¼ 0 : F M ðxb Þr ðxb Þ  S F M ðxb Þ  Ra + μðxb Þ ∂xb f ðxb Þ   (A3.6) ∂SðF M ðxb ÞÞ ¼0 + κ ðxb Þ r ðxb Þ  ∂F M ðxb Þ    ∂L nðxÞ ∂L 0 λðxÞ ¼ λðxÞ T ðxÞ c ¼ 0, λðxÞ  0,  0 (A3.7) ∂λðxÞ 2πxρðxÞ ∂λðxÞ ∂L ¼ 2πx½1  ρðxÞP ðxÞ + n0 ðxÞ ¼ 0 ∂μðxÞ

(A3.8)

∂L ∂SðF M ðxÞÞ ¼0 ¼ r ðxÞ  ∂κðxÞ ∂F M ðxÞ

(A3.9)

Now, substituting Eq. (A3.9) into Eq. (A3.3) yields

μðxÞ ∂2 SðF M Þ f ðxÞ ¼ κ ðxÞ ∂F M ðxÞ2 .

Substituting this into Eq. (A3.4) yields " # ∂2 S ðF Þ ∂f ðxÞ 1 0 0 P ðxÞ + ¼ 0 (A3.10) n ðxÞ  λ ðxÞ  2πx½1  ρðxÞκðxÞ ∂T ðxÞ f ðxÞ ∂F ðxÞ2 Because Eq. (A3.9) holds at any x, differentiating it with regard to xw and arranging this using ∂r/∂ T ¼  1/f yields dF ðxÞ ¼ dT ðxÞ

w ∂r ðxÞ dT ðxÞ

∂T ðxÞ dx

SðF ðxÞÞ dF ðxÞ  ∂ ∂F dx ¼ 0. ðxÞ2 2

1 ∂ SðF ðxÞÞ f ðxÞ ∂F ðxÞ2 2

(A3.11)

Differences in optimal regulation between closed and open cities

77

Next, differentiation of Eq. (A3.10) with respect to T(x) yields dF/dt ¼ P∂ f/∂ T. Using this with Eq. (A3.11) implies ∂2 SðF ðxÞÞ ∂f ðxÞ 1 ¼ 2 P ðxÞ ∂T ð x Þ f ð xÞ ∂F ðxÞ

(A3.12)

0 Substituting Eq. (A3.12) into Eq. (A3.10),Ð we obtain n0 (x) Ð xb0λ (x) ¼ 0. Intexb 0 grating this with regard to x, we obtain x λ ðmÞdm ¼ x n ðmÞdm, which leads to

λðxÞ ¼ nðxÞ

(A3.13)

noting that λ(xb) ¼ 0 and n(xb) ¼ 0. c 0 ðÞ ∂Tunit ðxÞ Next, from Eq. (A3.5), μ0 ðxÞ ¼ λðxÞ 2πxρ ðxÞ ¼ λðxÞ ∂nðxÞ . Substituting λ(x) from Eq. (A3.13) into this yields μ0 ðxÞdx ¼ nðxÞ ∂T∂nunitðxðÞxÞ dx, integrating that yields ðx ðx ∂Tunit ðmÞ μðxÞ ¼ μ0 ðmÞdm ¼ nðmÞ dm (A3.14) ∂nðmÞ 1 1 Note that Eq. (A3.14) holds under no FAR regulation (i.e., at the market equilibrium). We next explore how the social welfare changes according to an additional FAR over the market equilibrium FAR. Applying the envelope theorem to Eq. (A3.2), the change in welfare with regard to an increase in FG(x) while keeping FM(x) at the market equilibrium h i ∂SðF G Þ μðxÞ ∂L M level is ∂F∂W G ðxÞ ¼ ∂F G ðxÞ ¼ 2πx½1  ρðxÞ r ðxÞ  ∂F G ðxÞ + f ðxÞ . Substituting Eq. (A3.14) into this, we obtain   ∂W ∂SðF G Þ ¼ 2πx½1  ρðxÞ r ðxÞ  G ∂F G ðxÞ ∂F ðxÞ ðx 1 ∂Tunit ðmÞ nðmÞ  2πx½1  ρðxÞ dm f ðxÞ ∂nðmÞ

(A3.15)

1

This can be interpreted as follows. At the market equilibrium an additional FAR generates the first term in the left side of Eq. (A3.15), which expresses the additional payment to the landowner. But, at the market equilibrium, this term is zero as shown in Eq. (A3.9), implying that the resident’s willingness to pay for one unit of floor equals the marginal cost of producing

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one unit of the floor. The last term is interpreted as follows: 1/f(x) denotes the additional residents attributable to the additional unit of F(x). These additional residents increase the commuting cost along the route between 1 and x. This is a negative externality, which is represented in the term Ðx ∂Tunit ðmÞ G 1 nðmÞ ∂nðmÞ dm. Therefore, at any x 2 (1, xb], ∂ W/∂F (x) < 0. But, at x ¼ 1, ∂W/∂ FG(x) ¼ 0, as shown in Eq. (A3.15). Hence, excluding x ¼ 1, an open city needs maximum FAR regulation, which reduces the FAR from the market equilibrium FAR.

Mathematical note: Lagrangian method and optimal control theory In this book, we optimize land use policies applying the Lagrangian method to urban models with continuous distance. This section provides a concise description of the Lagrangian method for readers who are less familiar with such an application of the method. (1) Lagrangian method Each section formalizes problems maximizing the social welfare subject to market equilibrium constraints with respect to land use policies. These problems can be mathematically represented as follows. The welfare level W depends on a set of policies Y ¼ ðy1 , y2 , …, yN Þ. Policy makers maximize W(Y) subject to market equilibrium constraints g1 ðYÞ  0, …, and g1 ðYÞ  0. The optimal policies Y∗ are obtained as Y∗ ¼ arg max Y W ðYÞ subject to g1 ðYÞ  0, …, and gI ðYÞ  0. In the context of this book, the policy can be regarded as land use regulations such as spatially dependent FAR regulations. This constrained maximum problem can be represented, using the Lagrangian: L ¼ W ðYÞ +

I X

λi gi ðYÞ

(M.1)

i¼1

The first-order conditions (i.e., necessary conditions), which are called Kuhn-Tucker conditions, are shown in the succeeding text: I ∂L ∂W X ∂gi λi ¼ 0, and ¼ + ∂Y ∂Y ∂Y i¼1

(M.2)

Differences in optimal regulation between closed and open cities

79

∂L ¼ gi  0, λi  0 with complementary slackness (M.3) ∂λi 2 ∂L 3 2 ∂W 3 2 ∂g 3 i 6 ∂y1 7 6 ∂y1 7 6 ∂y1 7 6 7 6 7 6 7 6  7 6  7 6  7 ∂gi ∂L where ∂Y and ¼ 6 7, ∂W ¼ ¼ 6 7 6 7. Complementary slack∂Y 6  7 ∂Y 6  7 6  7 4 ∂L 5 4 ∂W 5 4 ∂g 5 i ∂yn ∂yn ∂yn ness can be represented by using a multiplication. For example, in the case of Eq. (M.3) λi∂L/∂ λi ¼ 0 represents the complementary slackness. The system of Eqs. (M.2), (M.3) can be used to find the optimal policy ∗ Y . However, constraint qualification is necessary for the Kuhn-Tucker condition to represent the optimal solution. There are several methods of constraint qualification. This section shows one method, which is called the regularity condition. This constraint qualification is explained as follows. First, we define a set I∗ 2 {i j gi(Y∗) ¼ 0}. In other words the constraints in set I∗ are all binding. In this case, expression (M.2) can be expressed as PI ∗ ∂gi ∂W i¼1 λi ∂Y . Using a matrix form, this is represented by ∂Y ¼

2

∂g1 6 ∂y1 6 6 ∂g1 6 where A  6 6 ∂y2 6 6 4 ∂g1 ∂yN

∂W ¼ Aλ ∂Y 3

(M.4)

∂g2 ∂gI ∗ ⋯ 2 3 ∂y1 ∂y1 7 7 λ1 ∂g2 7 6 λ2 7 7 ⋯ 7 and λ  6 7. ∂y2 4⋮5 7 7 7 λI ∗ 5 ∂g2 ⋯ ∂yN

A system (M.4) should have a solution of λ. The regularity condition is that a set of vectors that forms matrix A should be linearly independent. To prove that the regularity condition generates positive λ, we need the Minkowski-Farkas lemma, which is difficult to understand intuitively. So we skip the formal proof for this part. Note that matrix A is defined at optimal. Actually, this condition is a sufficient condition to satisfy constraint qualification. To exactly prove this, we need mathematical steps. But the necessity of this condition is straightforward. That is, if some vectors are linearly dependent, (M.4) does not have a set of optimal policies Y∗, which

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should have I∗ dimensions. Therefore a set of vectors, which forms matrix A, should be linearly independent. (2) Optimal control theory In this book, we optimize land use policies applying optimal control theory to urban land with continuous distance. Usually, optimal control theory is represented by Hamiltonian functions. This book uses Lagrangian functions in place of Hamiltonian functions. Actually, both methods can be used interchangeably because they are equivalent. The following proves the equivalency in the case when there is one constraint. We suppose here that the utility level u at location x depends on a set of policies y(x) and T(x). Next, T(x) follows a constraint T_ ¼ gðyðxÞ, T ðxÞÞ, and T(0) ¼ 0 and T(xb) 0. Note that a dot over a variable, hereafter, denotes differentiation Ð bof the variable with respect to u(y(x), T(x))dx. The optimal distance. Policymakers maximize W ¼ xx¼1 ∗ ∗ policies y(x) are obtained as fyðxÞ gx2ð1;xb Þ ¼ arg fyðxÞg b max W subject x2ð1;x

Þ

to T_ ¼ gðyðxÞ; T ðxÞÞ. In the context of this book, the utility level u at location x should be uniform regardless of x. But here, we ignore this condition and other conditions for simple discussion. The problem of maximizing the social welfare in a city from the center (x ¼ 1) to the city boundary (x ¼ xb) can be expressed by the following Lagrangian function: ð xb 

 uðyðxÞ, T ðxÞÞ + λðT Þ gðyðxÞ, T ðxÞÞ  T_ ðxÞ dx + μT ðxb Þ (M.5) L¼ x¼1

Here, the following relation holds: ð xb ð xb  λ_ ðxÞT ðxÞdx + λð0ÞT ð0Þ  λðxb ÞT ðxb Þ λðxÞT_ ðxÞdx ¼ x¼1

(M.6)

x¼1

Substituting Eq. (M.6) into Eq. (M.5) yields ð xb   L¼ uðyðxÞ, T ðxÞÞ + λðT ÞðgðyðxÞ, T ðxÞÞ + λ_ ðxÞT ðxÞÞ dx 1

+ λð1ÞT ð1Þ + λðxb ÞT ðxb Þ + μT ðxb Þ:

(M.7)

Differentiating with respect to y(x) and T(x), we have ∂u ∂g ∂H ðxÞ  λðxÞ ¼0,0¼ , at x 2 ½1; xb Þ ∂yðxÞ ∂yðxÞ ∂yðxÞ ∂u ∂g ∂H ðxÞ λ_ ðxÞ ¼   λðxÞ , λ_ ðxÞ ¼  , at x 2 ½1; xb Þ ∂T ∂T ∂T ðxÞ

(M.8) (M.9)

Differences in optimal regulation between closed and open cities

81

∂u ∂g  λðxb Þ  λðxb Þ + μ ¼ 0 and μT ðxb Þ ¼ 0 , λðxb ÞT ðxb Þ ¼ 0 ∂yðxb Þ ∂yðxb Þ (M.10) where Hamiltonian H(x) is defined as H(x)  u(y(x), T(x)) + λ(x)g(y(x), T(x)). Eqs. (M.8) and (M.9) imply that the discussion using Lagrangian Eq. (M.5) can be replaced by that using the corresponding Hamiltonian, which generates the same first-order conditions. Eq. (M.10)x is called the transversality condition. This explains the equivalency between the Lagrangian and the Hamiltonian methods. So, both methods are applicable to our land use regulation problems. However, land use purpose often discretely changes as the distance from the center increases. Indeed, Chapters 5 and 6 consider multiple discrete areas (e.g., business areas and residential areas). Such a case can be represented by the Lagrangian method easily because the Lagrangian function contains all the discrete areas with different land use purposes, whereas the Hamiltonian only considers a specific location.

References Arnott, R.J., 1979. Unpriced transport congestion. J. Econ. Theory 21 (2), 294–316. Arnott, R.J., MacKinnon, J.G., 1978. Market and shadow land rents with congestion. Am. Econ. Rev. 68 (4), 588–600. Bertaud, A., Brueckner, J.K., 2005. Analyzing building-height restrictions: predicted impacts and welfare costs. Reg. Sci. Urban Econ. 35 (2), 109–125. Brueckner, J.K., 1987. The structure of urban equilibria: a unified treatment of the MuthMills model. In: Mills, E.S. (Ed.), Handbook of Regional and Urban Economics. In: Vol. II. Elsevier Science Publishers, pp. 821–845. Brueckner, J.K., 2007. Urban growth boundaries: an effective second-best remedy for unpriced traffic congestion? J. Hous. Econ. 16 (3–4), 263–273. Fujita, M., 1989. Urban Economic Theory-Land Use and City Size. Cambridge University Press. Harberger, A.C., 1971. Three basic postulates for applied welfare economics: an interpretive essay. J. Econ. Literat. 9 (3), 785–797. Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511. Kanemoto, Y., 1977. Cost-benefit analysis and the second best land use for transportation. J. Urban Econ. 4 (4), 483–503. Kono, T., Joshi, K.K., 2012. A new interpretation on the optimal density regulations: closed and open city. J. Hous. Econ. 21 (3), 223–234. Kono, T., Kaneko, T., Morisugi, H., 2010. Necessity of minimum floor area ratio regulation: a second-best policy. Ann. Reg. Sci. 44 (3), 523–539. x

Note that ∂u/∂ y(xb)  λ(xb)∂ g/∂y(xb) ¼ 0 because Eq. (M.8) holds. μT(xb) ¼ 0 is the complementary condition for T(xb) 0.

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Pines, D., Kono, T., 2012. FAR regulations and unpriced transport congestion. Reg. Sci. Urban Econ. 42 (6), 931–937. Pines, D., Sadka, E., 1985. Zoning, first-best, second-best, and third-best criteria for allocation land for roads. J. Urban Econ. 17 (2), 167–183. Wheaton, W.C., 1974. A comparative static analysis of urban spatial structure. J. Econ. Theory 9 (2), 223–237. Wheaton, W.C., 1998. Land use and density in cities with congestion. J. Urban Econ. 43 (2), 258–272.

CHAPTER FOUR

Optimal land use regulation in a growing city

4.1 Introduction In a city with low population growth, a one-time decision on land use regulation to mitigate population density externalities (e.g., congestion and noise) may work over a long period. However, many cities, particularly those in developing countries, have a high population growth rate. In such cities, the population distribution and thereby the severity of population externalities at a location can change significantly over time. Therefore, to be effective, land use regulation also needs to change dynamically and concomitantly with population growth. In Chapter 2, we have explored a static closed-city model divided into two fixed-area zones, high-externality zone and low-externality zone, based on the severity of negative population density externality in each zone. Applying the model to obtain optimal floor area ratio (FAR) regulations, we have found that a maximum FAR regulation should be imposed in the high-externality zone along with a minimum FAR regulation in the low-externality zone. In this chapter, we extend the static model to take account of a growing population whereby the level of externalities would also change over time such that a present-day low-externality zone could end up becoming a high-externality zone in the course of time. Specifically, following Joshi and Kono (2009), we determine the socially optimal path of FAR regulation in the presence of negative population externalities that change dynamically with population growth. Like in Chapter 2, the model city is restricted against spatial expansion. But the model developed in this chapter has buildings constructed and replaced over time. The remainder of the chapter is organized as follows. Section 4.2 develops the model. Section 4.3 examines socially optimal FAR regulation. The conclusion is presented in Section 4.4. The final subsection provides Technical Appendix. Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00004-2

© 2019 Elsevier Inc. All rights reserved.

83

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Traffic congestion and land use regulations

4.2 The model 4.2.1 The city For simplicity, we assume that the city has only residents and no firms.a The city has a fixed area, but the total city population, given exogenously, increases over time because of natural growth and migration from other regions. For simplicity, population is identified with the number of households, which are assumed to be identical. In addition, the model has developer-cum-landowners.b The city is divided into two zones,c Zone h (high-externality zone) and Zone l (low-externality zone), based on the level of negative population density externality at an initial time. Although each zone in the model city could have both positive and negative externalities, we assume, for simplicity, that the net externality in each zone is negative.d Without any loss of generality, we assume that at an initial time t, the externality level in Zone h is more severe than in Zone l. Like the city area, the zonal area is also assumed to be fixed. Consequently, the population density in each zone is directly proportional to the zonal population. This means, with a growing zonal population, the magnitude of zonal externality also changes over time.

4.2.2 Household behavior At any time, households in the city choose to reside in either Zone h or Zone l. In equilibrium, utilities are equal between the two zones because no household will choose to live in a zone with lesser utility. The instantaneous household utility at time t in zone i (i 2 {h,l}) is a quasi-linear function, given as Vi(t) ¼ zi(t) + ui(t), where zi(t) is all consumer goods except floor space, and ui ðtÞ  ui ð fi ðt Þ, Ni ðt Þ, qi ðtÞÞ denotes the subutility function that depends on the floor space consumption per household fi(t), zone i population Ni(t), and a

However, the determinants of the property of FAR regulation are the population externality and deadweight losses in the floor space market (see Chapter 2). Consequently, if the residential floor space market is replaced by a business floor space market, the results are applicable to business zones as well. Conclusions of this chapter can be extended to business zones by replacing utility functions with the firms’ profit functions. b Introducing landowners as a separate agent would not change the social welfare function because land rent would simply pass from the developers to landowners with no net effect on social welfare. c The setting of two zones is for simplicity; multiple zones can also be grouped into two zones. d The concluding section of the chapter explains how the results are applicable to cases in which positive externality is greater than negative externality.

Optimal land use regulation in a growing city

85

exogenous zonal amenity level qi ðtÞ. Households derive higher utility from larger floor space consumption, that is, ∂ ui(t)/∂ fi(t) > 0. The negative population externality in zone i is expressed as ∂ ui(t)/∂ Ni(t) < 0, whereas q
i ðtÞ determines the equilibrium zonal population distribution at time t. Moreover, the following second-order properties are presumed: ∂2 ui ðtÞ=∂fi 2 ðtÞ < 0, ∂2 ui ðtÞ=∂Ni 2 ðtÞ < 0, and ∂2ui(t)/∂ fi(t)∂ Ni(t) ¼ 0. Each household has y(t) amount of money allocated for expenditures. For simplicity, the dynamic path of y(t) is given exogenously.e Each household maximizes utility by spending y(t) on zi(t) and fi(t). Then, the instantaneous utility-maximizing behavior of a household at time t is given as Eq. (4.1), where ri(t) is the unit floor rent paid to the developers: Vi ðtÞ ¼ max zi ðt Þ + ui ð fi ðt Þ, Ni ðt Þ, q
i ðtÞÞ zi ðtÞ, fi ðt Þ s:t: zi ðtÞ + ri ðtÞfi ðtÞ ¼ yðtÞ

(4.1)

The first-order condition of (4.1) yields a familiar result (4.2) that the marginal utility of floor space consumption equals the unit floor rent. The maximized instantaneous household utility in zone i at time t is given as Eq. (4.3), where fi(t), hereafter, denotes the optimal floor space consumption per household, in accordance with Eq. (4.2): ∂ui ðtÞ ¼ ri ðtÞ ∂fi ðtÞ

(4.2)

Vi ðtÞ ¼ yðt Þ  ri ðt Þfi ðtÞ + ui ð fi ðtÞ, Ni ðtÞ, qi ðt ÞÞ

(4.3)

4.2.3 Developers’ behavior Developers, who are also landowners in our model, combine housing capital and land to produce dwelling space under the FAR regulation. Some assumptions are made. First, developers are perfectly competitive. They are, therefore, price-takers and can be treated as an aggregate developer who maximizes profit from buildings constructed over time. Second, they have complete knowledge of the future evolution of rents. Developers demolish and construct buildings after a certain period because of increasing maintenance costs and changing market conditions, particularly those related to floor rent. The replacement timing can vary among plots even in the same zone. e

Although a dynamic path of y(t) can be chosen to maximize an individual’s life utility, it is not necessary for the purpose of this chapter to consider such choice behavior explicitly.

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Let each zone be divided into several plots, say m ^ plots, without loss of Pm^ generality, such that area of zone i equals m¼1 aim (i 2 {h,l}), where aim is the area of plot with index number m in zone i, where m 2 ð1, m ^ Þ. We assume that m ^ is sufficiently large such that the total floor area in a zone can be regarded as changing smoothly. Furthermore, although plot areas differ, each plot is sufficiently large to accommodate a building. Some assumptions are also made in relation to the construction of buildings. First, each plot has only one building at a time. Second, once built, no vertical or horizontal addition is made to the same building later. Third, the construction cost of a building depends only on its size. Finally, the agricultural land rent or opportunity cost of land is zero. Therefore, vacant lots do not exist in the city because the floor rent is always positive.f Fig. 4.1 depicts an overview of building construction over time on a representative plot aim. A new building constructed at time t on plot aim provides Fim(t) floor space, and is replaced at time t + L1im by a building that provides Fim(t + L1im) floor space area. Therefore, L1im denotes the replacement time of Fim(t) floor space with respect to time t. The subsequent replacement takes place at time t + L2im, and so on. In general, taking time t as the origin, the kth replacement on plot aim takes place at time t + Lkim, where k 2 (0, ∞) and Lkim ¼ 0 if k ¼ 0. The city can have buildings of different age groups at any given time. Let Fi(t) denote the total new floor space area in zone i at time t, given

Fig. 4.1 Floor replacement on a representative plot aimover time. (Source: Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511.)

f

Vacant lots can exist in the city if the agricultural land rent or opportunity cost of land is positive. However, the presence of vacant lots does not affect the conclusions of this chapter.

Optimal land use regulation in a growing city

87

P^ as Fi ðtÞ  mm¼1 Fim ðtÞ. Also, let Gi(t) denote the total existing floor Ð 1i space area in zone i at time t, given as Gi(t + L1i )  t+L τ¼t Fi(τ)dτ Pm^ Ð t + Lim1 1 ¼ m¼1 τ¼t Fim ðτÞdτ, where Li denotes the longest replacement time among buildings constructed at time t in zone i, that is,   Li1 ¼ max Li11 , Li21 , …, Li1m^ . The relation between Fi and Gi is depicted in Fig. 4.2 by two curves: the cumulative added floor space (the left curve) and the cumulative lost floor space (the right curve). In fact, Fi(t) is expressed by the increase in the left curve (or the slope of the left curve), while Gi(t) is expressed as the vertical distance between the two curves, at time t. The shaded area is added to the cumulative added floor space (the left curve) if Fi(t) is increased by dFi(t) at time t. The added floor space will be lost after the replacement time L1i ; so the shaded area is added to the cumulative lost floor space (the right curve) after t + L1i . In the end, Gi(τ) over τ 2 (t, t + L1i ) is increased by dFi(t) if Fi(t) is increased by dFi(t).   ^ im t + L k denote net profit from Fis(t + Lkis) floor space on plot Now, let Π im k aim over the period Lk+1 is , discounted back to time t + Lim, which is given as   ^ im t + L k  Π im

k+1 t +ð Lim

     k k ri ðτÞ  Pim τ  t  Lisk eσðτtLim Þ dτ Fim t + Lim

k τ¼t + Lim

   k  Si Fim t + Lim (4.4)

Fig. 4.2 Cumulative curves of added and lost floor space. (Source: Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511.)

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Traffic congestion and land use regulations

where the integral sums up the net rental income from Fis(t + Lkis) over period k Lk+1 is  Lis, i.e., the total rental income minus floor maintenance cost, denoted Pis(τ  t  Lkis), over that period, and discounts it to time t + Lkim, whereas Si(Fim(t + Lkim)) denotes the total construction cost of those floors. In Eq. (4.4), Pim(0)¼ 0, and ∂ Pim(τ  t  Lkis)/∂ τ > 0, where τ 2 (t + Lkim, t + Lk+1 im ), which implies that the maintenance cost increases from zero as the building ages.g The time-discounted net profit from Fis(t) floor space and its future replacements on plot aim, denoted Πim(t), is defined in Eq. (4.5) which gives the developers’ profit-maximizing behavior: ∞ X   ^ im t + L k eσLimk Π max Πim ðtÞ  im k + 1 Lim ðk2ð0,∞ÞÞ k¼0

(4.5)

where σ is the time-discount factor, which, for simplicity, is assumed to be constant over time. The optimal replacement time of Fim(t + Lkim) with respect to initial time t, hereinafter denoted Lk+1 im , where k 2 (0, ∞), is determined endogenously by solving the first-order condition of (4.5).h Note that the developers cannot control the building size under FAR regulation. The time-discounted net profit from all floors in new or old buildings at time t and their future replacements in both zones, denoted Π(t), is then given as

Πðt Þ 

2

X

4

∞ ð

ri ðτÞGi ðτÞeσ ðτtÞ dτ

i2fh, lg

2



∞ X 6 4 k¼0 ∞ ð



τ¼t t +ð Lisk + 1

3   σðτtLk Þ 7 σLk k im dτ e im e Pim τ  t  Lim 5

τ¼t + Lisk

#

(4.6)

Si ðFi ðτÞÞeσ ðτtÞ dτ

τ¼t

g

For simplicity, the maintenance cost is assumed to be independent of the building size. An example is the base maintenance cost. However, even without this assumption, an essentially similar conclusion would be obtained. h See Technical Appendix 4(1) for the optimality condition for L1im. A similar condition is obtained in Akita and Fujita (1982) (see also Miyao, 1987; Brueckner, 2000).

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Optimal land use regulation in a growing city

Because developers are perfectly competitive, and because they are also landowners, Π(t) is absorbed into the land rents.

4.2.4 Market clearing conditions The market clearing conditions are given in Eqs. (4.7)–(4.9) and explained as follows. First, the exogenous total city population is distributed between Zones h and l such that the marginal change in one zonal population at any time t, denoted Ni(t) (i 2 {h, l}), is counterbalanced by an equal change in the other zonal population: dN h ðtÞ ¼ dN l ðtÞ

(4.7)

Next, in equilibrium, the residential utilities, denoted Vi(t) (i 2 {h, l}), are equal between the two zones because no household will choose a lesserutility zone: Vh ðtÞ ¼ Vl ðtÞ

(4.8)

Finally, the total existing floor space in any zone (left side) equals the total floor space consumed by households in that zone (right side): Gi ðtÞ ¼ Ni ðtÞfi ðt Þ; i 2 fh,l g

(4.9)

4.3 Maximizing social welfare using FAR regulation 4.3.1 Social welfare Interpretation of the optimality of the FAR regulations depends on the objective function adopted. We adopt a Benthamite social welfare function which sums individual households’ utilities and developers’ profit. Accordingly, taking time 0 as the origin, the total time-discounted social welfare, denoted W(0), is given as ∞ ð

W ð0Þ 

X

Ni ðτÞVi ðτÞeστ dτ + Πð0Þ

(4.10)

τ¼0 i2fh,lg

where the integral discounts the sum of households’ utility over time, whereas Π(0) can be recalled from Eq. (4.6).

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Traffic congestion and land use regulations

4.3.2 Optimal dynamic FAR regulation Total social welfare W(0) can be increased by controlling Gi(τ)(τ 0), which in turn controls zonal population, and thereby affects household utility and zonal externality. On the other hand, controlling Gi(τ) also affects developers’ net income. Therefore, W(0) can be written as a function of Gh(τ) and Gl(τ), which are controllable variables under the FAR regulation. Optimality of the regulation can then be defined as the maximization of social welfare. More specifically, optimal FAR regulation in zone i at time t gives zonal floor space supply G∗i (t) such that Gi∗ ðtÞ ¼ arg max W ðGh ðtÞ, Gl ðtÞÞ subject Gh ðtÞ, Gl ðtÞ ði2fh,lgÞ ∗ to Eqs. (4.2), (4.3), (4.6)–(4.10). Here, Gi (t) implies the optimal total floor space in zone i. Because Gi(t) comprises previously constructed buildings, i.e., Fi(τ)(τ  t), any adjustment in G∗i (t) implies adjustment in the new buildings represented by Fi(t). Consequently, by controlling Fi(t) instead of Gi(t), the optimality condition for FAR regulation at any time t can be expressed as dW(0)/dFi(t) ¼ 0.i,j This optimal condition implies that the dynamic path of Fi(t) is optimal such that any infinitesimal deviation from the path of Fi(τ)(τ  t) does not change the social welfare. The marginal change in social welfare, denoted dW(0), can be derived from the utility function and market clearing conditions. Through the derivation process described in Technical Appendix 4(2), dW(0) is given as   ∂uh ðτÞ ∂ul ðτÞ  Nl ðτÞ dN h ðτÞ Nh ðτÞ dW ð0Þ ¼ ∂Nh ðτÞ ∂Nl ðτÞ τ¼0  X  ∂Si ðFi ðτÞÞ ri ðτÞdGi ðτÞ  dF i ðτÞ eστ dτ + ð τ Þ ∂F i i2fh,lg ∞ ð

i

(4.11)

This direct derivation of the necessary condition is slightly informal—the difficulty is that the term is of order dt, which is infinitesimal. One formal way to derive this expression is to model the economy over finite intervals (0, Δt), (Δt, 2Δt), (2Δt, 3Δt), …, assuming that the economy is constant within each interval, and then take the limit as Δt approaches zero. However, this process formally yields the same expression. j We assume that Fi(τ) > 0, which is plausible in the case of an increasing population. On the other hand, Gi(τ) comprises new floor space and previously built floor space that is constantly being demolished. Therefore, even if Fi(τ) > 0, Gi(τ) can be negative when the rate of cumulative building demolition exceeds the rate of cumulative building construction (see Fig. 4.2). This implies that Fi(τ) > 0 can also be compatible with the case of a decreasing population. However, if the population is decreasing steadily, we must take account of the corner solution, i.e., Fi(τ) ¼ 0, which, although not difficult to consider, is neglected in this paper.

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Optimal land use regulation in a growing city

Applying dW(0)/dFi(t) ¼ 0 to Eq. (4.11), the condition for optimal FAR regulation at any time t is achieved as Eq. (4.12), where L1i is the longest replacement time among buildings constructed at time t in zone i. Note that dFi(t) floors last up to t + L1i . Therefore, Gi(τ) is increased by dFi(t) over τ 2 (t, t + L1i ) as depicted in Fig. 4.2, and this increase in Gi(τ) also affects population distribution, i.e., dNi(τ), over the same period. Therefore, we obtain Eq. (4.12) from Eq. (4.11), as dW ð0Þ ¼ dF i ðt Þ

t +ðLi1 

τ¼t

 ∂uh ðτÞ ∂ul ðτÞ dN h ðτÞ σ ðτtÞ  Nl ðτÞ Nh ð τ Þ e dτ ∂Nh ðτÞ ∂Nl ðτÞ dF i ðtÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Marginal change in aggregate externality level due to change in F i ðtÞ

t +ðLi1

+

∂Si ðFi ðtÞÞ ¼ 0 ði 2 fh,lgÞ ∂Fi ðtÞ τ¼t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ri ðτÞeσðτtÞ dτ 

(4.12)

Deadweight loss in zonal floor space market

∂ui ðτÞ ði 2 fh, l gÞ. Expressed in Let E(τ)  Eh(τ)  El(τ), where Ei ðτÞ  Ni ðτÞ ∂N i ðτ Þ words, Ei(τ) denotes the severity of externality in zone i, and E(τ) represents the relative externality level between the two zonesk at time τ such that Ð t + L1 h ðτ Þ σ ðτt Þ dτ implies marginal E(τ) < 0 when Eh(τ) > El(τ).l So, τ¼t i E ðτÞ dN dF i ðtÞ e 1 change in the aggregate externality level over Li due to the marginal change in Fi(t). Ð t + L1i ri(τ)e σ(τt)dτ  ∂Si(Fi(t))/∂Fi(t). Expressed in Similarly, let Di(t)  τ¼t words, Di(t) is the difference between the time-discounted sum of unit floor rents over L1i and marginal construction cost of Fi(t) floors. In other words, Di(t) is the FAR regulation-induced marginal change in the deadweight loss in the corresponding floor space market, as depicted in Fig. 4.3. Eq. (4.12) gives the optimal condition for the FAR regulation. Specifically, optimal FAR regulation at time t in zone i (i 2 {h,l}) is achieved when Ð t + Li1 dN h ðτÞ σ ðτtÞ dτ is offset by Di(t). The optimal solution (4.12) τ¼t E ðτÞ dF i ðtÞ e requires updating the FAR regulation at all times in response to the changes in market conditions (e.g., rents), in addition to changes in local amenity

Because dNh(τ) ¼  dNl(τ), E(τ)dNi(τ) in (4.12) refers to the marginal change in the aggregate externality level in the city, whereas E(τ) alone refers to relative externality levels between the two zones. l By presumption, E(t) < 0, that is, the externality level in zone h is more severe than in zone l at an initial time t. However, the externality level in each zone can change dynamically such that at some point of time τ, E(τ)(τ > t) can also be positive, which would then imply that Eh(τ) < El(τ). k

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Traffic congestion and land use regulations

Fig. 4.3 Unit floor rent and marginal cost curve. (Source: Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511.)

levels. Note that even under the optimal FAR regulation, deadweight loss exists in the regulated market, which implies that the FAR regulation is second best because it cannot adjust the population density directly (see Chapter 2). The optimal solution (4.12) also provides a planning horizon for optimal FAR regulation; more specifically, to obtain optimal FAR regulation at time 1 h ðτ Þ t in zone i (i 2 {h,l}), it is sufficient to forecast E ðτÞ dN dF i ðt Þ for the period Li only. This is a useful result for urban planners because it is sufficient for them to consider the expected changes in zonal externality levels and market factors for a certain limited period given by L1i . We now explore the properties of optimal FAR regulation using Eq. (4.12). First, Di(t) is a function of Gi(t), as depicted in Fig. 4.3. At the market equilibrium without regulation, Di(t) ¼ 0, and the total zonal floor supply is denoted GM i (t). Under the constraint of Di(t) > 0, the optimal floor supply, denoted G∗i (t), falls below GM i (t), necessitating maximum FAR regulation. Likewise, when Di(t) < 0, G∗i (t) is greater than GM i (t), necessitating minimum FAR regulation. Next, factors such as interzonal migration, population density regulations, and/or investments in local amenities in the two zones can change the E(τ) path, leading to either E0 (τ) < 0 or E0 (τ) > 0, where E0 (τ)  ∂E(τ)/∂ τ. Here,

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Optimal land use regulation in a growing city

E0 (τ) < 0 implies that the marginal change in zonal externality level, attributable to the zonal population change, is increasing more in Zone h relative to Zone l at time τ. This occurs when the population growth rate is high in Zone h. This can also occur when a better local amenity improvement policy pertains in Zone l, which leads to a great decrease in j∂ ul(τ)/∂ Nl(τ)j than in 0 j∂ uh(τ)/∂ Nh(τ)j. A similar explanation holds for E (τ) > 0. Before proceeding further, we consider Presumption 4.1, which applies to most real situations.m Presumption 4.1 The signs of e σ(τt)dτ are the same.

R t+Ll1 Ð t+Lh1  σ(τt) dτ and τ¼t E(τ) τ¼t E(τ)e

Presumption 4.1 implies that the zonal externality levels summed over period L1l and L1h are either both positive or both negative. Note that the only difference between the two sides of the equation in Presumption 4.1 lies in theÐ integral periods. To explain cases where Presumption 4.1 holds, 1  σ(τt) l dτ is expressed as the term t+L τ¼t E(τ)e ð

ð

t + Ll1

E ðτÞe τ¼t

ð

t + Lh1 σ ðτt Þ

dτ ¼

t + Ll1

EðτÞe τ¼t

σ ðτtÞ

dτ +

EðτÞeσðτtLh Þ dτ 1

τ¼t + Lh1

(4.13) Note that the sign of the last term in the right side of Eq. (4.13) is the same as those of the other terms. Presumption 4.1 holds in any of the following three cases. Case C: L1h is significantly close to L1l regardless of the change in the sign of E(τ) within periods L1h and L1l . This means that the last term in the right side of Eq. (4.13) is negligible. Case U: the sign of E(τ) is unaltered within periods L1h and L1l in which case Presumption 4.1 directly holds. In contrast, Case C, L1h differs significantly from L1l . Case AE: the sign of E(τ) is altered earlier (i.e., soon after time t) and sufficiently before time t + L1h or t + L1l , whichever comes first. After the alteration, the sign of E(τ) is unaltered for the remaining periods within L1h and L1l . Similar to Case U, L1h differs significantly from L1l . Case AE refers to situations m

Presumption 4.1 is introduced at this stage to analyze more likely situations first. The violation of Presumption 4.1 is discussed in Section 4.3.4.

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Traffic congestion and land use regulations

in which the zonal externality levels would change within τ 2 (t, t + L1i ) such that Zone l would end up with more severe externality level than Zone h. Case C is expected to hold in most situations as both zones are likely to have similar economic conditions (e.g., household wages, household utility, building structures, or construction technologies) because of which developers would have no incentive to construct buildings differently in any particular zone. It is also important to note that even if L1h and L1l differ significantly or even if significant changes in zonal externality levels take place between t + L1h and t + L1l , such differences or changes are discounted to time t; therefore, the last term in Eq. (4.13) could be negligible in most cases. The example paths of E(τ) are depicted in Fig. 4.4 with reference to Cases C, U, and AE. Under Presumption 4.1, two outcomes regarding Ð 1i the dynamic E(τ) paths are obtained: Case C or Case U leads to t+L τ¼t E(τ) Ð 1i  σ(τt) e σ(τt)dτ < 0, whereas Case AE leads to t+L E(τ)e dτ > 0, where τ¼t i 2 {h,l}. Ð t+L1i E(τ)dNh(τ)/dFi(t) Using Eq. (4.12), which can be written as τ¼t  σ(τt) dτ + Di(t) ¼ 0 (i 2 {h, l}), and noting that dNh(τ)/dFi(t) > () GM i (t) when Di(t) > ( 0

Optimal FAR regulation at time t

h l

(+) ()

G∗h (t) < GM h (t) G∗l (t) > GM l (t)

h

()

G∗h (t) > GM h (t)

Minimum FAR regulation

l

(+)

G∗l (t) < GM l (t)

Maximum FAR regulation

Maximum FAR regulation Minimum FAR regulation

Optimal land use regulation in a growing city

Table 4.1 Optimal FAR regulation under Presumption 4.1 and initial condition Eh(t) < El(t). Cases Outcomes Zone Di(t) Floor supply

Source: Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511.

95

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Traffic congestion and land use regulations

Proposition 4.1 (Optimal dynamic FAR regulation) Assume Eh(t) < El(t) without any loss of generality where Ei(t ) < 0(i 2 {h,l}). The optimal FAR regulation at time t requires: (i) Maximum FAR regulation in Zone h and minimum FAR regulation in Zone l if Case C or Case U holds. (ii) Minimum FAR regulation in Zone h and maximum FAR regulation in Zone l if Case AE holds. The maximum FAR regulation in one zone and minimum FAR regulation in the other zone should be simultaneously effective or binding. In other words, maximum FAR regulation imposed in one zone with no regulation in the other zone is not optimal. This is because under Presumption 4.1, if Eq. (4.12) requires Dh(t) > 0, then simultaneously Dl(t) < 0, or vice versa. The necessity of both maximum and minimum FAR regulations has been explained in Chapter 2 although in a static-city case. A similar explanation applies in a growing city as well. Fig. 4.5 illustrates a case when G∗h (t) < GM h (t), which simultaneously implies G∗l (t) < GM l (t) (i.e., Case C or U). In this case, under optimal regulations (i.e., maximum and minimum FAR regulation in Zone h and Zone l, respectively), the floor supply is reduced (increased) in Zone h (Zone l). Consequently, households migrate from Zone h to Zone l, which affects the zonal externality levels and floor space demand and causes zonal floor rent to rise or fall.n The resultant total deadweight loss in both floor space

Fig. 4.5 Graphical interpretation of optimal FAR regulation (Case C or U). (Source: Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511.)

n

Fig. 4.4 presents a case in which floor rents decrease in both zones. Although not described in this chapter, it can be shown that, given Eq. (4.8), maximum (minimum) FAR regulation in zone h(l) can affect floor rents in three ways: (i) both rh, rl increase; (ii) both rh, rl decrease; or (iii) rh increases, but rl decreases.

Optimal land use regulation in a growing city

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markets is given as the sum of shaded areas Y and Z. If maximum FAR is imposed in Zone h leaving Zone l unregulated, the resultant migration from Zone h to Zone l may also reduce total externality in the economy, but the resultant deadweight loss in Zone l, given as the area X bound by a thick line in Fig. 4.4, would be greater than the deadweight loss achieved in the optimal case, that is, X > Y + Z.

4.3.3 Differences from a static city In a static city, the optimal FAR regulation in a zone at time t depends Ð t+L1i only E(τ) on the property of E(t), whereas in a growing city, the behavior of τ¼t  σ(τt) dτ dictates the optimality of the regulation. The result in the case of a e static city is straightforward: the optimal FAR regulation requires enforcement of minimum (maximum) FAR regulation in the low-externality (high-externality) zone. However, in a growing city, the opposite result can also be achieved as seen in Case AE (see Proposition 4.1(ii))—a maximum FAR regulation may be necessary in the low-externality zone (Zone l in our case) to protect it from a more severe level of externality in the future. Other distinctions lie in how unit floor rents are considered and how deadweight loss is calculated. In a growing city model, the deadweight loss in the floor space market depends on the sum of floor rents over the period L1i (i 2 {h, l}), which is discounted to the desired time (e.g., when regulations are to be enforced or updated) (see Fig. 4.4). In a static city only instantaneous unit floor rent is considered.

4.3.4 Exceptional cases Presumption 4.1 may not hold in the case of exceptionally longer or shorter optimal replacement time of buildings in one zone relative to the other zone, that is, L1h ≫ (or, ≪)L1l . We now examine how results vary from Proposition 4.1 in such cases. For the initial condition E(t) < 0, only the following two cases violate Presumption 4.1: Ð 1h Ð t+L1l  σ(τt) dτ < 0 and τ¼t E(τ) Exceptional Case (L1h ≪ L1l ): t+L τ¼t E(τ)e  σ(τt) dτ > 0. e Ð 1h Ð t+L1l  σ(τt) dτ > 0 and τ¼t E(τ) Exceptional Case (L1h ≫ L1l ): t+L τ¼t E(τ)e  σ(τt) dτ < 0. e As shown by example paths portrayed in Fig. 4.6, Exceptional Case (L1h ≪ L1l ) occurs when, with L1h ≪ L1l , the sign of E(τ), where τ 2 (t, t + L1l ), alters at a time closer to (i.e., just before or soon after) t + L1h, whereas Exceptional Case (L1h ≫ L1l ) occurs when, with L1h ≫ L1l , the sign of E(τ), where τ 2 (t, t + L1h), alters at a time closer to t + L1l .

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Fig. 4.6 Exceptions to Presumption 4.1. (Source: Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511.)

The replacement time of a building is determined mainly by its durability, which depends on the structural characteristics of the building (e.g., building materials). If the difference in the structural characteristics of buildings constructed in the two zones at time t is significant, then L1h ≪ L1l or L1h ≫ L1l can occur. If the zonal externality levels change in such a way that the lowexternality zone at initial time t (Zone l in our case) faces more severe level of externality within a short period relative to L1l Ðand at the same time Ð 1 t+L1l  σ(τt) h dτ < 0 and τ¼t E(τ)e σ(τt)dτ > 0 L1h ≪ L1l , it is possible that t+L τ¼t E(τ)e 1 1 simultaneously, which is the Exceptional Case (Lh ≪ Ll ). In this case, following Table 4.1, Di(t) > 0 for both i 2 {h,l}, requiring maximum FAR regulations in both zones. Similarly, in Exceptional Case (L1h ≫ L1l ), minimum FAR regulation in both zones would be the optimal solution. In the absence of Presumption 4.1, having maximum (or minimum) FAR regulation in both zones is not an unexpected result. This is because even under such regulation, one zone can have a higher permissible FAR than the other zone, which would induce interzonal migration and thereby lower aggregate externality levels. Moreover, because the floor space built at a certain time exists up to its replacement time, the FAR regulation at a certain time is also effective over such a period as a measure to mitigate externality. To optimize the FAR regulation over a long period, on some exceptional cases as discussed in this section, the optimal FAR regulation at a certain time can necessitate maximum (or minimum) regulation in both zones.

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99

4.4 FAR regulation in a growing monocentric city under traffic congestion externality We can extend the monocentric city models discussed in Chapter 3 to include traffic congestion externality in a dynamic setting using the results achieved in the preceding sections. This section presents a brief overview of the nature of optimal FAR regulation in a growing city—open and closed. We start at a time when all land within the city jurisdiction is still farmland. Then, taking time 0 as the origin, the first-ever building at location x is built at time L0(x), providing F(x, L0(x)) f loor area.o This building is replaced at time L1(x) by a new one that now provides F(x, L1(x)) floor area. Subsequent replacement takes place in a similar fashion. In general, with reference to time 0, the (k + 1)th construction or replacement at location x occurs at time Lk(x), where k ¼ 0, 1, 2, and so on.p The government determines dynamic path of optimal F(x, t) and accordingly announces optimal FAR binding at time t. Fig. 4.7 shows floor area supplied at location x (denoted by a solid line) against the floor area announced under the optimal dynamic path of FAR (denoted by a dashed line). Because of the durability of buildings, the regulated floor area at time t

Fig. 4.7 Existing floor area in the FAR-regulated city.

o p

L0(x) thus indicates time when land at location x is converted from agricultural to residential use. Note that the first construction at location x takes place at L0(x) (that is, when k ¼ 0) but the first replacement takes place at L1(x) (that is, when k ¼ 1).

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Fig. 4.8 Example of urban growth pattern in a closed monocentric city.

differs from that set by the government except at the construction times Lk(x). Fig. 4.8 shows a typical pattern of urban growth in a monocentric city over time. At any given time, population density and thereby building FAR decrease as we move away from the CBD edge (denoted as Distance in Fig. 4.8). Urban fringe areas increasingly undergo rural-tourban conversion as time passes. Buildings, once built, continue to exist for a long time. In other words, FAR of a building does not change for a long time. A static city is a snapshot of a growing city at any given time. Therefore, using results achieved earlier in Chapter 3, we can say that at any given time, a monocentric closed city requires increasingly upward (resp. downward) adjustment to the market density through minimum (resp. maximum) FAR regulation at the more central (resp. distant) locations. In a monocentric open city, only downward adjustment to the market density is required at locations beyond the CBD edge, which is obtained through maximum FAR regulation. However, in a dynamic setting, some additional differences arise between the closed and open city models.

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Specifically, in a growing closed city, optimal FAR regulation might require some locations between the center and boundary to switch from minimum to maximum FAR regulation and vice versa over time as illustrated in Fig. 4.8. This is because both deadweight loss and traffic congestion externality at a location are summed up over the use period of the building at that location, and their net effect dictates where minimum or maximum FAR regulations are needed (except in the most central or most outer locations where necessity of minimum FAR and maximum FAR regulations continue to hold). On the other hand, in a growing open city, maximum FAR regulation holds for all locations beyond the CBD edge at any time. Stringency of regulation, however, would change depending on how much population grows or how the city expands in a given period, and this applies in both open and closed cities.

4.5 Conclusion In this chapter, we have examined optimal FAR regulation in a growing two-zone fixed-area closed city in the presence of negative population externality. In Chapter 2, we considered a similar city model but in a static setup. Such a static-city model may be useful to analyze a city with low population growth because a one-time decision on land use regulation to mitigate population density externalities (e.g., congestion and noise) may work over a long period. However, in a city with a high population growth rate, the population distribution and thereby the severity of population externalities at a location can change significantly over time. Therefore, to be effective, land use regulation also needs to change dynamically and concomitantly with population growth. The results of this chapter are, therefore, useful for cities with growing population. In Chapter 2, we saw that in a two-zone city under negative population externality, optimal regulation requires enforcement of not only maximum FAR regulation in the high-externality zone but also minimum FAR regulation in the low-externality zone. In this chapter, we have found that the optimal FAR condition depends on how relative externality level between the zones changes over time. For example, if the low-externality zone is likely to be affected by a severe level of externality in the future under

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no intervention, the optimal policy could be to impose maximum FAR regulation now. In any case, minimum FAR regulation in one zone should be accompanied by maximum FAR regulation in the other zone or vice versa. It is difficult to forecast how externalities or market conditions would change over time. In this regard, we have found that to obtain optimal FAR regulation at any time, it is sufficient to forecast variables for a certain limited period given by the longest replacement time among buildings newly constructed at that time. Such a certain planning horizon for optimal FAR regulation is a useful result for urban planners. When the replacement times of buildings vary significantly between the two zones under exceptional cases, there may be situations where both zones need to be regulated under maximum (or minimum) regulation. This, however, is not an unexpected result because even under such regulation, one zone can have a higher permissible FAR than the other zone, which would induce interzonal migration and thereby lower aggregate externality levels. This chapter does not explicitly address positive population externality arising from the agglomeration of population and firms in the city. But as discussed in the concluding section of Chapter 2, the results of this chapter can also be extended to business zones by considering net population externality that remains negative as long as negative externality is more dominating; otherwise, the results can be interpreted oppositely by interchanging zones. For instance, instead of maximum FAR under negative externality, a zone might require minimum FAR under positive externality. Agglomeration effects can also be considered explicitly using the profit function of firms.

Technical Appendix 4 (1) The optimality condition for L1im Using Eq. (4.5), Πim(t) can be elaborated as in Eq. (A4.1). Then, upon applying ∂ Πim(t)/∂ L1im ¼ 0, the last term in Eq. (A4.1) vanishes, and after rearrangement of terms, the optimality condition for L1im is given as Eq. (A4.2). In words, L1im is optimal when the rental income from Fim(t) floors (left side) is just equal to the net income that might be gained

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from Fim(t + L1im) floors (right side). The general form of Eq. (A4.2) for any Lk+1 im (0  k < ∞) is obtainable using the same procedure: Πim ðtÞ ¼

∞ X   ^ im t + L k eσLimk Π im k¼0 1 t +ðLim

¼

½Fim ðtÞri ðτÞ  Pim ðτ  tÞeσ ðτtÞ dτ  Si ðFim ðt ÞÞ

τ¼t 2 t +ðLim

+

     σðτtL1 Þ σL 1 k 1 im e im ri ðτÞ  Pim τ  t  Lim e Fim t + Lim

1 τ¼t + Lim

∞ X    σL 1   k ^ im t + L k eσLimk Π e im + dτ  Si Fim t + Lim im

(A4.1)

k¼2

   1  σL 1     σL1 ∂Πim ðtÞ  1 1 1  Pim Lim e im  Fim t + Lim ri t + Lim e im ¼ Fim ðtÞri t + Lim 1 ∂Lim 2 t +ðLim   1    σL 1 ∂Pim τ  t  Lim 1 e im ¼ 0  eσ ðτtÞ dτ + σSi F^im t + Lim ∂τ 1 τ¼t + Lim

(A4.2)

(2) Derivation of Eq. (4.11) Using Eqs. (4.7), (4.8), (4.10), P dW(0) can be written P as Eq. (A4.3). It is also noteworthy that, because i2fh,lg Ni ðτÞ is fixed, i2fh,lg dN i ðτÞVi ðτÞ ¼ 0. The integral in Eq. (A4.3) shows the discounted marginal change in the total household utility summed over periods beyond time 0. P From Eq. (4.3) and using Eq. (4.2), dVi(τ) is solved by Eq. (A4.4). Then, i2fh,lg Ni ðτ ÞdV i ðτÞ can be expanded to yield Eq. (A4.5). Similarly, recalling Eq. (4.6), dΠ(0) in Eq. (A4.3) denotes the marginal change in developers’ net profit from all existing buildings at time 0 and their future replacements; this is defined in Eq. (A4.6). dLkis term does not appear by virtue of the first-order condition of Eq. (4.5) (i.e., by envelope theorem) because Lkis is optimal: ∞ ð

dW ð0Þ ¼

X

τ¼0 i2fh,lg

Ni ðτÞdV i ðτÞeστ dτ + dΠð0Þ

(A4.3)

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∂ui ðτÞ ∂ui ðτÞ df i ðτÞ + dN i ðτÞ  ri ðτÞdf i ðτÞ  dr i ðτÞfi ðτÞ ∂fi ðτÞ ∂Ni ðτÞ ∂ui ðτÞ dN i ðτÞ  dr i ðτÞfi ðτÞ ¼ ∂Ni ðτÞ

dV i ðτÞ ¼

  ∂uh ðτÞ ∂ul ðτÞ Ni ðτÞdV i ðτÞ ¼ Nh ðτÞ  Nl ðτÞ dN h ðτÞ ∂Nh ðτÞ ∂Nl ðτÞ i2fh, lg X  Ni ðτÞfi ðτÞdr i ðτÞ: i2fh, l g

(A4.4)

X

dΠð0Þ ¼

X i2fh, lg

2 4

∞ ð

στ

dr i ðτÞGi ðτÞe

∞ ð

dτ +

τ¼0

ri ðτÞdGi ðτÞeστ dτ

τ¼0

3

∞ ð

 τ¼0

(A4.5)

(A4.6)

∂Si ðFi ðτÞÞ dF i ðτÞeστ dτ5 ∂Fi ðτÞ

Next, the floor space equilibrium Eq. (4.9) is rewritten as Eq. (A4.7), which states that summing over the same period and discounted to time t, developers’ rental income from the overall zonal floor space supply is equal to the rental payment by households for consumption of the same. Finally, substituting Eqs. (A4.4)–(A4.7) into Eq. (A4.3) yields Eq. (4.11): Gi ðτÞdr i ðτÞeστ ¼ Ni ðτÞfi ðτÞdr i ðτÞeστ

(A4.7)

References Akita, T., Fujita, M., 1982. Spatial development processes with renewal in a growing city. Environ. Plan. A 14 (2), 205–223. Brueckner, J.K., 2000. Urban growth models with durable housing: an overview. In: Huriot, J.-M., Thisse, J.-F. (Eds.), Economics of Cities. Cambridge University Press, Cambridge, pp. 263–289. Joshi, K.K., Kono, T., 2009. Optimization of floor area ratio regulation in a growing city. Reg. Sci. Urban Econ. 39 (4), 502–511. Miyao, T., 1987. Urban growth and dynamics. In: Miyao, T., Kanemoto, Y. (Eds.), Urban Dynamics and Urban Externalities. Harwood Academic Publishers, Chur, Switzerland, pp. 1–42.

CHAPTER FIVE

Optimal land use regulations in a city with business areas

5.1 Introduction Previous chapters have targeted land use regulations in residential areas only. However, business zones also can be regulated to enhance positive externalities. In business zones, concentration of workers enhances communication and thus facilitates exchange of innovative ideas (see Rauch, 1993; Ciccone and Hall, 1996; Duranton and Puga, 2001; Moretti, 2004). To make use of such positive externalities and to mitigate negative externalities such as traffic congestion, governments can intervene in the urban space market through simultaneous imposition of multiple regulations on building size and lot size and by zoning the city into different land uses. In real cities, however, such land use regulations have rarely been implemented or seriously considered. This chapter explores how multiple land use regulations should be simultaneously imposed across a city in the presence of agglomeration economies and traffic congestion following Kono and Joshi (2018). In particular, we consider a monocentric city with three distinct land use zones—consisting of business, condominiums, and detached houses in that order—which closely resemble land use observed in real-world cities. The city has agglomeration economies in the business zone and traffic congestion across the city. Applying optimal control theory to the continuous city with the three distinct zones, we obtain optimal density regulation that changes continuously in each distinct zone. The rigorous derivation is shown in Technical Appendix, but the sketch of the derivation process is demonstrated in the main text. We separately treat floor area ratio (FAR) regulation and lot size (LS) regulation because building-size regulation such as FAR regulation necessarily generates deadweight loss caused by the regulation itself (see Chapter 2), whereas LS regulation has no deadweight losses (see Wheaton, 1998). Under FAR regulation, households can choose their Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00005-4

© 2019 Elsevier Inc. All rights reserved.

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optimal floor size within the regulated buildings. That is, FAR regulation controls population density indirectly, whereas LS regulation does this directly. In addition, we design optimal regulations on multiple zonal boundaries between the business zone, condominium zone, and detached housing zone. Section 5.2 develops a model, and Section 5.3 examines optimal regulations. Section 5.4 presents conclusion. The final subsection provides a technical appendix that shows mathematical treatment of such three-zone model to derive the optimal conditions on land use regulation. We show by how much the building size, the LS, and the zone size should differ from those found at the market equilibrium. The differences are composed of empirically observable economic variables. This result can readily be used to evaluate whether the current density and zonal regulations are optimal or not.

5.2 The model 5.2.1 The city The model city is closed, monocentric, and linear with a width of unity and size defined by m 2 [MH, MH], where m denotes distance from the city center. As depicted in Fig. 5.1, which shows only the right-hand side of the symmetrical city, the city is divided into the following three zones in the given order: (i) the central business district (CBD) or business zone, Zone B (m 2 [0, MB]), consisting of office buildings; (ii) the condominium zone, Zone C (m 2 [MB, MC]); and (iii) extending to the city boundary, the lot housing zone, Zone H (m 2 [MC, MH]), consisting of single-family houses.

Fig. 5.1 The model city. (Source: Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598.)

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107

Zone B and Zone C are regulated by FAR regulation. We assume that all buildings are built on lots of equal size, which is normalized to unity. Therefore the FAR of a building is equivalent to its total floor supply area. Let Fj ( j 2 {B, C}) denote the FAR of a building in Zone j. Likewise the lots within Zone H are regulated; let fH denote the lot consumption per household in the zone.
identical households divided equally The city is inhabited by 2N between the two halves of the symmetrical city. One member of each household commutes to the CBD where all firms, and therefore jobs, are located. In other words the city population is identified with the number
is exogenously fixed. of households. The city is closed, implying that 2N Buildings in Zones B and C are constructed by developers, whereas we ignore housing capital in Zone H, assuming that land is directly consumed by the residents. We assume so-called public land ownership under which residents share the city land equally. Hereafter, we basically model the right half of the city unless it is unavoidable to model both sides. We consider two types of externalities: (i) agglomeration economies that arise from communication between firms in Zone B and (ii) traffic congestion across the city. To address these two externalities, FAR regulation, LS regulation, and zonal regulation on zonal boundaries are imposed. The policy variables are (1) FAR at each location in Zone B and Zone C, that is, Fj(m), ( j 2 {B, C}, m 2 [0, MC]) (2) lot size at each location in Zone H, that is, fH(m), (m 2 [MC, MH]); and (3) three zonal boundaries, that is, Mk (k 2 {B, C, H}). Firm density in Zone B and population density in Zone C are adjusted only by FAR regulation, whereas in Zone H, only LS regulation adjusts population density.

5.2.2 Firms’ behavior All firms are located within Zone B, and they have identical production functions. We model single-worker production that uses floor area as an input and labor. The production function is expressed as AX( fB), where A is the communication-based factor productivity function, fB is the perfirm floor area, and X( fB) is the partial production function.a Following Borukhov and Hochman (1977), O’Hara (1977), and Ogawa and Fujita (1980), we assume that each worker communicates inelastically a

As Borukhov and Hochman (1977) note, single-worker production is not so specific. If the production function is expressed as AΓ(Q, l), where Γ(Q, l) is one-degree homogeneous production function, Q is the total floor space for a firm, and l is the labor size, then we obtain a production function with one unit of labor, given by AX( fB) ¼ AΓ(Q/l, 1), where fB  Q/l.

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with workers in the other firms. Although inelastic communication is less realistic, such inelastic bilateral communication trips can represent agglomeration economies in the sense that firms would concentrate more on saving social communication trip costs. Although we focus only on the right side of the CBD, firms on the right-hand side communicate with firms all over the CBD including those on the left-hand side. For each firm the number of trips to each other firm during a certain period is normalized to one without loss of generality. With the number
of total workers and thus the number of single-worker firms being 2N,
the total communication trips for each firm is 2N  1. In this case,
which implies that A
is constant because 2N
 1Þ  A,
 1 is A ¼ Að2N constant. The profit for a firm at m, denoted π(m), is then given by
ðfB Þ  gðmÞ  w ðmÞ  rB ðmÞfB , m 2 ½0,MB  π ðmÞ ¼ AX

(5.1)

where g(m), w(m), and rB(m) denote the communication trip cost, wage, and floor rent for the firm at location m, respectively. The communication trip cost g(m) is defined as follows. A worker at m communicates with a worker at x at the cost of τ j x  mj, where jx  mj is the distance between the communicating firms and τ is the constant unit-distance cost. The worker communicates with all other workers, so the total communication trip cost borne by a worker at m, say G(m), is given by ð MB GðmÞ ¼ n0B ðxÞ½τjx  mjdx, m 2 ½0,MB  (5.2) MB

where ÐnB0 (x)  ∂ nB/∂ x, which denotes worker density at x,b and 0 nB(m)  m 0 nB (x)dx is the number of total workers working at the firms located between the CBD center and location m. The communication trip cost G(m) is physically determined by the supply-side condition (or transport capacity). Workers must pay at least the supply-side cost, but they could be paying more (e.g., by driving inefficiently slowly or by consuming more fuel for unnecessary acceleration). Therefore the relation between G(m) and the actual payment g(m) is expressed as an inequality conditionc: gðmÞ  GðmÞ, m 2 ½0,MB  b c

(5.3)

An apostrophe on a variable, hereafter, denotes the derivative of the variable with respect to distance. A similar method is used in Kono and Joshi (2012) and Kono and Kawaguchi (2017).

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Land use regulations with business areas

One may think that this inequality is not necessary because a rational worker does not pay more than the supply-side cost. Indeed, this expression is going to hold as an equality due to the rational worker’s optimal behavior. Nevertheless, this inequality expression works later for determining the sign of the shadow price (or Lagrange multiplier) of this constraint.d Differentiating the right-hand side of Eq. (5.2) with respect to m yieldse dGðmÞ  G0 ðmÞ ¼ 2nB ðmÞτ, m 2 ½0,MB  dm and an initial condition is obtained as ð MB n0B ðmÞτmdm Gð0Þ ¼ 2

(5.4)

(5.5)

0

Rent bidding among firms yields π(m) ¼ 0. The bid rent is expressed as  
ðfB Þ  gðmÞ  wðmÞ AX , m 2 ½0,MB  (5.6) rB ðgðmÞ, w Þ ¼ max fB fB The first-order condition yields
ðfB Þ + gðmÞ + w ðmÞ ¼ 0
∂X  AX fB A ∂fB

(5.7)

Thus fB(m) is a function of g(m) and w(m). As Eq. (5.7) shows a firm considers only its private communication costs, whereas its proximity to other firms may allow the other firms to economize on their communication costs. This is what Kanemoto (1990) calls “locational externality.” Locational externalities are technological externalities generated by firms through their location selection. Kanemoto (1990) shows locational externalities existing in the case of bilateral trading between firms, say firm A and firm B, as shown in Fig. 5.2. In this situation, when firm A determines its location, it compares advantages (e.g., a reduction in transportation costs) and disadvantages (e.g., an increase in rents) of the location. For example, a move of firm A toward firm B will benefit not only firm A but also firm B through a reduction in transportation costs for both firms. This is a technological externality. d

Because the multiplier implies the value of the supply-side transport cost, the sign should be negative, implying a natural result that an increase in the transport cost has a negative welfare effect. Thus, when we use an inequality condition, this sign is straightforwardly determined using the Kuhn-Tucker condition. Ð e d MB 0 dm MB nB ðxÞ½τ jx  mjdx ¼ ½nB ðMB Þ + nB ðmÞ  ½nB ðMB Þ  nB ðmÞτ ¼ 2nB ðmÞτ, m 2 [0, MB].

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Fig. 5.2 The mechanism generating locational externalities.

In other words the selection of a location by firm A is not socially optimal due to the existence of locational externalities. From the social viewpoint the combined advantages for both firm A and firm B should be compared with the disadvantages for firm A alone when location of firm A is determined. So the market equilibrium locations are sparser than the optimal locations. Such locational externality can be adjusted by land use regulations. In the real world, there are other agglomeration economy factors such as nonmarket-based communications. However, basically, the locations of firms in the existence of agglomeration economy factors are sparser than the optimal locations. Accordingly, our setting can represent other such agglomeration economy factors basically when we target the effect of agglomeration economies on optimal land use regulations. Finally the wage at each location should compensate for the commuting cost. Let TB(m) denote the commuting cost from MB to m; then, wage w(m) should follow (5.8) wðmÞ ¼ w ðMB Þ + TB ðmÞ

5.2.3 Developers’ behavior Developers supply buildings in Zones B and C under FAR regulation. Let π dj ( j 2 {B, C}) denote developers’ profit from the construction of a building in Zone j, which is given by           π dj mj ¼ Fj mj rj mj  Sj Fj ðmÞ  Rj mj , (5.9) j 2 fB,Cg, mB 2 ½0,MB , mC 2 ½MB ,MC  where Sj(Fj(mj)) denotes the total construction cost of FAR-regulated floor area Fj(mj). Likewise, rj and Rj denote floor rent and land rent, respectively. Note that buildings are constructed on lots of equal size normalized to one. Considering perfectly competitive developers, the zero-profit condition is given by π dj (m) ¼ 0 ( j 2 {B, C}), which yields   Rj ðmÞ ¼ Fj ðmÞrj ðmÞ  Sj Fj ðmÞ , j 2 fB,Cg, mB 2 ½0,MB , mC 2 ½MB ,MC  (5.10)

Land use regulations with business areas

111

5.2.4 Commuting cost—An external factor To consider congestion externalities, we adopt Bureau of Public Roads (BPR) functions, which are frequently used in related papers and used throughout this book. For simplicity the commuting cost is divided into two parts: that within the residential area (i.e., Zone C and Zone H) and that within the business area (i.e., Zone B), denoted by T() and TB(), respectively (see Fig. 5.1). The unit-distance commuting cost within the residential area, T(m), borne by the resident at location m has the following condition:  
 nðmÞ γ N dT ðmÞ 0 , m 2 ½MB ,MH   T ðmÞ  ξ + δ ρðmÞ dm

(5.11)

where n(m) is the total commuter population residing beyond the CBD edge
 nðmÞ is the total number of commuters passup to location x, and thus N ing through location m on the way to the CBD. Likewise, ξ is the free-flow commuting cost factor; δ and γ are positive parameters; and ρ(m) is the road capacity at location m, given exogenously. Eq. (5.11) uses an inequality condition. The left-hand side is the demand-side cost, which is paid by a commuter, while the right-hand side is the supply-side cost, which is determined physically. Similar to the communication trip cost in the business area, commuters could be paying more than the supply-side cost.f Next, TB(m) is defined as   dT B ðmÞ nB ðmÞ γ 0 , m 2 ½0,MB   TB ðmÞ  ξ  δ dm ρðmÞ

(5.12)

where nB(m) denotes number of firms located between the city center and location m. The right-hand side in Eq. (5.12) implies that there is congestion caused by commuting trips. Note that as distance m from the center decreases, the traffic volume, implied by nB(m), also decreases because most commuters would have already reached their firms.g f

The discussion following Eq. (5.3) also applies here; we replace “rational worker” with “rational commuter.” Note that this unnecessary commuting cost is not equal to congestion pricing. The payment for congestion pricing will be returned to the society, but the unnecessary commuting cost is lost by the society. g This is why Eq. (5.12) differs from Eq. (5.11) in formulation. In Eq. (5.11), N
 nðmÞ denotes commuter population residing beyond location m (m 2 [MB, MH]). In Eq. (5.12), nB(m) denotes commuter population working at firms located over [0, x] (m 2 [0, MB]) that would cross location m while commuting to and from the firms.

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5.2.5 Household behavior Each household worker earns wage w per period by working in the CBD. The household’s expenditure comprises commuting, housing, and nonhousing commodity costs. Private cars are the only mode of commuting. For simplicity, we assume a quasilinear utility function for households living in Zone C and H, denoted VC and VH, respectively, which is expressed as Vi(mi) ¼ ui( fi(mi)) + zi(mi),h where i 2 {C, H}, mC 2 [MB, MC], and mH 2 [MC, MH]. Here, uC and uH denote household utility derived from the consumption of floor space fC(m) and LS fH(m), respectively, and zi is the numeraire nonhousing commodity.  The income constraint is expressed as zi + fi ri ðmÞ ¼ w ðmÞ + 1=N
Φ T ðmÞ  TB ðmÞ ði 2 fC,H gÞ, where T(m) is the round-trip commuting cost to the CBD edge borne by a household residing at location m, and Φ is the total profit from the land, that is, total differential land rent. Note that ½1= N
Φ on the right-hand side of Eq. (5.12) implies the assumption of public ownership of land. Using Eq. (5.8) the income constraint can be simplified as
Φ  T ðmÞ (i 2 {C, H}). Then, Φ is expressed as zi + fi ri ðmÞ ¼ w ðMB Þ + ½1= N Φ¼

ð MB 0

½RB ðmÞ  Ra dm +

ð MC MB

½RC ðmÞ  Ra dm +

ð MH

½rH ðmÞ  Ra dm

MC

(5.13) where Ra is the agricultural rent and rH is the land rent in Zone H.

5.2.6 Market clearing conditions and definitions The equality of utilities among locations and market clearing conditions are shown here. First, Eq. (5.14) implies that the household utility is equal everywhere because households are indifferent regarding locations. VC ðmC Þ ¼ VH ðmH Þ  V , mC 2 ½MB ,MC , mH 2 ½MC ,MH 

(5.14)

Population function n(m) and commuting cost T(m) are both continuous at MC but are not necessarily smooth. To clearly distinguish these functions h

Because marginal utility with respect to income is constant over locations under the assumption of a quasi-linear utility, the derivation would be simple. In contrast, Pines and Kono (2012) use a general utility function (i.e., u( f, z)) to obtain the optimal FAR regulation. Even in the latter case, the formula describing optimal regulation will essentially be the same. Moreover, if the change in marginal utility with respect to income does not change much according to the regulations, the same properties regarding optimal regulation are obtained.

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before and after MC, we define population functions as nC(m) and nH(m) and commuting costs as TC(m) and TH(m), where the subscripts represent corresponding zones. Next, Eqs. (5.15), (5.16) express market equilibrium in floor space in Zone B and C, respectively, implying that the total floor space consumed is balanced by total floor space supplied. Next, Eq. (5.17) shows that, in Zone H, the households at m consume fH(m) area of lot; therefore the total area consumed is equal to the unit land area supplied. Floor space in Zone B: fB ðgðmÞ, w ðmÞÞn0B ðmÞ ¼ FB ðmÞ, m 2 ½0,MB 

(5.15)

Floor space in Zone C: fC ðmÞn0C ðmÞ ¼ FC ðmÞ where n0C ðmÞ 

∂nC ðmÞ , m 2 ½MB , MC  ∂m

(5.16)

Lot supply in Zone H: n0H ðmÞfH ðmÞ ¼ 1 where n0H ðmÞ 

∂nH ðmÞ , m 2 ½MC , MH  ∂m

(5.17)

Finally, as shown in Eq. (5.18), because one household member works in the CBD, the total number of workers (left side) is equal to the household pop
(right side). ulation N Labor population: ð MB
n0B ðmÞdm ¼ N (5.18) 0

5.3 Optimal land use regulations 5.3.1 Maximizing social welfare The objective of optimal regulations can be denoted as Eq. (5.19) using the social welfare W composed of total utilities:
V W ¼N

(5.19)

where V is defined in Eq. (5.14). Optimal FAR at each location in Zones B and C, that is, Fj(m), ( j 2 {B, C}; m 2 [0, MC]); optimal LS at each location in Zone H, that is, fH(m), (m 2 [MC, MH]); and three optimal zonal boundaries MB, MC, and MH are

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achieved by maximizing W subject to the market equilibrium Eqs. (5.1)– (5.19). Note that regulation controlling MH is same as urban growth boundary (UGB). The optimal control problem is expressed in the following Lagrangiani:
V L¼N   ð MC FC ðmÞ 1 + κC ðmÞ wðMB Þ + Φ + uC ðfC ðmÞÞ  V  fC ðmÞrC ðmÞ  TC ðmÞ dm
fC ðmÞ N MB   ð MH 1 1 w ðMB Þ + Φ + uH ðfH ðmÞÞ  V  fH ðmÞrH ðmÞ  TH ðmÞ dm κ H ðmÞ +
fH ðmÞ N MC 2 ðM ðM B

C

3

½FB ðmÞrB ðmÞ  SB ðFB Þ  Ra dm  ½FC ðmÞrC ðmÞ  SC ðFC Þ  Ra dm 7 6Φ 6 7 0 MB η6 ð MH 7 4 5 ½rH ðmÞ  Ra dm  MC

ð MB + 0

  ðM ð MB B  ϕB ðmÞ G0 ðmÞ  2nB ðmÞτ dm + θ Gð0Þ  2 n0B ðmÞτmdm + ωðmÞ½ gðmÞ 0

0

      ð MB nB ðmÞ γ FB ðmÞ 0 0  nB ðmÞ dm GðmÞdm + λB ðmÞ TB ðmÞ + ξ + δ μB ðmÞ dm + ρðmÞ fB ðmÞ 0 0       ð MC ð MC
 nC ðmÞ γ FC ðmÞ N 0 0 dm + + λC ðmÞ TC ðmÞ  ξ  δ μC ðmÞ  nC ðmÞ dm ρðmÞ fC ðmÞ MB MB  γ     ð MH ð MH
1 0 ðmÞ  ξ  δ N  nH ðmÞ dm + λH ðmÞ TH μH ðmÞ +  n0H ðmÞ dm ρðmÞ fH ðmÞ MC MC ð MB

+ ςn ½nH ðMC Þ  nC ðMC Þ + ςT ½TH ðMC Þ  TC ðMC Þ

(5.20)

where rB(m)  rB(g(m), w(MB) + TB(m)), fB(m)  fB(g(m), w(MB) + TB(m)), fC(m)  fC(rC(m)), and Sj(Fj)  Sj(Fj(m)) ( j 2 {B, C}). Likewise, κ i(m), η, θ, ω(m), ϕB(m), ψ B(m), λk(m), μk(m), ςn, and ςT (i 2 {C, H}, k 2 {B, C, H}) are shadow prices. Because communication between firms takes place bilaterally, the conÐ MB 0 straint, G(0) ¼ 2 0 nB (m)τmdm from Eq. (5.5), is necessary in Eq. (5.20). i

The Lagrangian considers only the right-hand side of the city. However, communication costs in the right side of the city also depend on the left side. Eqs. (5.4), (5.5) include the left-side firms’ communication costs. So, we can first consider dividing them by two for inclusion in the Lagrangian. But the city is symmetrical, so the left side’s symmetrical change should be considered, which can be done by multiplying the equations by two because the communication trips are inelastic. In conclusion, we can consider Eqs. (5.4), (5.5) as they are. This Lagrangian can also be replaced by Hamiltonian, which generates the same first-order conditions.

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As Tauchen and Witte (1984) and Fujita and Thisse (2013) show, the total communication cost in the CBD is expressed by double integrals, so the Lagrangian should consider this constraint specifically in addition to the 0 constraint of G (m). This necessity is also intuitive because the boundary condition TB(0) depends on the endogenous labor distribution. The other boundary conditions are as follows. For Zone B, nB(0) ¼ 0, and
. For Zone C, nC(MB) ¼ 0. For Zone H, nH ðMH Þ ¼ N.
nB ðMB Þ ¼ N Regulations affect social welfare W through changes in (i) agglomeration economies arising from communication in the business zone due to the distribution of firms, (ii) deadweight loss in the floor space and LS market due to the distribution of residences, and (iii) commuting costs. The first-order conditions presented in Technical Appendix 5(1) show the relationships among distortions caused by the regulations, agglomeration economies, and congestion. We interpret these relationships in Technical Appendix 5(2).

5.3.2 Optimal FAR and LS regulations This section obtains important properties of the optimal regulations on FAR and LS. First, we consider shadow prices μk(m)(k ¼ B, C, H), which directly show how and by how much the FAR or lot regulations should be imposed. Following Lemma 5.1, which is presented in Technical Appendix 5(3), the motion of μk(m) is illustrated in Fig. 5.3. As shown in Fig. 5.2, two cases arise in Zone B, depending on whether communication costs are greater or smaller than traffic congestion costs: more specifically, whether 2τ + δγ(m)γ1/ρ(m)γ < 0 (Case I) or 2τ + δγ(m)γ1/ ρ(m)γ > 0 (Case II), where m 2 [0, MB]. Case I implies that either communication cost τ or the transportation capacity ρ(m) is sufficiently large, resulting into low traffic congestion, whereas Case II arises when either τ or ρ(m) is significantly small, resulting into severe congestion.j As explained in Technical Appendix 5(3), the sign of μj(mj)( j 2 {B, C},   ∂Sj ðFj ðmj ÞÞ mB 2 [0, MB], mC 2 [MB, MC]) is the reverse of the sign of rj mj  ∂F m , jð jÞ which denotes distortion in the floor space market caused by FAR   ∂Sj ðFj ðmj ÞÞ regulation as shown in Fig. 5.4A. Note that rj mj  ∂F m ¼ 0 if jð jÞ j

Case I is likely to emerge near the city center because the communication cost for business people is generally high, whereas the number of commuters near the city center would be close to zero. Moreover, in most developed cities with high wages, Case (I) is likely to hold in rather broad areas.

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Fig. 5.3 The motion of shadow prices μk(m). (Source: Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598.)

Fig. 5.4 Deadweight loss due to (A) FAR regulation (left) and (B) LS regulation (right). Note: Superscripts ‘†’ and ‘∗’, respectively, refer to the market equilibrium and optimal cases. (Source: Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598.)

the FAR is unregulated and determined by the market. FAR greater (resp.   ∂Sj ðFj ðmj ÞÞ smaller) than the market FAR implies rj mj  ∂F m < 0 (resp. jð jÞ   ∂Sj ðFj ðmj ÞÞ rj mj  ∂F m > 0). Likewise the sign of μH(m) (m 2 [MC, MH]) is the jð jÞ reverse of the sign of ∂uH∂fðHfðHmðÞmÞÞ  rH ðmÞ, which denotes distortion in the lot supply market due to LS regulation as shown in Fig. 5.4B. Following Lemma 5.1 presented in Technical Appendix 5(3) and Fig. 5.3, we achieve Proposition 5.1 (see Technical Appendix 5.3 for further explanation).

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Proposition 5.1 (Optimal FAR and LS regulation in the presence of optimal zonal boundaries) (1) Business zone: nB ðmÞγ1 < 0 at any m 2 [0, MB]: The optimal firm • Case I implying 2τ + δγ ρðmÞγ density is higher (resp. lower) in the more central (resp. peripheral) locations relative to the market firm density, requiring minimum (resp. maximum) FAR regulation. nB ðmÞγ1 > 0 at any m 2 [0, MB]: The optimal • Case II implying 2τ + δγ ρðmÞγ firm density is lower (resp. higher) in the more central (resp. peripheral) locations relative to the market firm density, requiring maximum (resp. minimum) FAR regulation. (2) Condominium zone: The optimal population density is higher (resp. lower) in the more central (resp. peripheral) locations relative to the market population density, requiring minimum (resp. maximum) FAR regulation. (3) Lot housing zone: The optimal population density is lower relative to the market population density across the zone, requiring minimum LS regulation. It is important to recall that Proposition 5.1 holds when optimal zonal boundaries are simultaneously imposed, which are presented later in Proposition 5.2. Results of both propositions are presented in Fig. 5.5.

Fig. 5.5 Optimal regulations. Note: In Zone B, FAR regulations in bold letters hold in Case I, and those in parenthesis hold in Case II. Regarding the area regulation a decrease in the zone area hold in Case I, and an increase in the zone area holds in Case II. (Source: Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598.)

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The implication of Proposition 5.1 is explained as follows. First the combination of maximum and minimum FAR regulations in the business zone results in an efficient labor distribution to optimize the total welfare composed of deadweight loss in the floor market and agglomeration benefits in the CBD. To achieve a certain labor distribution, if only conventional “maximum FAR regulation” is imposed, the total deadweight loss arising from FAR regulation would be greater than that achieved from the combination of maximum and minimum FAR regulations, as also explained in Chapter 2. Whether the minimum or maximum FAR regulation should be enforced in the central (or peripheral) locations in the business zone depends on the trade-off between agglomeration economies and traffic congestion costs. Next, in the condominium zone, the optimal policy of minimum FAR regulation at the central locations and maximum FAR regulation at the peripheral locations would shift population in favor of more central locations and thereby reduce traffic congestion caused by commuters from distant locations. See Chapter 2 for a detailed discussion. In the lot housing zone, however, the optimal regulation requires enforcement of minimum LS regulation across the zone. The optimal regulation addresses deadweight losses (see Fig. 5.3B). Given that the city also has UGB regulation (explained in the next section) that prevents sprawl, minimum LS regulation reduces supply of housing lots and thereby decreases population in the suburb. This would ultimately reduce traffic congestion across the condominium zone and lot housing zone. It is important to note what minimum LS regulation achieves in our model and how. Although minimum allowable LS in the suburb is prevalent in most countries, especially in the United States, our interpretation of the necessity of minimum LS regulation is different. The objective of such regulation as being practiced is to promote low-density development, but such a policy contributes to urban sprawl (Pasha, 1996) and thus increases congestion costs. That is why Pines and Sadka (1985) and Wheaton (1998) suggest maximum LS regulation in the central area. But our model is able to alleviate congestion externality through minimum LS regulation and without sprawl because it allows options for additional higher density in the central area through minimum FAR regulation. Such result is achieved because of simultaneous consideration of multiple regulations. Finally, Lemma 5.1 (in Technical Appendix 5(3)) shows that the optimal conditions of the regulations are composed of observable variables only.

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This is useful for policy evaluators to check whether the current regulations are optimal or not.

5.3.3 Optimal zonal boundaries We now examine how a change in zonal boundaries affects the social welfare under regulated FAR and LS. From Lemma 5.2 in Technical Appendix 5(4), which summarizes the related first-order conditions of the welfare maximization, we achieve Proposition 5.2. Proposition 5.2 (Optimal zonal boundaries in the presence of optimal FAR regulation and LS regulation) (1) Business zone: Recalling the cases stated in Proposition 5.1(1), the optimal zone area is smaller than the market equilibrium zone area in Case I, whereas the result is ambiguous in Case II. (2) Condominium zone: The zone boundary should be shrunk relative to the market boundary. (3) Lot housing zone: The zone boundary, which also acts as the city boundary, should be shrunk relative to the market boundary. The implication of Proposition 5.2(1) is explained as follows. Whether the business zone should be more compact or larger than the market zone area depends on the trade-off between agglomeration economies and traffic congestion costs. An enlargement of the business zone decreases agglomeration economies resulting in a decrease in welfare. Simultaneously, it implies that the business area becomes closer for all residents. Accordingly, congested commuting distances decrease, resulting in an increase in welfare. The net effect is clear in Case I that an enlargement of the business zone would certainly reduce welfare, but the net effect is ambiguous in Case II. Proposition 5.2(2) implies that contraction of Zone C decreases the deadweight loss at the outer edge of Zone C by   ∂SC ðFC ðMC ÞÞ rC ðMC Þ  FC ðMC Þ but increases the same at the inner edge ∂FC ðMC Þ   ∂uH ðfH ðMC ÞÞ of Zone H by  rH ðMC Þ . The first exceeds the latter in abso∂fH ðMC Þ lute value. This can also be explained using the equation in Lemma 5.2(2) (see Technical Appendix 5(4)). Proposition 5.2(3) implies that a marginal expansion of Zone H decreases social welfare. Expanding Zone H by a unit area means supplying additional lots, thereby increasing the population at the city edge by 1/fH(MH).

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Because the city is closed, such expansion results in relocation of some households from the outer edge of Zone C to the edge of the city, noting that fH(m) (m2 [MC, MH]) is fixed by LS regulation. Such relocation increases the commuting cost in Zone H. The relocation decreases the deadweight loss   ∂SC ðFC ðMC ÞÞ fC ðMC Þ at the outer edge of Zone C by rC ðMC Þ  , where ∂FC ðMC Þ fH ðMH Þ fC(MC)/fH(MH) implies decrease in the building size at the outer edge of Zone C because of the relocation of 1/fH(MH) number of residents. This can also be explained using the equation in Lemma 5.2(3). Finally, we explore whether the total area of the city decreases or not. Although the areas of Zone C and Zone H should be shrunk relative to the market boundary, the optimal size of Zone B is ambiguous in Case II. When the congestion costs are large enough compared with agglomeration benefits, a larger Zone B is welfare improving because it reduces traffic congestion by reducing the distance between Zone B and the residential area. In such case an optimal city can be larger than a market city if +ΔMB  ΔMC  ΔMH > 0, where ΔMk (k 2 {B, C, H}) denotes optimal change (“+” if expansion and “” if shrinkage) in the corresponding zonal boundary relative to the market equilibrium. Lemma 5.2 in Technical Appendix 5 provides optimal conditions for zone boundaries, which are composed of only observable variables. This is useful for policy evaluators to check whether the current regulations are optimal or not.

5.4 Numerical simulation 5.4.1 The setup This subsection presents some numerical examples to demonstrate how social welfare in our model changes with FAR and/or zonal boundaries. This helps understand the property of optimal land use regulation, theoretically achieved in this paper, in a quantitative manner. However, the numerical simulation does not completely trace our propositions. The base simulation model is a market equilibrium model, not our maximization problem (see Section 5.3.1).k Our propositions show what properties the optimal regulations possess, compared with the market equilibrium, while k

Programs solving the market equilibrium are composed of simultaneous equations only. In contrast, programs solving the optimal solution maximize the social welfare subject to multiple equations (conditions or constraints). In other words the second-best optimum is a constrained nonlinear optimization subject to numerous constraints multiplied by the number of blocks that our three-zone model city is divided into. We were able to numerically solve the market equilibrium only.

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our numerical simulations only show how much a certain level of difference in the level of regulation from the market equilibrium changes the welfare. Regarding the main parameters such as income and the functional forms such as the utility function and the traffic congestion function, we follow the numerical setup of Kono et al. (2012). The parameters are set for the solution to represent a real situation as closely as possible, using real data or the established parameters. We briefly explain the reasons for setting the parameters in the succeeding text. However, it was impossible to calibrate all the parameters using real data. So, some parameters are set arbitrarily. The three zones in the city are divided into small blocks of equal size ε. The blocks are indexed by i in the residential area composed of Zones C and H, where i ¼ 1 is the residential block adjacent to zone B. In Zone B, i ¼ 1 is the city center block. The inner edge of block i is given by mi ¼ 1 + ε[i  1], where mi now represents the distance variable for the corresponding block. We consider a hypothetical city in Japan with 2 million households, that
¼ 200,000. is, N Zones C and H We first solve Zones C and H. The subutility function uk ( fk (mk )), 00 k 2 {C, H}, is specified as uC( fC(m)) ¼ α ln fC and u( fH) ¼ α ln(1/φ  fH), where α ¼ 0.12y, representing 12% of the household income y, which is
Φ, and φ is a positive constant; we adopt defined as y  w ðMB Þ + ½1=N φ ¼ 0.0015. 00

00

00

The maximization of household utility yields rC ¼ α/fC and rH ¼ α/fH, which leads to the following demand functions: fC ¼ exp(kC) and fH ¼ φ exp(κH), where kC  [VC  y + TC + α]/α and κ H  [VH  y + TH + α]/α. The floor production function is specified as FC ¼ ϖ[S(FC)]β, where ϖ ¼ exp(5.87907) and β ¼ 0.749734, both of which are estimated from 112 samples of Japanese buildings; S(FC) is measured in Japanese yen (JPY)/year, whereas FC represents FAR. When FAR is not regulated, rC  ∂SC(FC)/∂FC ¼ 0, which yields FC ¼ θ[θβrC]β/[1β]. Household density is given by PC ¼ FC/fC and PH ¼ 1/fH in the respective zone. Land rent in Zone C is obtained from the zero-profit condition as RC ¼ FCrC  SC(FC). Assuming 20% allocation of land for nonresidential purpose denoted Ð MC by ρ ¼ 0.2, the differential land Ð MH rent is given by ΦC ¼ [1  ρ] MB [RC(m)  Ra]dm and ΦH ¼ [1  ρ] MC [rH(m)  Ra]dm in the respective zone, where Ra denotes agricultural land rent, equal to 20,000,000 JPY per square km.

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Both income y and household utility V are set exogenously, whereas the total commuting cost from block i, Ti, is derived as follows: T1 ¼ 0; 00 Ti+1 ¼ Ti + ε[Tunit k i + ptoll k i], where Tunit k i, k 2 {C, H}, implies unit cost of commuting and ptoll k i denotes congestion toll. We assume 225 round trips per year, and we use a free-flow commuting cost of 17.6 JPY per km, which is estimated by the Ministry of Land, Infrastructure, Transport, and Tourism (MLIT, 2010). Tunit k i is specified as a BPR function using the Japan Society of Civil Engineers (JSCE, 2003) parameters relevant to urban roads in Japan: " "  2:82 ## 56:78 n^i Tunit k00 i ¼ 2  225  17:6 + 1 + 0:48 Υk00 30 00

00

00

00

00

where ΥC, ΥH are road capacity in respective zones given as ΥC ¼ 152, 240 and ΥH ¼ 76, 120; n^i denotes population beyond block i given by P∗ n^i ¼ ρ ik¼i εϑk , and i∗ denotes the outermost block such that n^i∗ ¼ 0. The congestion toll is calculated as ptoll k00 i ¼ n^i ∂Tk00 i =∂^ ni .
and is carried The iterative process starts at i ¼ 1 with T1 ¼ 0 and n^1 ¼ N out conditional on the value of V that should satisfy the equilibrium conditions. In the laissez-faire and toll-regime (or Pigouvian tax) cases, the iteration stops when i reaches a value i∗ such that n^i  0 and n^i + 1 < 0, indicating that
is just accommodated within mi∗ ¼ MH. The the household population N increment in n^i is simply given by n^i + 1 ¼ n^i  ε½1  ρPi . We then check the equilibrium condition rHi∗ ¼ Ra. The iteration is repeated by adjusting V until the equilibrium condition is achieved within a reasonable degree of accuracy. Note that the boundary of Zone C is defined by the condition: RCi rHi and RC(i+1) < rH(i+1). Also recall that income y remains assumed. y is adjusted while solving Zone B as explained next. Zone B Zone B is also divided into several blocks of size ε, which can be adjusted for more accuracy as explained later. Different from the setting in the residential zones, the total number of blocks, say j, is assumed beforehand (this implies that MB is assumed in advance). Then, i ¼ j, j  1, … , 2, 1 denotes the successive blocks. At the outermost block within Zone B where i ¼ j,
and calculating backward, nB(i1) ¼ nBi  [1  ρ]εPBi where nBði¼jÞ ¼ N, PBi denotes firm density given by PB ¼ FB/fB. Note that nBi denotes commuter population working at firms located between the city center and

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block i. As distance from the center to the block decreases, the number of commuting trips, implied by nBi, also decreases because most commuters would have already reached their firms. The way commuting cost is calculated is also different from the setup in the residential zones. Recall that the commuting cost TB is measured from the CBD edge, MB. So, TB(i¼j) ¼ 0, and TB(i1) ¼ TBi + ε[Tunit Bi + ptoll Bi], where Tunit Bi is given by "

"  2:82 ## 56:78 nB Tunit Bi ¼ 2  225  17:6 + 1 + 0:48 ΥB 30 where ΥB ¼ 318,560. The congestion toll is calculated as ptoll Bi ¼ ni ∂ Tunit Bi/∂ nBi. Besides MB and income y, wage w(MB) is also assumed beforehand. Adjustments are made to these variables unless the following three conditions are met simultaneously within a reasonable degree of accuracy: : (i) RBj ¼ RCj (adjust ε for more accuracy); : (ii) WB ¼y  NΦ , where Φ ¼ ΦB + ΦC + ΦH; ΦB is total differential land rent in Zone B, and [ΦC + ΦH] is derived from the calculation related to the residential zones; and :

(iii) y¼ N1 +

hÐ

 Ð MB MB 0 0 nB ðmÞ AX ð fB Þ  gðmÞ  TB ðmÞ  0 SðFB ðmÞÞdm  Ra MB

i

1 ½ΦC + ΦH  N

(adjusting y would start a new iteration for the residential zones as well). The last condition is derived from the resource constraint that is obtained by combining equations of rB (from maximization of utility) and RB (from zeroprofit condition) and integrating the result over Zone B. The components of the condition based on the resource constraint are derived next (index i is suppressed where there is no ambiguity). 0:05 (i) Agglomeration: A ¼ χ ½2N  1 , where χ is a positive scale factor. We use χ ¼ 1,500,000. (ii) Communication: G(MB) ¼ 2τN MB, and G_ ðmÞ ¼ 2τnB ðmÞ > 0ðm > 0Þ: Accordingly, Gi+1 ¼ Gi + 2τnBiε, and by backward calculation, Gi ¼ Gi+1  2τnBiε. In fact, MB is assumed beforehand to facilitate this backward calculation. Allowing round trips for communication, τ ¼ 2  17.6 (JPY/km).

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(iii) Production: Assuming a one-degree homogeneous Cobb-Douglas function, AΓ ðFB l Þ ¼ AFB αB l 1αB , where l denotes total number of workers, and αB ¼ 0.4, a constant. Then, applying this to our  αB FB Γ ðFB l Þ one-labor firms, X ðfB Þ ¼ l ¼ ¼ ½fB αB . This also yields l ∂X/∂ fB ¼ αB[ fB]αB1. (iv) Floor rent and consumption: Solving zero-profit condition, we
½∂X ðfB Þ=∂fB  ¼ αB ½fB αB 1 , and fB ¼ ½½gðmÞ + wðmÞ= get rB ¼ A
½1  αB 1=αB . ½A (v) Floor production function: FB ¼ ϖ[S(FB)]β, where ϖ ¼ exp(5.87907) and β ¼ 0.749734; S(FB) is measured in JPY/year, whereas FB represents FAR. When FAR is not regulated, rB  ∂SB(FB)/∂FB ¼ 0, which yields FC ¼ θ[θβrB]β/[1β]. (vi) Land rent: RB ¼ FBrB  SB(FB).

5.4.2 Numerical results For the residential zones, we adopt ε ¼ 4 km, whereas the block size in Zone B is kept flexible for reasons explained earlier. We present a total of 10 examples including the laissez-faire case, Pigouvian tax regime, and examples with exogenously set FARs and/or UGB. The Pigouvian tax regime internalizes two kinds of externalities addressed in our paper, namely, traffic congestion and “locational
are fixed because externality.” Agglomeration economies arising from A the population size is exogenously fixed. Regarding the “locational externality” whereby a firm considers only its private communication costs, the social cost is calculated by doubling the communication costs. The welfare achievable by this regime is the first best. In Example 1, residential zone size, (MH  MB), is set at 44 km, representing a UGB case. The Example 2 (20, 10, 15, 30, 2) refers to the case where, relative to the laissez-faire case, the FARs at the central locations of Zone B are increased by 20% whereas the FARs at the farther locations of Zone B are decreased by 10%; the FARs at the central locations of Zone C are increased by 15% whereas those at the farther locations are decreased by 30%; and the LS across Zone H is increased by 2%. The same setup is repeated in another example (Example 3) with MH  MB set at 44 km, representing FAR with UGB policy. The results under different cases are summarized in Table 5.1. The social
V. welfare is expressed in terms of household utility given W ¼ N

Zonal boundary (km)

Policy regime

MB

MC 2 MB

MH 2 MB

MH 2 MC

Social welfare per household (V)

Welfare gain (%)

Laissez-faire Pigouvian Ex. 1 residential zone size (MH  MB) set at 44 km Ex. 2 (20, 10, 15, 30, 2) Ex. 3. (20, 10, 15, 30, 2) with residential zone size (MH  MB) set at 44 km

0.354 0.176 0.355

38 40 38

50 44 44

12 4 6

2,606,090 2,709,630 2,660,970

100.00 53.00

0.396 0.390

34 34

50 44

16 10

2,662,850 2,667,450

54.82 59.26

Land use regulations with business areas

Table 5.1 Zonal boundaries and welfare change.

Note: Ex.: Example.

125

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Traffic congestion and land use regulations

Examples show that compared with a single regulation, a set of appropriately chosen regulations leads to higher welfare gain. Densification at the central locations of Zones B and C relative to the laissez-faire case is welfare inducing. Moreover, compactness of the city (or zones) leads to increase in welfare. Earlier in this chapter, we have shown that optimal regulations compose of minimum FAR regulation at the more central locations and maximum FAR regulation at the farther locations in Zone C, followed by minimum LS regulation in Zone H. Figs. 5.6–5.8 illustrate these results (compare Firm density at Zone B

Number of firms per sq. km

9,000,000

Pigouvian tax regime

8,000,000 Laissez faire 7,000,000 Example 3 (20, -10, 15, -30, 2) with residential zone size set at 44 km

6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 0.00

0.01

0.02

0.03

0.04

0.05

Distance from the city center (km)

Fig. 5.6 First density at the central locations of Zone B under selected policy regimes.

Household density at central locations in Zone C

Number of households per sq. km

29,000

Pigouvian tax regime 27,000

Laissez faire

25,000

Example 3 (20, -10, 15, -30, 2) with residential zone size set at 44 km

23,000 21,000 19,000 17,000 15,000 0

1

2 3 Distance from Zone B edge (km)

4

Fig. 5.7 Household density in Zone C under selected policy regimes.

5

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Land use regulations with business areas

Household density in Zone H No. of households per sq. km

100

Pigouvian tax regime

95

Laissez faire

90

Example 3 (20, -10, 15, -30, 2) with residential zone size set at 44 km

85 80 75 70 65 60 34

36

38

40

42

44

46

48

50

52

Distance from Zone B edge (km)

Fig. 5.8 Household density in Zone H under selected policy regimes.

laissez-faire case with Pigouvian tax regime). Example 3, with the highest relative welfare gain, is included in Figs. 5.6–5.8 for comparison with Pigouvian tax regime. Note that we have found ambiguity in terms of the effect of FAR regulation and boundary regulations on social welfare in the case of Zone B. So regarding Zone B, our numerical examples should be interpreted accordingly.

5.5 Conclusion In this chapter, we have simultaneously optimized multiple regulations—on building size, LS, and zonal boundaries—in a monocentric city with office buildings, condominiums, and single-family dwellings in that order in distinct but adjoining districts. We demonstrate the necessity of both minimum and maximum FAR regulation in both business zone and condominium zone. Although Chapter 2 shows this result in residential zone, this chapter considers office buildings and residential buildings simultaneously. Regarding the optimal FAR regulation in the business zone, if agglomeration economies are relatively dominant over traffic congestion costs (such as in developed cities), it is more likely that the optimal policy requires enforcement of minimum FAR regulation at the more central locations and maximum FAR regulation at more peripheral locations. The same

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Traffic congestion and land use regulations

applies nonambiguously in the case of the condominium zone, followed by minimum LS regulation in the suburb. In the presence of optimal FAR regulation and LS regulation, we also explore optimal zonal boundaries. The optimal size of the business zone is ambiguous depending on the trade-off between agglomeration economies and traffic congestion costs, but the outer boundaries of the condominium and housing lot zones should be shrunk. The objective of minimum LS regulation as practiced is to promote lowdensity development, but such a policy contributes to urban sprawl and thus increases congestion costs. However, our model is able to alleviate congestion externality through minimum LS regulation and without sprawl because it allows options for additional higher density in the central area through minimum FAR regulation. Such result is achieved because of simultaneous consideration of multiple regulations.

Technical Appendix 5 Traditionally, urban models have been solved by applying optimal control theory to the continuous area (typically, residential area). The model in this chapter can also be solved with optimal control theory. But one important distinct feature from the traditional urban models is that our urban model is composed of three distinct zones (i.e., business, condominium, and detached housing zones). Hence, we have to combine the three distinct zones, each of which has continuous area, using the boundary conditions between the adjacent zones. (1) Lagrangian and Pontryagin’s maximum principle To obtain the first-order conditions, we integrate the Lagrangian in Eq. (5.20) by parts. After that, differentiating the Lagrangian with regard to the policy variables and the endogenous variables, we obtain the firstorder conditions (A5.1)–(A5.29). These expressions use the following relations: ∂rB/∂ g ¼  1/fB and ∂ rB/∂ w ¼  1/fB. Also note that an apostrophe denotes derivative with respect to distance from the center:   ∂L ∂SB ðFB ðmÞÞ 1 ¼ 0 : η r B ð mÞ  + μB ðmÞ ¼ 0, m 2 ½0,MB  ∂FB ðmÞ ∂FB ðmÞ fB ðmÞ (A5.1)

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Land use regulations with business areas

  ∂L ∂SC ðFC ðmÞÞ 1 ¼ 0 : η rC ðmÞ  + μ C ð mÞ ¼ 0, m 2 ½MB ,MC  ∂FC ðmÞ ∂FC ðmÞ f C ð mÞ (A5.2) ∂L ¼0: ∂fH ðmÞ   1 ∂uH ðfH ðmÞÞ 1 ¼ 0, m 2 ½MC ,MH   rH ðmÞ  μH ðmÞ κ H ðmÞ fH ðmÞ ∂fH ðmÞ fH ðmÞ2 (A5.3) where, for deriving Eqs. (A5.2), (A5.3), Eqs. (5.16), (5.17) are used respectively, in addition to Eq. (5.14): ∂L ¼0: ∂rC ðmÞ κC ðmÞFC + ηFC ðmÞ  μC ðmÞ

FC ðmÞ df C ðmÞ ¼ 0, m 2 ½MB ,MC  (A5.4) fC ðmÞ2 dr C ðmÞ

∂L ¼ 0 : κ H ðmÞ + η ¼ 0, m 2 ½MC ,MH  ∂rH ðmÞ ∂L
 ¼0:N ∂V

ð MC

FC ðmÞ dm  fC ðmÞ

ð MH

(A5.5)

1 dm ¼ 0 f H ð mÞ

(A5.6)

ð MC ð MH ∂L FC ðmÞ 1 κC ðmÞ κ H ðmÞ ¼0: dm + dm ð m Þ ð ∂w ðMB Þ f f C H mÞ MB MC ð MB ð MB FB ðmÞ FB ðmÞ ∂fB ðmÞ dm + dm ¼ 0 μB ðmÞ η fB ðmÞ2 ∂w ðMB Þ 0 fB ðmÞ 0

(A5.7)

MB

κ C ð mÞ

MC

κ H ðmÞ

where for the last term, note that ∂ w(m)/∂ w(MB) ¼ 1 by virtue of Eq. (5.8): ∂L FB ðmÞ FB ðmÞ ∂fB ðmÞ (A5.8) ¼ 0 : η  μB ðmÞ  λ0B ðmÞ ¼ 0 2 ð m Þ ∂TB ðmÞ fB ðmÞ ∂T fB ðmÞ B ∂L ¼ 0 : λB ð0Þ ¼ 0 (A5.9) ∂tB ð0Þ ð  ð MH ∂L 1 MC FC ðmÞ 1 dm + dm  η ¼ 0 (A5.10) κ ðmÞ κ H ð mÞ ¼0:
MB C ∂Φ N fC ðmÞ fH ðmÞ MC

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Traffic congestion and land use regulations

∂L FB ðmÞ FB ðmÞ ∂fB ðmÞ + ωðmÞ  μB ðmÞ ¼ 0 : η ¼ 0, m 2 ½0,MB  ∂gðmÞ fB ðmÞ fB ðmÞ2 ∂gðmÞ (A5.11) ∂L ¼ 0 : ϕ0B ðmÞ  ωðmÞ ¼ 0, m 2 ½0,MB  ∂GðmÞ ∂L ¼ 0 : ϕB ð0Þ + θ ¼ 0 ∂Gð0Þ

(A5.12) (A5.13)

where for deriving this condition, Eq. (A5.12) is used noting that the latter holds when m ! 0: ∂L ¼ 0 : ϕB ðMB Þ ¼ 0 ∂GðMB Þ

(A5.14)

∂L FC ðmÞ 0 ¼ 0 : κ C ðmÞ  λC ðmÞ ¼ 0, m 2 ½MB ,MC  ∂TC ðmÞ fC ðmÞ

(A5.15)

∂L 1 ¼ 0 : κ H ðmÞ  λ0 ðmÞ ¼ 0, m 2 ½MC ,MH  ∂TH ðmÞ fH ðmÞ H

(A5.16)

∂L ¼ 0 : λC ðMC Þ  ςT ¼ 0 ∂TC ðMC Þ

(A5.17)

∂L ∂L ¼ 0 : λH ðMC Þ + ςt ¼ 0 and ¼ 0 : λH ðMH Þ ¼ 0 ∂TH ðMC Þ ∂TH ðMH Þ (A5.18) ∂L ½nB ðmÞγ1 ¼ 0 : 2τ½θ  ϕB ðmÞ + μ0B ðmÞ + λB ðmÞδγ ¼ 0, m 2 ½0,MB  ρðmÞγ ∂nB ðmÞ (A5.19) ∂L ½N  nC ðmÞγ1 + μ0C ðmÞ ¼ 0, m 2 ½MB ,MC  ¼ 0 : λC ðmÞδγ ρðmÞγ ∂nC ðmÞ (A5.20) ∂L ½N  nðmÞγ1 + μ0H ðmÞ ¼ 0, m 2 ½MC ,MH  ¼ 0 : λH ðmÞδγ ρðmÞγ ∂nH ðmÞ (A5.21) ∂L ¼ 0 : μH ðMC Þ + ςn ¼ 0 (A5.22) ∂nH ðMC Þ

Land use regulations with business areas

∂L ¼ 0 : μC ðMC Þ  ςn ¼ 0 ∂nC ðMC Þ

131

(A5.23)

  γ 
∂L FB ðMB Þ N  λB ðMB Þ ξ + δ ¼ 0 : ½FB rB ðMB Þ  SB ðFB Þ  Ra  + μB ðMB Þ ∂MB fB ðMB Þ ρðMB Þ   γ 
FC ðMB Þ N λC ðMB Þ ξ+δ  ½FC rC ðMB ÞSC ðFC Þ Ra μC ðMB Þ ¼0 fC ðMB Þ ρðMB Þ (A5.24)

To derive (A5.24), we use ϕB(MB) ¼ 0 from Eq. (A5.14) and η ¼ 1 from Eqs. (A5.6), (A5.10): ∂L ¼ 0 : ½FC ðMC ÞrC  SC ðFC Þ  Ra   ½rH ðMC Þ  Ra  ∂MC FC ðMC Þ 1 + μC ðMC Þ  μH ðMC Þ ¼0 fC ðMC Þ fH ðMC Þ

(A5.25)

To derive (A5.25), we used λC(MC) ¼ λH(MC) by virtue of Eqs. (A5.17), (A5.18): ∂L 1 ¼0 ¼ 0 : ½rH  Ra  + μH ðMH Þ ∂MH fH ðMH Þ ∂L ∂L  0,ωðmÞ  0, ωðmÞ ¼ 0, m 2 ½0,MB  ∂ωðmÞ ∂ωðmÞ λB ðmÞ

∂L ∂L ¼ 0, λB ðmÞ  0, 0 ∂λB ðmÞ ∂λB ðmÞ

(A5.26) (A5.27) (A5.28)

noting the sign of the corresponding constraint in Eq. (5.13): λi ðmÞ

∂L ∂L ¼ 0, λi ðmÞ  0,  0, i 2 fC,Hg ∂λi ðmÞ ∂λi ðmÞ

(A5.29)

The first-order conditions with respect to shadow prices, except for ω(m), λB(m), λC(m), and λH(m), are suppressed because they are obvious. (2) Interpretation of optimal FAR regulation and LS regulation The following discussion interprets optimal conditions of FAR regulations and LS regulation by zone. These conditions can be explained with Harberger’s welfare formula, which measures the welfare change in a distorted economy (see Harberger, 1971).

132

Traffic congestion and land use regulations

a. FAR regulation in the business zone First, solving Eqs. (A5.1), (A5.8), (A5.9) and using η ¼ 1 from Eq. (A5.6) and

 ðm  ∂SB ðFB ðxÞÞ FB ðxÞ ∂fB Eq. (A5.10) yields λB ðmÞ ¼ nB ðmÞ + rB ðxÞ  dx, ∂FB ðxÞ fB ðxÞ ∂TB 0 where λB(m) is the shadow price for the commuting cost within the business zone. Next, substituting Eq. (A5.12) into Eq. (A5.11) to cancel out ω(m) and then integrating the result and using boundary condition ϕB(MB) ¼ 0 from ð MB FB ðxÞ ∂fB Eq. (A5.14) yield ϕB ðmÞ ¼ ½N
 nB ðmÞ  μB ðxÞ dx: From fB ðxÞ2 ∂g m Eq. (A5.13), θ ¼ ϕB(0), whereas ϕB(0) is obtained from the aforementioned ð MB FB ðxÞ ∂fB equation involving  ϕB(m), thereby yielding θ ¼ N
+ μB ðxÞ dx: fB ðxÞ2 ∂g 0 Substituting these three equations regarding λB(m),  ϕB(m), and θ as well as

Eq. (A5.1) into Eq. (A5.19) yields ΩB  μ0B ðmÞdm

2 3  6  ðm  ∂SB ðFB ðxÞÞ FB ðxÞ ∂fB ½nB ðmÞγ1 7 6 7 rB ðxÞ dx 62τ + δγ ¼ nB ðmÞ 7dm, 4|{z} ∂FB ðxÞ fB ðxÞ ∂g ρðmÞγ 5 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 0

>0

m 2 ð0,MB  (A5.30)

Note that μ0B ðmÞdm ¼ μB ðm + dmÞ  μB ðmÞ, where μB(m) is the shadow price for the floor area containing FB/fB number of workers at location m. Therefore Eq. (A5.30) implies a change in the social welfare from the relocation of one worker from location m to location m + dm due to a change in the FAR regulation. Recall that we have supposed a symmetrical distribution of firms with respect to the center and have focused only on the right side of the city. In fact the relocation should take place symmetrically on both sides. The combined effect of the relocation of one worker on communication costs and traffic congestion is 2τ + δγ[nB(m)]γ1/[ρ(m)]γ in Eq. (A5.30). The first term [2τ] is related to the increase in the communication cost for firms located in [0, MB]. The relocation of one worker from m to m + dm increases the communication cost of the firms located in [0, m] but decreases the communication cost of the firms located in [m, MB]. Likewise the symmetrical left-side relocation of one worker from m to mdm increases the communication cost of the firms located in both [0, m] and [m, MB]. Summing up the total communication cost for all firms located

133

Land use regulations with business areas

in [0, m] increases by 2nB(m)τ and that for all the firms over [m, MB] increases
 nB ðmÞ½τ  τ ¼ 0. As a result the change in the total communication by ½N cost for the firms in the right side of the city is 2nB(m)τdm. Furthermore the change in the communication cost for the firms located at x 2 [0, m] affects the respective floor market deadweight loss caused by FAR regulation, which is expressed as  ðm  ∂SB ðFB ðxÞÞ FB ðxÞ ∂fB ðxÞ  rB ðxÞ  dx  ½2τ, noting that expresses ∂FB ðxÞ fB ðxÞ ∂gðxÞ 0 the change in the deadweight loss in the floor market (see Fig. 5.3). Next the term δγ[nB(m)]γ1/ρ(m)γ in Eq. (A5.30) is the saving in traffic congestion cost for the firms located over [0, m]. This change in traffic congestion costs also affects the respective floor market distortions caused by FAR regulation, which is expressed as 

ðm  0

rB ðxÞ 

 ½nB ðmÞγ1 ∂SB ðFB ðxÞÞ FB ðxÞ ∂fB ðxÞ dx  δγ . ∂FB ðxÞ fB ðxÞ ∂gðxÞ ½ρðmÞγ

This concludes the explication of Eq. (A5.30). Next, from Eq. (A5.1), ΨB  μ0B ðmÞdm

  ∂SB ðFB ðm + dmÞÞ ¼ fB ðm + dmÞ rB ðm + dmÞ  ∂FB ðm + dmÞ   ∂SB ðFB ðmÞÞ + fB ðmÞ rB ðmÞ  ∂FB ðmÞ

(A5.31)

recalling that η ¼ 1 from Eqs. (A5.6), (A5.10). The relocation of one worker implies relocation of fB units of floor space. Accordingly, Eq. (A5.31) implies the total change in deadweight loss in the floor market, which arises from the FAR regulation at m + dm and m. At the optimal condition the social welfare change due to the change in the communication cost, expressed as ΩB in Eq. (A5.30), should be balanced with the change in the deadweight loss in the floor market expressed as ΨB in Eq. (A5.31). This is the explication of the first-order conditions with respect to FAR regulation in the business zone. b. FAR regulation in the condominium zone First, noting that η ¼ 1, combination of Eq. (A5.4) and Eq. (A5.15), by FC ðmÞ μC ðmÞ FC ðmÞ df C ðmÞ cancelling κ C,  ¼ 0. yields λ0C ðmÞ + fC ðmÞ fC ðmÞ fC ðmÞ2 dr C ðmÞ μ ðmÞ ∂SC ðFC ðmÞÞ Rearrangement of Eq. (A5.2) yields  C ¼ rC ðmÞ  . fC ðmÞ ∂FC ðmÞ Substituting this into the earlier equation and integrating the result with respect to m, we obtain

134

ð MC m

Traffic congestion and land use regulations

λ0C ðxÞdx ¼ 

ð MC m

n0C ðxÞdx +

 ð MC  ∂SC ðFC ðxÞÞ 0 df ðxÞ nC ðxÞ C rC ðxÞ  dx, ð x Þ ∂F dT C C ð xÞ m

which leads to
 nC ðmÞ + λC ðmÞ ¼ ½N

ð MC  m

where

relationships are used: 1 df df C
 nðMC Þl and  λC ðMC Þ ¼ λH ðMC Þ ¼ N ¼ C , which is obtained fC dr C dT C from differentiating Eq. (5.16).m Recalling the form of the Lagrangian in Eq. (5.20), [λC(m)] on the lefthand side of Eq. (A5.32) expresses the shadow price for a unit of commuting time at m (m 2 [MB, MC]). Eq. (A5.32) is easily interpreted as follows.n The first term on the righthand side of Eq. (A5.32) is the direct effect of total increase in commuting
 nC ðmÞ. The second time for all commuters passing through m, that is, N term is the effect of change in the per-capita floor area consumption, dfC/dTC, beyond m, which is induced by the increase in commuting cost TC(x) (x 2 [m, MC]). The change in the per-capita floor area consumption, 0 dfC/dTC, multiplied by nC (x) gives the change in the total floor area ∂SC ðFC ðxÞÞ is the marginal F(x) at x. As explained earlier the term rC ðxÞ  ∂FC ðxÞ change in the deadweight loss (or distortion) caused by the FAR regulation (see Fig. 5.4). If FC is determined in perfect competition, ∂SC ðFC ðxÞÞ , and correspondingly the second term on the rightrC ðxÞ ¼ ∂FC ðxÞ hand side of Eq. (A5.32) is zero. However, when the floor area is regulated, ∂SC ðFC ðxÞÞ ; that is, the second term in Eq. (A5.32) is not zero. rC ðxÞ 6¼ ∂FC ðxÞ l

the

 ∂SC ðFC ðxÞÞ 0 df ðxÞ rC ðxÞ  nC ðxÞ C dx ∂FC ðxÞ dT C ðxÞ (A5.32)

following


 nC ðMC Þ is proved simply by using Eqs. (A5.5), (A5.16)–(A5.18). λH ðMC Þ ¼ N h i ∂fC ∂uC C Totally differentiating Eq. (5.16) yields dTC ¼  ∂u ∂fC  rC ∂rC dr C + fC dr C , where ∂fC ¼ rC from the dfC dr C 1 first-order condition of utility maximization. So, we obtain  fC ¼ dTC , which leads to  f1C drdfCC ¼ dT . C

m

n

A similar equation to Eq. (A5.32) appears in Kanemoto (1977), Arnott and MacKinnon (1978), Arnott (1979), Pines and Sadka (1985), and Pines and Kono (2012). As explained in the main text, μC(m) expresses the distortion arising from the FAR regulation in our model. In contrast, in the case of those previous papers, except for Pines and Kono (2012), the distortion arises from the allocation of land between road and residential areas, which is fixed in the present model. Simultaneously controlling FAR with road area is shown in Chapter 6 in which cordon pricing is implemented.

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Land use regulations with business areas

Next, we interpret the optimality condition of FAR regulation by combining the first-order conditions. Substituting λC(m) from Eq. (A5.32) into Eq. (A5.20) yields
 nC ðmÞγ1 ½N
 nC ðmÞ ½N ρðmÞγ (A5.33)  
 nC ðmÞγ1 ð MC ½N ∂SC ðFC ðxÞÞ 0 dfC rC ðxÞ  dx nC ðxÞ + δγ ρðmÞγ dT C ∂FC ðxÞ m

ΩC  μ0C ðmÞdm ¼ δγ


 nC ðmÞγ1 =ρðmÞγ ¼ ∂TC ðmÞ=∂n0C ðmÞ. where δγ ½N Next, differentiating Eq. (A5.2) with respect to m yields.   ∂SC ðFC ðm + dmÞÞ 0 ΨC  μC ðmÞdm ¼ fC ðm + dmÞ rC ðm + dmÞ  ∂FC ðm + dmÞ   ∂SC ðFC ðmÞÞ + fC ðmÞ rC ðmÞ  ∂FC ðmÞ (A5.34) Because both ΩC and ΨC are equal to μC0 (m)dm, the right-hand side of Eq. (A5.33) should be equal to the right-hand side of Eq. (A5.34) for the optimality of the FAR regulation; that is, ΩC  ΨC ¼ 0. A similar case is presented in detail in Kono and Joshi (2012). We present only a brief explanation here as follows. The relation ΩC  ΨC ¼ 0 for the optimality of FAR regulation implies that with the relocation of one person from m to m + dm, the welfare increase associated with the deadweight loss in the FAR regulation, that is, ΨC, should be cancelled out by the welfare increase associated with the increased commuting cost, that is, ΩC. Importantly, “ΩC  ΨC” is compatible with Harberger’s welfare formula, which measures the welfare change in a distorted economy (see Harberger, 1971). c. LS regulation in the lot housing zone In the detached housing zone, population density is directly adjusted, whereas in the case of FAR regulation, even if the building sizes are adjusted, per-capita floor area cannot be controlled by the government. Therefore the second term in Eq. (A5.33) exists in the case of FAR regulation but not in the case of LS regulation. To check the difference, we can derive μ0H (m) using Eqs. (A5.5), (A5.16), (A5.18), (A5.21), as
 nH ðmÞγ1

½N μ0H ðmÞ ¼ δγ

ρðmÞγ


 nH ðmÞ ½N

(A5.35)

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Traffic congestion and land use regulations

Eq. (A5.35) does not have a term corresponding to the second term in Eq. (A5.33). However, the other first-order conditions are essentially the same. (3) Lemma 5.1 and Proof of Proposition 5.1 Lemma 5.1 corresponds to the mathematical expressions of Proposition 5.1. Lemma 5.1 (Optimality condition for FAR regulation and LS regulation in the presence of optimal zonal boundaries.o (1) Business zone:The sign of μB(m)(m 2 [0, MB]) depends on the sign of nB ðmÞγ1 , which can be positive or negative; however, there is at least 2τ + δγ ρðmÞγ one location where μB(m) changes sign. Two cases arise: nB ðmÞγ1 • Case (i) implying 2τ + δγ < 0 at any m 2 [0, MB]: μB(m) > 0 at ρðmÞγ  ^ ^ any m 2 0, m and μB(m) < 0 at any m 2 m,MB ; nB ðmÞγ1 : • Case (ii) implying 2τ + δγ γ > 0 at any m 2 [ε, MB], where ε ¼ 0: ρðmÞ  ^ ^ μB(m) < 0 at any m 2 ε, m and μB(m) > 0 at any m 2 m,MB .  _ (2) Condominium zone: μC(m) > 0 at any m 2 MB , m and μC(m) < 0 at any _ _ m 2 m ,MC and μC m ¼ 0. (3) Lot housing zone: μH(mH) < 0 at any m 2 (MC, MH] and μH(MC) ¼ μC(MC). Lemma 5.1 is proved as follows. (1) Business zone: We prove Lemma 5.1(1) in two steps. First, noting that ð MB FB ðmÞ
combination of Eqs. (A5.6), (A5.7) yields η ¼ 1 and dm ¼ N, 0 fB ðmÞ ð MB 0

o

μ B ð mÞ

FB ðmÞ ∂fB dm ¼ 0 fB ðmÞ2 ∂w

(A5.36)

Note that market land rents between two adjacent zones are not equalized in the formulation of the Lagrangian in Eq. (5.20) or subsequent derivations. This should show that Proposition 5.1 is not derived with zonal boundaries set at market equilibrium.

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This implies that the solution of μB(m) has one of the following two patterns: pattern (1) μB(m) is positive at some m and negative at other m, where m 2 [0, MB], or pattern (2) μB(m) is zero all over m 2 [0, MB]. Next, we will analyze the sign of μB0 (m) in Eq. (A5.30). Noting that ∂ fB/∂ w ¼ ∂ fB/∂ g based on Eq. (5.7), substituting Eq. (A5.36) into the ð MB FB ðxÞ ∂fB
+ dx (from Technical Appendix 5 μB ðxÞ equation θ ¼ N fB ðxÞ2 ∂g 0
.
, and substituting into (A5.13) yields ϕB ð0Þ ¼ N (2)-a) yields θ ¼ N Note that ϕB(MB) ¼ 0 from (A5.14). Next, Eqs. (A5.12), (A5.27) show ϕ0B ðmÞ < 0 at any m 2 (0, MB) when ∂ L/∂ ω(m) ¼ 0 due to the complementary slackness. Therefore θ  ϕB(m) > 0, m 2 (0, MB]. This explains why the first parenthesis in Eq. (A5.30) is positive. It thus turns out that " # γ1 ½ n ð m Þ  B the sign of μB0 (m) is the same as that of 2τ + δγ , which can ρðmÞγ be either positive or negative. This concludes the proof of Lemma 5.1(1). (2) Condominium zone: A similar explanation as in the case of the business zone applies. Noting that η ¼ 1, combination of Eqs. (A5.4), (A5.5), ð MC FC ðmÞ ∂fC (A5.6) yields μC ðmÞ dm ¼ 0. To satisfy this the solution fC ðmÞ2 ∂rC MB of μC(m) has one of the following two patterns: pattern (1) μC(m) is positive at some m and negative at other m, m 2 [MB, MC], or pattern (2) μC(m) is zero all over m 2 [MB, MC]. From Eq. (A5.20), ½N  nC ðmÞγ1 , m 2 (MB, MC). Accordingly, μ0C ðmÞ ¼ λC ðmÞδγ ρðmÞγ μ0C (m) < 0 because λC(m) > 0 from Eq. (A5.29), where m 2 (MB, MC). Therefore, we can exclude pattern (2) of the solution of μC(m). Finally, continuous μC(m) satisfies Lemma 5.1(2). (3) Lot housing zone: From Eqs. (A5.5), (A5.16), noting the condition ÐM
 nH ðmÞ. Plugging this into λH(MH) ¼ 0, λH ðmÞ ¼ m H fH 1ðxÞ dx ¼ N  
 nH ðmÞ γ N 0 Eq. (A5.21) yields μH ðmÞ ¼ δγ < 0,m 2 ðMC ,MH Þ. ρðmÞ Eqs. (A5.22), (A5.23) imply μH(MC) ¼ μC(MC), and as proved earlier,   ðm
 nðxÞ γ N μC(MC) < 0 . Therefore μH(m) ¼ μC(MC) δγ dx < 0, ρðxÞ MC m 2 [MC, MH). The results μ0H ðmÞ < 0 and μH(m) < 0 prove Lemma 5.1(3).

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Proof of Proposition 5.1 Proposition 5.1 is derived from Lemma 5.1 as follows. From Eqs. (A5.1), (A5.2), it is evident that the sign of μj(m)    ∂Sj Fj ðmÞ . Thus, com( j 2 {B, C}) is the reverse of the sign of rj ðmÞ  ∂Fj ðmÞ bining Eq. (A5.1) with Lemma 5.1(1) while also using η ¼ 1 from Eqs. (A5.6), (A5.10) yields Proposition 5.1(1). Likewise, combining Eq. (A5.2) and Lemma 5.1(2) using η ¼ 1 again yields Proposition 5.1(2). The combination of Eq. (A5.3) with Lemma 5.1(3), noting that κ H(m) ¼ ∂uH ðfH ðmÞÞ  rH ðmÞ < 0 for any 1 from Eq. (A5.5) given η ¼ 1, leads to ∂fH ðmÞ m 2 (MC, MH], thereby yielding Proposition 5.1(3). (4) Lemma 5.2 and Proof of Proposition 5.2 Lemma 5.2 corresponds to the mathematical expressions of Proposition 5.2. Lemma 5.2 (Optimal condition on zonal boundaries in the presence of optimal FAR regulation and LS regulation) (1) Business zone: ½FB rB ðgB ðMB Þ, wÞ  SB ðFB Þ  ½FC rC ðMB Þ  SC ðFC Þ ¼   ∂SB ðFB ðMB ÞÞ FB ðMB Þ rB ðMB Þ  ∂FB ðMB Þ   ∂SC ðFC ðMB ÞÞ FC ðMB Þ,  rC ðMB Þ  ∂FC ðMB Þ   ∂SB ðFB ðMB ÞÞ fB ¼ μB ðMB Þ. In Case (i), where rB ðMB Þ  ∂FB ðMB Þ μB(MB) > 0, and in Case (ii), μB(MB) < 0.p Likewise,   ∂SC ðFC ðMB ÞÞ rC ðMB Þ  fC ¼ μC ðMB Þ < 0; ∂FC ðMB Þ (2) Condominium zone: ½FC ðMC ÞrC  SC ðFC Þ  Ra   ½rH ðMC Þ  Ra      ∂SC ðFC ðMC ÞÞ ∂uH ð fH ðMC ÞÞ FC ðMC Þ  rH ðMC Þ  , ¼ rC ðMC Þ  ∂FC ðMC Þ ∂fH ðMC Þ

p

See Lemma 5.1(1) for definition of Cases (I) and (II).

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  ∂SC ðFC ðMC ÞÞ fC ðMC Þ ¼ μC ðMC Þ; where rC ðMC Þ  ∂FC ðMC Þ   ∂uH ðfH ðMC ÞÞ rH ðMC Þ  fH ðMC Þ ¼ μH ðMC Þ; and ∂fH ðMC Þ μC(MC) ¼  μH(MC) > 0. (3) Lot housing zone: rH  Ra ¼    ð MH  N  nðmÞ γ 1 ∂SC ðFC ðMC ÞÞ fC ðMC Þ + rC ðMC Þ  , δγ dm  ρðmÞ fH ðMH Þ ∂FC ðMC Þ fH ðMH Þ MC   ∂SC ðFC ðMC ÞÞ fC ðMC Þ ¼ μC ðMC Þ > 0. where rC ðMC Þ  ∂FC ðMC Þ Lemma 5.2 is proved as follows. Lemma 5.2(1) is derived from the combination of Eqs. (A5.1), (A5.4), (A5.24), (A5.32) using η ¼ 1 and ð MC FC ðmÞ ∂fC ðmÞ μC ðmÞ dm ¼ 0 (from the proof of Lemma 5.1(2)). FollowfC ðmÞ2 ∂rC ðmÞ MB
and, from Technical Appendix 5(2)-b, ing relations are also used: λC ðMB Þ ¼ N
λB ðMB Þ ¼ N . The inequality conditions involving μk(k ¼ {B, C, H}) are obtained from Lemma 5.1. Next, Lemma 5.2(2) is derived from the combination of Eqs. (A5.2), (A5.4), (A5.5), (A5.25). μC(MC) ¼ μH(MC) is obtained from Eqs. (A5.22), (A5.23). μC(MC) < 0 is from Lemma 5.1(2). Finally, Lemma 2(3) is derived from Eq. (A5.26) with   ðm
 nðxÞ γ N μH ðmÞ ¼ μC ðMC Þ  δγ dx, which is obtained in the proof ρðxÞ MC of Lemma 5.1. This concludes the proof of Lemma 5.2. We now compare the optimal zonal boundaries with the market boundaries. If the boundary of Zone B is determined by the market, then FBrB(g(MB), w)  SB(FB) ¼ FCrC(MB)  SC(FC) where the land rents are equal between Zones B and C. Lemma 5.2(1) shows that, in Case I, FBrB(g(MB), w)  SB(FB) > FCrC(MB)  SC(FC) because the right-hand side of the first equation in Lemma 5.2(1) is greater than zero. However, in Case II, whether FBrB( g(MB), w)  SB(FB) is greater or less than FCrC(MB)  SC(FC) is ambiguous because the right-hand side of the first equation in Lemma 5.2(1) can be either negative or positive. Likewise, if the boundary between Zone C and Zone H is determined by the market, then FC(MC)rC(MC)  SC(FC) ¼ rH(MC). Lemma 5.2(2) shows that, in the optimal case, the sign of [FC(MC)rC  SC(FC)  Ra] 

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[rH(MC)  Ra] is the same as that of the right-hand side of the equation in Lemma 5.2(2). The right-hand side can be arranged into

   ∂SC ðFC ðMC ÞÞ ∂uH ðfH ðMC ÞÞ FC ðMC Þ   rH ðMC Þ ¼ fC ðMC Þ½rC ðMC Þ ∂FC ðMC Þ ∂fH ðMC Þ ∂SC ðFC ðMC ÞÞ ½nC ðMC Þ  nH ðMC Þ, using the second equation in Lemma 5.2  ∂FC ðMC Þ



rC ðMC Þ 

(2). This implies that the right-hand side is greater than zero because



 ∂SC ðFC ðMC ÞÞ > 0 as denoted in Lemma 5.2(2) and ∂FC ðMC Þ nH(MC)] > 0 because, by definition, a condominium has more rC ðMC Þ 

[nC(MC) 

households than a detached house. Correspondingly, in the optimal case, [FC(MC)rC  SC(FC)  Ra]  [rH(MC)  Ra] > 0. Finally, if the city boundary is determined in the market, then rH(MH) ¼ Ra. Lemma 5.2(3) shows that in the optimal case, rH(MH) > Ra because the right-hand side of the equation in Lemma 5.2(3) is greater than zero.

References Arnott, R.J., MacKinnon, J.G., 1978. Market and shadow land rents with congestion. Am. Econ. Rev. 68 (4), 588–600. Arnott, R.J., 1979. Unpriced transport congestion. J. Econ. Theory 21 (2), 294–316. Borukhov, E., Hochman, O., 1977. Optimum and market equilibrium in a model of a city without a predetermined center. Environ. Plan A 9 (8), 849–856. Ciccone, A., Hall, R., 1996. Productivity and the density of economic activity. Am. Econ. Rev. 86 (1), 54–70. Duranton, G., Puga, D., 2001. Nursery cities: urban diversity, process innovation, and the life cycle of products. Am. Econ. Rev. 91 (5), 1454–1477. Fujita, M., Thisse, J.F., 2013. Economics of Agglomeration: Cities, Industrial Location, and Globalization. Cambridge University Press. Harberger, A.C., 1971. Three basic postulates for applied welfare economics: an interpretive essay. J. Econ. Lit. 9 (3), 785–797. Japanese Society of Civil Engineering, 2003. Theory and Applications of Traffic Demand Forecasting. Maruzen (in Japanese). Kanemoto, Y., 1977. Cost-benefit analysis and the second best land use for transportation. J. Urban Econ. 4 (4), 483–503. Kanemoto, Y., 1990. Optimal cities with indivisibility in production and interactions between firms. J. Urban Econ. 27 (1), 46–59. Kono, T., Joshi, K.K., 2012. A new interpretation on the optimal density regulations: closed and open city. J. Hous. Econ. 21 (3), 223–234. Kono, T., Joshi, K.K., 2018. Spatial externalities and land use regulation: an integrated set of multiple density regulations. J. Econ. Geogr. 18 (3), 571–598. Kono, T., Kawaguchi, H., 2017. Cordon pricing and land-use regulation. Scand. J. Econ. 119 (2), 405–434. Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630.

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Ministry of Land, Infrastructure, Transport and Tourism (MLIT), 2010. http://www.mlit. go.jp/road/ir/hyouka/plcy/kijun/bin-ekiH20_11 (in Japanese). Moretti, E., 2004. Workers’ education, spillovers, and productivity: evidence from plantlevel production functions. Am. Econ. Rev. 94 (3), 656–690. O’Hara, D.J., 1977. Location of firms within a square central business district. J. Polit. Econ. 85 (6), 1189–1207. Ogawa, H., Fujita, M., 1980. Equilibrium land use patterns in a non-monocentric city. J. Reg. Sci. 20 (4), 455–475. Pasha, H.A., 1996. Suburban minimum lot zoning and spatial equilibrium. J. Urban Econ. 40 (1), 1–12. Pines, D., Kono, T., 2012. FAR regulations and unpriced transport congestion. Reg. Sci. Urban Econ. 42 (6), 931–937. Pines, D., Sadka, E., 1985. Zoning, first-best, second-best, and third-best criteria for allocation land for roads. J. Urban Econ. 17 (2), 167–183. Rauch, J.E., 1993. Productivity gains from geographic concentration of human capital: evidence from the cities. J. Urban Econ. 34 (3), 380–400. Tauchen, H., Witte, A.D., 1984. Socially optimal and equilibrium distributions of office activity: models with exogenous and endogenous contacts. J. Urban Econ. 15 (1), 66–86. Wheaton, W.C., 1998. Land use and density in cities with congestion. J. Urban Econ. 43 (2), 258–272.

CHAPTER SIX

Introducing cordon pricing in a regulated city

6.1 Introduction As discussed in earlier chapters, cities impose land use regulations to control urban externalities including traffic congestion. In addition, several cities around the world have also implemented cordon pricing as a more direct measure to mitigate traffic congestion. Indeed, London, Milan, Oslo, Singapore, and Stockholm, which impose cordon or area pricing,a all impose land use regulations. This chapter addresses how cordon pricing should be imposed along with land use regulations in the form of floor area ratio (FAR) regulation and urban growth boundary (UGB) regulation. In particular, this chapter clarifies a mechanism among all the related distortions caused by not only traffic congestion but also cordon pricing and land use regulation themselves, in a continuous space. From the mechanism, we derive some important theoretical properties of optimal regulation and cordon pricing, which are different from current understanding. In a cordon pricing scheme, a vehicle is charged a fixed toll when it passes a specified cordon line in the inbound and/or outbound direction. Cordon pricing cannot generally correct negative externality completely, unlike the first-best congestion pricing. In addition, it is not an easy task for governments to rationally determine the toll rate and the location of the cordon line because of the need to take account of the changes in distortions in the related market. Land use regulations, which can control population density, are also effective against urban externalities including traffic congestion, as we have shown in the previous chapters. An important point is that both cordon pricing and land use regulations are second-best policies and, therefore, both can be imposed simultaneously to increase the welfare. a

The main objective of cordon pricing as practiced is not necessarily to mitigate congestion. For instance, the cordon pricing in Oslo was designed primarily for revenue generation.

Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00006-6

© 2019 Elsevier Inc. All rights reserved.

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Second-best pricing policies, such as cordon pricing, have been explored in many studies, as reviewed by Lindsey and Verhoef (2001). Sullivan (1983) examines a second-best policy in which the toll is proportional to the distance traveled, based on the general equilibrium simulation model. Kraus (1989) numerically calculates welfare gains from various pricing regimes, including cordon pricing. More recently, Anas and Hiramatsu (2013) have simulated the effect of cordon pricing in a computable general equilibrium model representing Chicago. Against such simulation models, Mun et al. (2003) and Li et al. (2014) analytically obtain the optimal toll and location of cordon pricing. However, they exogenously fix population densities in all locations. In the long run, population density should be endogenously determined so that the utility level is identical among locations. Verhoef (2005) considers urban density endogenously and shows that the relative welfare losses from second-best pricing, compared with first-best pricing, are surprisingly small. However, these papers exploring congestion pricing do not consider any land use regulations that are imposed in cities simultaneously. Among related papers, only Tikoudis et al. (2018) consider both road tolls and FAR regulation. However, their analysis is based on numerical simulations. This chapter derives theoretical properties based on Kono and Kawaguchi (2017). The remainder of this chapter is organized as follows. Section 6.2 develops a monocentric city model with cordon pricing. Section 6.3 explains the optimal toll and location of the cordon line under optimal FAR regulation. Section 6.4 explores optimal FAR and UGB regulations with cordon pricing and presents a numerical example. Section 6.5 concludes the paper. The final subsection provides technical appendix.

6.2 The model 6.2.1 The city The city is circular and symmetric along any radial axis. The residential area expands from x ¼ 1 at the central business district (CBD) edge to x ¼ xb at the city boundary. The city is divided into several concentric bands of equal width. The total floor supply per unit area of land at x, that is, in the xth band, is denoted F(x), which represents FAR at x. At distance x, a fraction ρ(x) of the land is used for roads; the amount of land available for housing at x from the center of the CBD is given by 2πx[1  ρ(x)].

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We consider a closed city. The city land is under public ownership, and the land is rented by developers to build dwelling units. The total revenue of land rent Φ can be expressed as ð xb (6.1) Φ ¼ 2πxRðxÞdx 1

where R(x) is the per unit land rent. City residents must pay rent Θ for land to be used for roads, which is given as ð xb (6.2) Θ ¼ 2πxρðxÞRðxÞdx 1

For simplicity, we assume that agricultural rent (i.e., opportunity cost of land) is zero. The city government imposes FAR regulation on all buildings and UGB regulation to control traffic congestion externality. In addition, the government introduces cordon pricing in the residential area. The cordon line is established at x ¼ xc ðxc 2 ð1, xb ÞÞ; each vehicle passing the cordon line pays a toll p. The geographic configuration of the city is shown in Fig. 6.1.

6.2.2 Household behavior The city is inhabited by N identical households. For simplicity, we define population in terms of the number of households. Residents have the following utility function: v ¼ υðz, f Þ

Fig. 6.1 City configuration.

(6.3)

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where υ(z, f ) is a function of the numeraire composite goods z (which include all consumer goods except floor space) and housing square footage f. The net revenue from land rent, Φ  Θ, and the sum of the cordon tolls, denoted Ptoll, are equally distributed among all residents. The total earning per resident, denoted ΠN, is given as ΠN  [Φ  Θ +Ptoll]/N.b Each household earns income y per period. The household rents floor space from developers at price r per unit floor space per period. The income constraint is expressed as z + rf ¼ y  T ðxÞ + ΠN

(6.4)

where T(x), which depends on traffic volume, is the commuting cost for a resident at x. Floor rent r equals the maximum floor rent bid by a household as a result of competition among residents. Mathematically, such behavior is expressed as max r ¼ f ,z

y  T ðxÞ + ΠN  z subject to Eq: ð6:3Þ f

(6.5)

Solving Eq. (6.5) and using utility level v(x) at location x yields r ðxÞ ¼ r ðy  T ðxÞ + ΠN , vðxÞÞ and f ðxÞ ¼ f ðy  T ðxÞ + ΠN , vðxÞÞ (6.6)

6.2.3 Developers’ behavior under FAR regulation Developers must follow the FAR regulation set by the city government. Construction of floor space per unit of land area at each location x, denoted F(x), requires S(F) where S is the capital-to-land ratio; the price of the capital S is normalized at unity. The sum of the developers’ net profits from the total floor space supply in the city, denoted Π, is given as ð xb Π ¼ 2πx½1  ρðxÞ½F ðxÞr ðxÞ  SðF ðxÞÞ  RðxÞdx (6.7) 1

Under FAR regulation the level of F(x) is exogenously given to the developers. Further, because the developers are perfectly competitive, their own profit is zero. The land rent at each location x is then given by RðxÞ ¼ F ðxÞr ðxÞ  SðF ðxÞÞ b

(6.8)

This assumption may seem unrealistic, but it helps us to focus on the welfare effects of cordon pricing and land use regulations, ignoring the effects of distribution of the revenue from the rent and the toll.

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6.2.4 Traffic flow and commuting cost We assume that commuters use only automobiles and that there is a continuum of radial roads running into the CBD. Because commuters from more distant locations join traffic while moving towards the CBD, traffic volume increases towards central locations. Thus the unit cost of commuting for a commuter passing location x depends on the traffic flow across x. The number of households residing beyond x is denoted n(x). Then, since the number of people in a household is one by assumption, n(x) also represents the number of commuters passing through x. We adopt the following commonly used functional form of the commuting cost to cross x. The level of traffic congestion at each location is determined by the ratio of the volume of commuters’ trips to road capacity. Specifically the unit-distance commuting cost, Tunit(x), to cross x is expressed as the function c(n(x)/[2πxρ(x)]), where both the first derivative c 0 () and the second derivative c00 () are positive. A commuter at x pays the total cost T(x). The unit-distance commuting cost T 0 (x) paid by commuters satisfies   nðxÞ 0 T ðxÞ  Tunit ðxÞ  c (6.9) 2πxρðxÞ where the apostrophe expresses differentiation with respect to distance. Eq. (6.9) uses an inequality condition. This treatment is seen in Chapter 5 too; so, we give only a simple explanation here. The left-hand side, T 0 (x), is the demand-side cost paid by a commuter, while the right-hand side, c(), is determined physically. The inequality condition (6.9) implies that the commuter can spend more than the supply-side cost. However, a rational commuter does not pay more than the supply-side cost; so, this expression will hold as an equality as a result of utility maximization behavior. We set the inequality condition (6.9) because this treatment is useful later to determine the sign of the Lagrange multiplier for this constraint using the Kuhn-Tucker condition, as shown in Technical Appendix 6(1).c

6.2.5 The model under cordon pricing We incorporate cordon pricing into the model. Recall that the cordon line is located at x ¼ xc. Variables (Tunit(x), T(x), F(x), n(x), and ρ(x)) in the model are separated into the area inside the cordon line (x 2 [1, xc)) and the area outside it ðx 2 ðxc , xb Þ, with subscripts “i” denoting “inside” and “o” denoting c

Specifically in deriving λj 0 as expressed in Eq. (A6.13) in Technical Appendix 6(1).

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“outside.” The number of households residing beyond the band at x, nj(x) where j 2 {i, o}, is defined as in Eqs. (6.10), (6.11): ð xc Fi ðmÞ dm ni ðxÞ ¼ 2πm½1  ρi ðmÞ fi ðmÞ x ð xb (6.10) Fo ðmÞ 2πm½1  ρo ðmÞ dm; x 2 ½1, xc Þ + fo ðmÞ xc no ðxÞ ¼

ð xb x

2πm½1  ρo ðmÞ

Fo ðmÞ dm; x 2 ðxc , xb  fo ðmÞ

(6.11)

The total commuting cost Tj(x)( j 2 {i, o}) is defined as in Eqs. (6.12) and (6.13)d:  ðx ðx  ni ðmÞ (6.12) Ti ðxÞ  Tunit i ðmÞdm ¼ c dm; x 2 ½1, xc Þ 2πmρðmÞ 1 1 ð xc ðx To ðxÞ  Tunit i ðmÞdm + Tunit o ðmÞdm + p   ð xc 1 ðxcx  (6.13) ni ðsÞ no ðmÞ ¼ c dm + c dm + p; x 2 ðxc , xb  2πmρðmÞ 2πmρðmÞ xc 1 At location xc, because ni(xc) ¼ no(xc), commuting time is equal on either immediate side of the cordon line. So the commuting cost Tj(x) differs only by the size of the toll p: To ðxc Þ ¼ Ti ðxc Þ + p

(6.14)

6.2.6 Market equilibrium conditions Because households can choose a residential location, household utility is equal to a common level V at any location, the level of which is endogenously determined: vðxÞ ¼ V 8x 2 ½1, xb 

(6.15)

The population constraint at each location, which is obtained by differentiating Eqs. (6.10) and (6.11) with respect to x, should satisfy h i Fj ðxÞ 0  ; j 2 fi, og nj ðxÞ ¼ 2πx 1  ρj ðxÞ  (6.16) f y  Tj ðxÞ + ΠN , vðxÞ d

The inequality condition in Eqs. (6.12) and (6.13) can clarify the sign of the Lagrange multipliers for these constraints using the Kuhn-Tucker condition.

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Next, Eq. (6.17) gives the boundary condition of traffic flow at the cordon line as stated earlier: ni ðxc Þ ¼ no ðxc Þ (6.17) Finally, we consider the nonlabor income constraint. Because the volume of traffic passing the cordon line is no(xc), the total revenue from tolls, Ptoll ¼ no(xc)  p. Substituting this and Eqs. (6.1), (6.2), (6.6), and (6.8) into ΠN  [Φ  Θ + Ptoll]/N, we obtain ð xc N ΠN ¼ 2πx½1  ρi ðxÞ½Fi ðxÞr ðy  Ti ðxÞ + ΠN ; vðxÞÞ  SðFi ðxÞÞdx 1 ð xb + 2πx½1  ρo ðxÞ½Fo ðxÞr ðy  To ðxÞ + ΠN ; vðxÞÞ  SðFo ðxÞÞdx xc

+ no ðxc Þp (6.18)

6.3 Optimal cordon pricing and location The welfare level W depends on the location of the cordon line and toll level and on the land use regulations. The optimal location of cordon line, toll level, FAR regulation, and UGB regulation are given by maximizing the total utility of households subject to Eqs. (6.12)–(6.18). We obtain the optimal solution using the Lagrangian function as followse:       ð xb ð xc NV ni ðxÞ no ðxÞ 0 0 dx + dx L¼ λi ðxÞ Ti ðxÞ  c λo ðxÞ To ðxÞ  c + η 2πxρi ðxÞ 2πxρo ðxÞ 1 xc   ð xc Fi ðxÞ + + n0i ðxÞ dx μi ðxÞ 2πx½1  ρi ðxÞ ð ð f y  T i xÞ + ΠN ;V Þ 1   ð xb Fo ðxÞ 0 + no ðxÞ dx + μo ðxÞ 2πx½1  ρo ðxÞ f ðy  To ðxÞ + ΠN ;V Þ xc ð 2 3 xc N ΠN + 2πx½1  ρi ðxÞ½Fi ðxÞr ðy  Ti ðxÞ + ΠN ;V Þ  SðFi ðxÞÞdx 6 7 1 7 +6 4 ð xb 5 + 2πx½1  ρo ðxÞ½Fo ðxÞr ðy  To ðxÞ + ΠN ;V Þ  SðFo ðxÞÞdx + no ðxc Þp xc

+ ϕ½To ðxc Þ  Ti ðxc Þ  p + ψ ½no ðxc Þ  ni ðxc Þ (6.19)

e

This Lagrangian does not include the constraints that would obviously break the constraint qualification for Lagrangian, such as complementary slackness conditions. We assume that there are solutions that satisfy such constraint qualification. See the Mathematical Note in Chapter 3 for constraint qualification.

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where η is the Lagrange multiplier with regard to the nonlabor income constraint. In other words, η is the shadow price for the constraint for ΠN (see Eq. 6.18). Likewise, λj(x) and μj(x), ( j 2 {i, o}) are the Lagrange multipliers for commuting cost and population at location x, respectively; ϕ and ψ are the Lagrange multipliers for the boundary conditions of commuting cost and traffic flow at the cordon line, respectively. Note that in the Lagrangian, v(x) has been substituted by V using Eqs. (6.15), (6.16), and (6.18). Dividing the original Lagrangian function by η, the Lagrangian is measured in terms of composite goods.f The other four boundary conditions are as follows: λo ðxb Þ ¼ 0,g Ti(1) ¼ 0, ni(1) ¼ N, and no ðxb Þ ¼ 0. After integrating the Lagrangian by parts, we derive the first-order conditions for the optimum policy, as shown in Technical Appendix 6(1), which lead to Proposition 6.1 (see Technical Appendix 6(2) for proof ). Proposition 6.1 (Optimal toll level with optimal FAR regulation) When optimal FAR regulation is imposed, the optimal toll p satisfies     ∂SðFi ðxc ÞÞ ∂SðFo ðxc ÞÞ fi ðxc Þ  ro ðxc Þ  fo ðxc Þ p ¼ ri ðxc Þ  ∂Fi ðxc Þ ∂Fo ðxc Þ

(6.20)

Proposition 6.1 implies that the optimal toll should be equal to the difference in the deadweight loss per household in the floor area market across the cordon line. This is because household migration across the cordon line would not improve social welfare in the optimal case. The distortions   ∂SðFj ðxc ÞÞ rj ðxc Þ  ∂Fj ðxc Þ ( j 2 {i, o}), which are commonly discussed throughout this book, are shown in Fig. 3.2 in Chapter 3. Note that the toll p is also a distortion generated by migration across the cordon line. Because each term in Eq. (6.20) can be observed, this relation is useful for practically checking whether the toll is optimal ex post. Note that rj(xc) and fj(xc) ( j 2 {i, o}) are directly observable. Likewise, ∂S(Fj(xc))/∂Fj(xc) ( j 2 {i, o}) is the marginal cost for floor space production. If the building supply is perfectly competitive, the supply curve is equal to the marginal cost; so, it is observable. Even if building supply is not perfectly competitive, the cost function of F can be estimated from the data of F together with the production cost. Next, arranging and calculating the first-order condition (A6.9) with respect to the cordon location xc, we obtain Proposition 6.2, which shows f g

Note that ΠN is measured in terms of composite goods (see Eq. 6.4). Because xb is unconstrained, To ðxb Þ is also unconstrained. Accordingly, λo ðxb Þ ¼ 0.

Introducing cordon pricing in a regulated city

151

the condition for optimal location of the cordon line when optimal FAR is imposed, and road width is equal across the cordon line; that is, ρi(xc) ¼ ρo(xc). See Technical Appendix 6(3) for the proof. Proposition 6.2 (Optimal location of the cordon line with optimal FAR regulation) When optimal FAR is imposed and ρi(xc) ¼ ρo(xc), the optimal location of cordon line, xc, satisfies     ∂SðFi ðxc ÞÞ ∂SðFo ðxc ÞÞ Ri ðxc Þ  Ro ðxc Þ ¼ Fi ðxc Þ ri ðxc Þ   Fo ðxc Þ ro ðxc Þ  ∂Fi ðxc Þ ∂Fo ðxc Þ (6.21) The left-hand side of Eq. (6.21) is the difference in land rent across the cordon line. We can interpret the right-hand side in the following manner. ∂SðFj ðxc ÞÞ As explained earlier, rj ðxc Þ  ∂Fj ðxc Þ ( j 2 {i, o}) is the distortion (or the marginal deadweight loss per unit floor area) caused by FAR regulation. That is, the right-hand side of Eq. (6.21) expresses the difference in the deadweight loss in the floor area market across the cordon line. Interestingly, Eqs. (6.20) and (6.21) do not include traffic congestion externality. This is because FAR regulation optimizes the population density all over the city although it cannot achieve the first best. Importantly the net distortion in the floor area market, either inside or outside the cordon line, becomes zero because of the optimal FAR regulation, as Technical Appendix 6(5) shows. As a result, both Propositions 6.1 and 6.2 are based only on the properties across the cordon line.

6.4 Optimal land use regulation under cordon pricing 6.4.1 Characterization of optimal FAR and UGB regulation The relation between FAR distortions and traffic congestion is explained in Technical Appendix 6(4). We next obtain the property of the motion of μj(x) ( j 2 {i, o}), the shadow price for population density at x, 2πx[1  ρj(x)]Fj(x)/fj(x). The motion of μj(x) is shown in Fig. 6.2 (see Lemma 6.2 in Technical Appendix 6(5)). In words, inside the cordon line, μj(x) monotonically decreases from a positive value at the CBD edge to a negative value at the cordon line. Similarly, outside the cordon line, μj(x) monotonically decreases from a positive value at the cordon line up to a

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Traffic congestion and land use regulations

Fig. 6.2 Optimal cordon toll and location of cordon line. (Source: Kono, T., Kawaguchi, H., 2017. Cordon pricing and land-use regulation. Scand. J. Econ. 119(2), 405–434.)

negative at the UGB. The difference in the levels across the cordon line is the cordon toll p.h From Eq. (A6.2) in Technical Appendix 6(1), μj ðxÞ ¼   ∂SðFj ðxÞÞ fj ðxÞ rj ðxÞ  ∂Fj ðxÞ , which implies that if μj(x) is positive (resp. negative), rj ðxÞ 

∂SðFj ðxÞÞ ∂Fj ðxÞ

if rj ðxÞ 

is negative (resp. positive). Next, from Fig. 3.2 in Chapter 3,

∂SðFj ðxÞÞ ∂Fj ðxÞ

is positive (resp. negative), the FAR should be smaller (resp.

larger) than the market FAR. Therefore, following Fig. 6.3 (left side), the FAR should be larger than the market FAR for any x 2 ½1, x^1 Þ and x 2 ðxc , x^2 Þ, whereas the FAR should be smaller than the market FAR for any x 2 ðx^1 , xc Þ and x 2 ðx^2 ; xb . This leads to Proposition 6.3. Proposition 6.3 (Optimal FAR regulation with cordon pricing) Inside the cordon line, the optimal FAR regulation requires minimum (resp. maximum) FAR regulation at locations closer to the CBD edge (resp. the cordon line), whereas outside the cordon line, the optimal FAR regulation requires minimum (resp. maximum) FAR regulation at locations closer to the cordon line (resp. the city boundary). Proposition 6.3 can be explained using Fig. 6.3 (left-hand side). Accordingly, inside the cordon line, an increase in the floor area supply at locations closer to the CBD edge (resp. the cordon) decreases (resp. increases) the total congestion externality in the closed city. Likewise, outside the cordon line, h

The most intriguing property of μj(x) ( j 2 {i, o}) is that the sign alternates not only inside the cordon line but also outside it. As in the previous research on optimal density regulation (e.g., Kanemoto, 1977; Kono et al., 2012; Pines and Kono, 2012), μj0 (x) < 0, where μj0 (x) is the derivative of μj(x), but the alternation of the sign both inside and outside the cordon line is novel and specific to this model considering cordon pricing.

Introducing cordon pricing in a regulated city

153

Fig. 6.3 Optimal land use regulations with optimal and infinitesimal toll. (Source: Kono, T., Kawaguchi, H., 2017. Cordon pricing and land-use regulation. Scand. J. Econ. 119(2), 405–434.)

an increase in floor area supply at locations closer to the cordon (resp. the city boundary) decreases (resp. increases) the total congestion externality. The right-hand side of Fig. 6.3 shows the motion of optimal floor area when the toll is set at an infinitesimal level (see Pines and Kono, 2012) in contrast to the optimal toll in this study (left-hand side of Fig. 6.3). Intuitively, if a cordon toll is levied, people migrate to the area inside the cordon line to avoid the toll. However, such migration is inefficient from the viewpoint of population distribution over the city. Therefore, just outside the cordon line, minimum FAR regulation, which increases population density, is necessary. In addition, we can prove the following proposition (see Technical Appendix 6(6) for a formal proof ). Proposition 6.4 (Continuity of optimal FAR at cordon line) Optimal FAR is continuous at the cordon line although maximum (resp. minimum) FAR is imposed immediately inside (resp. outside) the cordon line. Intuitively, in the optimal case, land would be efficiently used. So the FAR should be same in the infinitesimal vicinity of the cordon line on either side. In contrast, because ri(xc) 6¼ ro(xc), housing footage f changes discontinuously across the cordon line, and the population density changes discontinuously across the cordon line. Next, we compare the optimal UGB with the market city boundary. Dividing both sides of the first-order condition (A6.12) in Technical Appendix 6(1) by 2πxb ½1  ρo ðxb Þ, we obtain μo ðxb Þ

Fo ðxb Þ + ½Fo ðxb Þro ðxb Þ  SðFo ðxb ÞÞ ¼ 0 fo ðxb Þ

(6.22)

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Traffic congestion and land use regulations

where μo ðxb Þ < 0 from Fig. 6.1. Accordingly, we obtain the following condition: Fo ðxb Þro ðxb Þ  SðFo ðxb ÞÞ ¼ Rðxb Þ > 0. Noting that agricultural rent (i.e., opportunity cost of land) is supposed to be zero for simplicity,i we obtain Proposition 6.5. Proposition 6.5 (Optimal UGB with cordon pricing) The optimal city boundary is closer to the CBD than the market city boundary. Previous studies such as Brueckner (2007) and Kono et al. (2012), which do not consider cordon pricing, obtain similar result as obtained in Proposition 6.5.

6.4.2 Numerical example of optimal cordon pricing and FAR regulation This section numerically simulates the obtained properties of optimal cordon pricing and FAR regulation using the simulation model from Kono et al. (2012). The city is divided into discrete bands of equal width (¼ 1.0 km). However, to demonstrate optimal FAR regulations clearly, the first band is equally divided into two bands (i.e., 0.5-km bands). The bands are indexed by k, where k ¼ 1 at the CBD edge. We specify the utility function and the floor production cost function as v(z, f ) ¼ z + 10,000 + α ln f j and S ¼ [F/θ]1/β, respectively, where α and θ are positive multiplicative factors and 0 < β < 1. Next the total commuting cost Tj ( j 2 {i, o}) with cordon pricing is expressed as T1 ¼ 0, Tk ¼ T1 +

k X

εk0 Tunit k0 ðxk 2 ½1,xc ÞÞ,

k0 ¼2

Tk ¼ T1 +

k X

εk0 Tunit k0 + p ðxk 2 ðxc , xb Þ

(6.23)

k0 ¼2

P where εk is the width of band k and xk ¼ kk0 ¼1εk0 is the distance from the CBD. The unit-distance commuting cost Tunit k is set per kilometer, expressed as   nk γ (6.24) Tunit k ¼ ω + δ 2πxk ρ i j

Positive agricultural rent can also yield the same result. “10,000” is added only to make utility value positive. Note that any monotone transformation is inessential.

Introducing cordon pricing in a regulated city

155

where ω is a positive parameter that expresses commuting cost incurred while driving in the case of no congestion; γ and δ are congestion exponent and multiplicative factor, respectively, both of which are positive. Likewise, nk denotes the total population beyond band k given as nk ¼

k* X k0 ¼k

 ½1  ρπ x2k0 + 1  x2k0 Dk0

(6.25)

where k* denotes the outermost band, such that nk* ¼ 0, and Dk0 is the population density. Finally the social welfare is expressed as W¼

k* X

 ½1  ρπ x2k + 1  x2k ½Dk V + Rk  Ra  + nxc p,

(6.26)

k¼1

where Ra is agricultural rent and nxc denotes total population beyond xc. The iterative process begins at k ¼ 1 with T1 ¼ 0 and n1 ¼ N and is implemented conditionally on the value of V, which should satisfy the equilibrium conditions. Parameters are set in the following manner. The total number of households N is set at 400,000. The income per household per year is set at $40,000 as in Brueckner (2007). The housing parameter α in the utility function is set at 8000, thereby implying 20% of the income. Next, setting ρ ¼ 0.2 as in Brueckner (2007), 20% of the land in each band is allocated for roads. Further, Ra is set at $150,000 per square kilometer. Parameter β is set at 0.80, and the multiplicative factor θ is set at 0.00001. We set the number of trips to the CBD as 225 round trips per year per person, average speed as 30 km/hour, and commuting cost including commuting time as US$15/hour. Then, assuming one worker per household, ω turns out to be US$225. Likewise, we set γ ¼ 1.2 and δ ¼ 0.001. The simulation results are presented in Table 6.1. Four cases are presented: laissez-faire; toll regime; optimal cordon pricing with no FAR and UGB regulations; and optimal cordon pricing, FAR regulation, and UGB regulation. With regard to the welfare gains, optimal cordon pricing with no FAR and UGB regulations obtains 79.8% of the welfare gain of the first-best toll regime. Such a high gain is comparable with that obtained in Mun et al. (2003) and Verhoef (2005). With optimal FAR and UGB regulations, optimal cordon pricing yields even more welfare gain—96% of the first best. The total welfare gain under optimal cordon pricing with no FAR and UGB

156

Table 6.1 Numerical results. xb (km)

W (109 $/year)

Welfare gain (%)

p ($/year)

F(1)

F(xc 2 1)

F(xc)

F(xb)

R(1) (107 $/ km2/year)

Laissez-faire Toll regime (first best) Optimal cordon pricing with no FAR and UGB regulationsa Optimal cordon pricing, FAR regulation, and UGB regulationa

17 15 16

3.4536 3.5934 3.5651

– 100 79.8

– – 2325

19.3 42.8 36.7

– – 21.6

– – 4.36

0.473 0.466 0.474

1.794 4.860 4.021

15

3.5878

96.0

2364

43.1

15.2

6.76

0.457

3.939

a The optimal cordon pricing is at 2.0 km from the CBD edge. Source: Kono, T., Kawaguchi, H., 2017. Cordon pricing and land-use regulation. Scand. J. Econ. 119(2), 405–434.

Traffic congestion and land use regulations

Cases

Introducing cordon pricing in a regulated city

157

regulations and optimal cordon pricing, FAR regulation, and UGB regulation is, respectively, $111.5 million and $134 million; so the annual gain per household is, respectively, $278.9 and $335.4.k According to this simulation the combination of FAR, UGB, and cordon pricing is thus significantly effective. These high efficiency scores are attributable to the assumption in the model that floor area is the only choice variable for residents. The relative efficacy of the cordon toll and FAR regulation might decline if workplace location, travel mode, vehicle occupancy, route, and departure time, among others, are endogenous as in the real world. As demonstrated in Proposition 6.3, the optimal FAR regulation under cordon pricing involves the enforcement of minimum and maximum FAR regulation both inside and outside the cordon line. This is verified by the result depicted in Fig. 6.4.l

Fig. 6.4 Numerical results. (Source: Kono, T., Kawaguchi, H., 2017. Cordon pricing and land-use regulation. Scand. J. Econ. 119(2), 405–434.)

De Palma et al. (2006) show the estimated welfare gain per capita worth €19–€319 in some hypothetical toll policies in several cities. l The properties of the optimal toll and cordon line, shown in Propositions 6.1 and 6.2, are not clearly shown in this simulation because the model of this simulation is spatially discrete, whereas the theoretical result is based on a continuous model. Taking Proposition 6.1 as an example, although the numerical ∂SðFj ðxÞÞ simulation derives the optimal p, the price distortion in the floor market, rj ðxÞ  ∂Fj ðxÞ , can be shown k

in the discrete zones on either side of the cordon line. If the width of each band is infinitesimal, Proposition 6.1 could be accurately verified; however, the calculation burden would be heavy.

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Traffic congestion and land use regulations

6.5 Conclusion In this chapter, we have investigated the optimal combination of FAR regulation, UGB regulation, and cordon pricing (the toll level and the location of the cordon) in a closed monocentric city. In addition, we provide a numerical simulation to confirm the obtained theoretical properties of optimal cordon pricing and FAR regulation. First, under optimal building size regulation, we show that the optimal toll level can be represented by the difference in the per-household distortion in the floor area market across the cordon line. At the optimal location of the cordon line, the rent difference across the cordon line is equal to the difference in the deadweight loss per unit land area caused by FAR regulation. Next, with regard to optimal FAR regulation, we present a distinct result from cities without cordon pricing as shown in Fig. 6.2 (left side). Optimal FAR regulation requires minimum FAR regulation and maximum FAR regulation on both inside and outside the cordon line. In addition to land use regulations and cordon pricing, governments can also adopt the policy of road widening. This chapter also derives the cost-benefit method under the second-best policies. The obtained results are applicable to cities where cordon pricing and land use regulations are simultaneously imposed as in Milan, Oslo, Singapore, and Stockholm. In this study, we did not assume any other means of transport except automobiles. However, congestion is also applicable to bus systems. In addition, many cities have no railroads, and even if they do, railroads are sparsely located within the city. The results of this study are applicable to areas without railroads within cities. Future research could take multiple modes of transport into account.

Technical Appendix 6 Traditionally, urban models have been solved by applying optimal control theory to the continuous area (typically the residential area). The model in this chapter can also be solved with optimal control theory. But one important feature distinct from the traditional urban models is that the model in this chapter is composed of two distinct zones (i.e., residential areas closer to the CBD and residential areas close to the suburb). Hence, we have to combine the two distinct zones, each of which has continuous area, using the boundary conditions between the adjacent zones.

Introducing cordon pricing in a regulated city

159

(1) First-order conditions of the Lagrangian Integrating the Lagrangian function (6.19) by parts, we obtain   ð xc ð xc NV ni ðxÞ xc 0 dx L¼ + ½λi ðxÞTi ðxÞ1  λi ðxÞTi ðxÞdx  λi ðxÞc η 2πxρi ðxÞ 1 1   ð xb ð xb no ðxÞ xb 0 dx + ½λo ðxÞTo ðxÞxc  λo ðxÞTo ðxÞdx  λo ðxÞc 2πxρo ðxÞ xc xc ð xc Fi ðxÞ dx + μi ðxÞ2πx½1  ρi ðxÞ ð ð f y  T i x Þ + ΠN ; V Þ 1 ð xb Fo ðxÞ μo ðxÞ2πx½1  ρo ðxÞ dx + f ðy  To ðxÞ + ΠN ;V Þ xc ð xb ð xc xc 0  μi ðxÞni ðxÞdx + ½μi ðxÞni ðxÞ1  μ0o ðxÞno ðxÞdx + ½μo ðxÞno ðxÞxxbc 1 xc ð xc + ½N ΠN + 2πx½1  ρi ðxÞ½Fi ðxÞr ðy Ti ðxÞ + ΠN ;V Þ SðFi ðxÞÞdx 1 ð xb + 2πx½1  ρo ðxÞ½Fo ðxÞr ðy To ðxÞ+ΠN ; V Þ SðFo ðxÞÞdx + no ðxc Þp xc

+ ϕ½To ðxc Þ  Ti ðxc Þ  p + ψ ½no ðxc Þ  ni ðxc Þ (A6.1) Differentiating Eq. (A6.1) with respective variables, we obtain Eqs. (A6.2)– (A6.16).m We use the following relational expressions: ∂ rj/∂ Tj(x) ¼  1/fj, ∂ rj/∂ ΠN ¼ 1/fj, and ∂ fj/∂ΠN ¼  ∂ fj/∂ Tj, where j 2 {i, o}. For simplicity, r(y  Tj(x) + ΠN, V) and f(y  Tj(x) + ΠN, V) are denoted as rj(x) and fj(x), respectivelyn:    μj ðxÞ ∂S Fj ðxÞ ∂L ¼0: ¼  rj ðxÞ  (A6.2) fj ðxÞ ∂Fj ðxÞ ∂Fj ðxÞ h i F ðxÞ ∂f ðxÞ ∂L j j ¼ 0 : λ0j ðxÞ  μj ðxÞ2πx 1  ρj ðxÞ + n0j ðxÞ ¼ 0 (A6.3) 2 ∂Tj ðxÞ fj ðxÞ ∂Tj ðxÞ Apart from these equations, ∂L/∂ V also holds because V is an endogenous variable. However, we do not use ∂L/∂V to derive propositions in this study. hÐ Ðx ∂f ðxÞ ∂f ðxÞ x n Note that, in Eq. (A6.12), we use ∂Tjj ðxÞ ¼ ∂Πj N and 1 + N1 1 c 2πx½1  ρi ðxÞ FfiiððxxÞÞ dx + xcb 2πx½1  ρo ðxÞ m

Fo ðxÞ fo ðxÞ dx ¼ 0.

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Traffic congestion and land use regulations

∂L ¼ 0 : λi ðxc Þ  ϕ ¼ 0 ∂Ti ðxc Þ

(A6.4)

∂L ¼ 0 : λo ðxc Þ + ϕ ¼ 0 ∂To ðxc Þ

(A6.5)

! nj ðxÞ ∂L 1 0 ¼ 0 : λj ðxÞc  μ0j ðxÞ ¼ 0 ∂nj ðxÞ 2πxρj ðxÞ 2πxρj ðxÞ

(A6.6)

∂L ¼ 0 : μi ðxc Þ  ψ ¼ 0 ∂ni ðxc Þ

(A6.7)

∂L ¼ 0 : μo ðxc Þ + p + ψ ¼ 0, ∂no ðxc Þ

(A6.8)

∂L ¼ 0 : 2πxc ½1  ρi ðxc Þ½Fi ðxc Þri ðxc Þ  SðFi ðxc ÞÞ ∂xc  2πxc ½1  ρo ðxc Þ½Fo ðxc Þro ðxc Þ  SðFo ðxc ÞÞ + n0o ðxc Þp   + ϕ To0 ðxc Þ  Ti0 ðxc Þ + ψ n0o ðxc Þ  n0i ðxc Þ ¼ 0

(A6.9)

∂L ¼ 0 : no ðxc Þ  ϕ ¼ 0 (A6.10) ∂p ð ∂L 1 xc Fi ðxÞ ∂fi ðxÞ μi ðxÞ2πx½1  ρi ðxÞ ¼0: dx ∂ðN Π N Þ N 1 fi ðxÞ2 ∂Ti ðxÞ (A6.11) ð 1 xb Fo ðxÞ ∂fo ðxÞ μ ðxÞ2πx½1  ρo ðxÞ + dx ¼ 0 N x o fo ðxÞ2 ∂To ðxÞ ∂L Fo ðxb Þ ¼ 0 : μo ðxb Þ2πxb ½1  ρo ðxb Þ fo ðxb Þ ∂xb (A6.12) + 2πxb ½1  ρo ðxb Þ½Fo ðxb Þro ðxb Þ  SðFo ðxb ÞÞ ¼ 0 " !# nj ðxÞ ∂L ∂L ¼ 0; λj ðxÞ  0, λj ðxÞ ¼ λj ðxÞ Tj0 ðxÞ  c 0 2πxρj ðxÞ ∂λj ðxÞ ∂λj ðxÞ (A6.13)

h i F ðxÞ ∂L j ¼ 0 : n0j ðxÞ + 2πx 1  ρj ðxÞ ¼0 fj ðxÞ ∂μj ðxÞ

(A6.14)

∂L ¼ To ðxc Þ  Ti ðxc Þ  p ¼ 0 ∂ϕ

(A6.15)

Introducing cordon pricing in a regulated city

∂L ¼ 0 : no ðxc Þ  ni ðxc Þ ¼ 0 ∂ψ

161

(A6.16)

(2) Proof of Proposition 6.1 Arranging the first-order conditions (A6.4), (A6.5), (A6.10), and (A6.16) yields Lemma 6.1. Lemma 6.1 λi ðxc Þ ¼ ni ðxc Þ ¼ λo ðxc Þ ¼ no ðxc Þ ¼ ϕ

(A6.17)

Here, [λj(x)] expresses the shadow price for unit-distance commuting cost at x. In other words, it expresses the marginal change in welfare according to the marginal change in the unit-distance commuting cost. Eq. (A6.17) expresses that the shadow price for the unit-distance commuting cost at the cordon line is equal to the traffic flow at the cordon line because an additional unit of commuting time increases the commuting time for all commuters crossing the cordon line. Integrating both sides of Eq. (A6.3) from x to xc ( j ¼ i) and from x to xb ( j ¼ o) with the use of Eq. (A6.2) and boundary conditions, λo ðxb Þ ¼ 0 and no ðxb Þ ¼ 0, we obtain   ð  h i F ðmÞ ∂f ðmÞ ∂S Fj ðmÞ j j λj ðxÞ ¼ nj ðxÞ + rj ðmÞ  2πm 1  ρj ðmÞ dm fj ðmÞ ∂Tj ðmÞ ∂Fj ðmÞ Xj j 2 fi; og,Xi ¼ ½x; xc ,Xo ¼ ½x;xb 

(A6.18)

The first term on the right-hand side of Eq. (A6.18) is the direct effect caused by the increase in the commuting cost for all commuters crossing location x,   ∂SðFj ðmÞÞ that is, n(x). Note that rj ðmÞ  ∂Fj ðmÞ in the second term is the marginal change in the deadweight loss caused by FAR regulation (see Fig. 6.1). Optimal FAR regulation is affected by the change in commuting cost. In this sense the second term in Eq. (A6.18) implies indirect effect caused by the increase in commuting cost. Note that an increase in the commuting cost in the inner zone x 2 [1, xc] will affect the commuting cost of the residents living in the outer zone, x 2 ½xc , xb , too. However, interestingly, as the integral term of Eq. (A6.18) shows, [λi(x)] includes this indirect effect only in the inner zone Xi ¼ [x, xc] but excludes this indirect effect within the outer zone (note that Xi ¼ [x, xc], i.e., Xi \ ðxc , xb  ¼ ;). This is because cordon pricing is imposed in this model.

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Other studies such as Kanemoto (1977) and Kono et al. (2012), which do not impose cordon pricing, do not include this effect. The mechanism related to this distinct aspect is explained in Technical Appendix 6(5). Next, from Eqs. (A6.7) and (A6.8), we obtain ψ ¼ μi ðxc Þ ¼ μo ðxc Þ  p

(A6.19)

fj ðxc Þ Likewise, from Eq. (A6.2), we get μj ðxc Þ ¼    ∂SðFj ðmÞÞ rj ðmÞ  ∂Fj ðmÞ ð j 2 fi, ogÞ. Substituting this into Eq. (A6.19), we obtain Proposition 6.1. (3) Proof of Proposition 6.2 Using Eqs. (6.16), (A6.19), and (A6.2), the term ψ[no0 (xc)  ni0 (xc)] can be expressed as ψ





n0o ðxc Þ  n0i ðxc Þ

  ∂So ðFo ðxc ÞÞ ¼ 2πxc ½1  ρo ðxc ÞFo ðxc Þ ro ðxc Þ  ∂Fo ðxc Þ   ∂Si ðFi ðxc ÞÞ  2πxc ½1ρi ðxc ÞFi ðxc Þ ri ðxc Þ   n0o ðxc Þp ∂Fi ðxc Þ (A6.20)

Substituting Eq. (A6.20) into Eq. (A6.9) and using Fj(xc)rj(xc)  S(Fj(xc)) ¼ Rj(xc), we obtain 2πxc ½1  ρi ðxc ÞRi ðxc Þ  2πxc ½1  ρo ðxc ÞRo ðxc Þ   ∂So ðFo ðxc ÞÞ + 2πxc ½1  ρo ðxc ÞFo ðxc Þ ro ðxc Þ  ∂Fo ðxc Þ    ∂Si ðFi ðxc ÞÞ  2πxc ½1  ρi ðxc ÞFi ðxc Þ ri ðxc Þ  + ϕ To0 ðxc Þ  Ti0 ðxc Þ ¼ 0 ∂Fi ðxc Þ (A6.21)

Next, we show that ϕ[To0 (xc)  Ti0 (xc)] ¼ 0 when ρi(xc) ¼ ρo(xc). Using

 n ðx Þ Tj0 ðxc Þ ¼ c 2πxjc ρ cðxc Þ , (j 2 {i, o}), we obtain j

     0 no ðxc Þ ni ðxc Þ 0 ϕ To ðxc Þ  Ti ðxc Þ ¼ ϕ c c 2πxc ρo ðxc Þ 2πxc ρi ðxc Þ

(A6.22)

Because no(xc) ¼ ni(xc) and ρo(xc) ¼ ρi(xc), we obtain ϕ[To0 (xc)  Ti0 (xc)] ¼ 0. Using this in Eq. (A6.21) and dividing both sides of the resultant equation

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by 2πxc[1  ρi(xc)] or 2πxc[1  ρo(xc)] (both are equal), we prove Proposition 6.2. (4) Relation between FAR distortions and congestion Rearranging Eq. (A6.6) with Eq. (A6.19) yields Aj  μj0 (x)dx ( j 2 {i, o}), which is the left-hand side of Eq. (A6.23). Next, differentiating Eq. (A6.2) with respect to x yields Bj(x)  μj0 (x)dx ¼ μj(x + dx)  μj(x), which is the right-hand side of Eq. (A6.23). Note that [μj(x + dx)  μj(x)](¼μj0 (x)) implies increase in welfare associated with the migration of one person from location x to location x + dx. Because both Aj and Bj are equal to μj0 (x)dx, Aj ¼ Bj, which is expressed as Eq. (A6.23): ! n ð x Þ 1 j dx  c0 2πxρj ðxÞ 2πxρj ðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} "

∂Tj ðxÞ=∂nj ðxÞ

#   ð  h i F ðmÞ ∂f ðmÞ ∂S Fj ðmÞ j j rj ðmÞ  nj ðxÞ + 2πm 1  ρj ðmÞ dm fj ðmÞ ∂Tj ðmÞ ∂Fj ðmÞ Xj       ∂S Fj ðx + dxÞ ∂S Fj ðxÞ + fj ðxÞ rj ðxÞ  ; ¼ fj ðx + dxÞ rj ðx + dxÞ  ∂Fj ðx + dxÞ ∂Fj ðxÞ j 2 fi; og, Xi ¼ ½x; xc , Xo ¼ ½x; xb 

(A6.23)

Under optimal FAR regulation, Eq. (A6.23) holds. This shows that, by marginally moving residents away from the CBD, the welfare change associated with the deadweight loss in FAR regulation could be canceled with the welfare loss of the increased commuting cost arising from the relocation of residents. More specifically the left-hand side of Eq. (A6.23) expresses the welfare change with respect to an increase in commuting cost caused by the relocation c 0 ðÞ of a resident from x to x + dx (i.e., 2πxρ ðxÞ dx). This is composed of the direct j

effect of congestion externality and the indirect effect on the deadweight loss in the floor area market caused by the change in the per capita floor area beyond x. The first term in the bracket on the left-hand side of Eq. (A6.23) represents the number of residents beyond x. The second term is the change in social cost caused by FAR regulation associated with the change in commuting cost. On the other hand the right-hand side of Eq. (A6.23) can be interpreted as follows. First, if population at location x decreases by one resident, the total floor area Fj(x) decreases by fj(x). Accordingly the deadweight loss in

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  ∂SðFj ðxÞÞ the floor area market decreases by fj ðxÞ rj ðxÞ  ∂Fj ðxÞ (see Fig. 3.2 in Chapter 3). Second, in the same manner, if the population at location x +dx increases by one resident, the deadweight loss in the floor area market   ∂SðFj ðx + dxÞÞ increases by fj ðx + dxÞ rj ðx + dxÞ  ∂Fj ðx + dxÞ . To sum up the welfare     ∂SðFj ðx + dxÞÞ ∂SðFj ðxÞÞ change is fj ðx + dxÞ rj ðx + dxÞ  ∂Fj ðx + dxÞ + fj ðxÞ rj ðxÞ  ∂Fj ðxÞ . (5) Proof of μj(x) motion illustrated in Fig. 6.2 Substituting x ¼ xc into Eq. (A6.18) ( j ¼ o) and using λo(xc) ¼ no(xc) in Lemma 6.1 and (A6.2), we obtain ð xb xc

μo ðxÞ Fo ðxÞ ∂fo ðxÞ 2πx½1  ρo ðxÞ dx ¼ 0 fo ðxÞ fo ðxÞ ∂To ðxÞ

(A6.24)

Next, arranging Eq. (A6.11) with Eq. (A6.24) yields ð xc 1

μi ðxÞ Fi ðxÞ ∂fi ðxÞ 2πx½1  ρi ðxÞ dx ¼ 0 fi ðxÞ fi ðxÞ ∂Ti ðxÞ

(A6.25)

Using Eqs. (A6.24) and (A6.25), μi(x) and μo(x) have only two patterns of solution, because Fj(x)/fj(x) > 0 and ∂ fj/∂Tj > 0. In Pattern 1, μi(x) should be positive for some x 2 [1, xc) and negative for other x 2 [1, xc). Moreover, μo(x) should be positive for some x 2 ðxc ; xb  and negative for other x 2 ðxc ;xb . In Pattern 2, μj(x) is always zero anywhere in the city. Now, we recall that λj > 0 holds,o as shown in Eq. (A6.13). Using Eq. (A6.6), we obtain μj0 (x) < 0. Therefore only Pattern 1 holds. Moreover, Eqs. (A6.7), (A6.8) yield μi(xc) ¼ μo(xc)  p. These discussions lead to Lemma 6.2. The motion of μj(x) described in Lemma 6.2 is illustrated in Fig. 6.2. Lemma 6.2 (1) There exists x^1 in (1, xc ) such that μi(x) > 0 for any x 2 ½1; x^1 Þ and μi(x) < 0 for any x 2 ðx^1 , xc Þ and μi ðx^1 Þ ¼ 0, (2) there exists x^2 in ðxc ; xb Þ such that μo(x) > 0 for any x 2 ðxc , x^2 Þ and μo(x) < 0 for any x 2 ðx^2 ;xb  and μo ðx^2 Þ ¼ 0, and (3) μi(xc) ¼ μo(xc)  p. o

λj > 0 holds at any x, as shown in Eq. (A6.13). Intuitively, considering λj(x) ¼ ∂ W/∂Tj(x), λj should be positive because an increase in the unit-distance commuting cost decreases welfare.

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Introducing cordon pricing in a regulated city

We now interpret Eqs. (A6.24) and (A6.25). To interpret Eq. (A6.24), we suppose that at the optimal toll level, the government increases the toll by dp. The increase in the toll increases the commuting cost for all no(xc) commuters who reside outside the cordon line and changes the distortion in the floor market. Because commuting trips are inelastic, the traffic congestion distortion for no(xc) commuters does not change. At the optimal level the change in congestion distortion caused by the change in toll, which is zero as shown in the right-hand side of Eq. (A6.24), should be canceled by the change in distortion in the floor market, which is the left-hand side of Eq. (A6.24). Next, as expressed in Eq. (A6.11), any change in total distortion in the floor area market over the entire city, which is caused by a change in ΠN (the lump-sum revenue distributed to all residents), is zero at the optimal condition. Note that ΠN affects housing demand through an income effect, which also changes the distortion. Reducing Eq. (A6.24) from Eq. (A6.11) shows that the total distortion in the floor area market inside the cordon line is zero, as expressed in Eq. (A6.25). The earlier discussion intuitively shows why Eqs. (A6.24) and (A6.25) hold. Next, μj0 (x) < 0 as shown earlier. This implies that each side of Eq. (A6.23) is negative. As the earlier proof shows, combining μj0 (x) < 0 with Eqs. (A6.24) and (A6.25) yields Lemma 6.2, which directly implies that minimum FAR regulation and maximum FAR regulation are required both inside and outside the cordon line as shown in Proposition 6.3. (6) Proof of Proposition 6.4 Substituting Fj(xc)rj(xc)  S(Fj(xc)) ¼ Rj(x) into Eq. (A6.18) and rearranging yield ∂Si ðFi ðxc ÞÞ ∂So ðFo ðxc ÞÞ  Si ðFi ðxc ÞÞ ¼ Fo ðxc Þ  So ðFo ðxc ÞÞ (A6.26) Fi ðxc Þ ∂Fi ðxc Þ ∂Fo ðxc Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ΛðFi Þ

ΛðFo Þ

Next, we define the left-hand side of Eq. (A6.26) as Λ(Fi) and the right-hand side as Λ(Fo). Differentiating Λ(Fj) with regard to Fj ( j 2 {i, o}), we have   ∂ΛðFj Þ ∂2 Sj ðFj ðxc ÞÞ . Because S00 (Fj) > 0, Λ(Fj) is a monotonically ∂Fj ¼ Fj ðxc Þ ∂F ðx Þ2 j

c

increasing function. Therefore, only if Fi(xc) is equal to Fo(xc), Λ(Fi) ¼ Λ(Fo) can hold. This implies Fi(xc) ¼ Fo(xc), which leads to Proposition 6.4.

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References Anas, A., Hiramatsu, T., 2013. The economics of cordon tolling: general equilibrium and welfare analysis. Econ. Transp. 2 (1), 18–37. Brueckner, J.K., 2007. Urban growth boundaries: an effective second-best remedy for unpriced traffic congestion? J. Hous. Econ. 16 (3–4), 263–273. De Palma, A., Lindsey, R., Niskanen, E., 2006. Policy insights from the urban road pricing case studies. Transp. Policy 13 (2), 149–161. Kanemoto, Y., 1977. Cost-benefit analysis and the second best land use for transportation. J. Urban Econ. 4 (4), 483–503. Kono, T., Kawaguchi, H., 2017. Cordon pricing and land-use regulation. Scand. J. Econ. 119 (2), 405–434. Kono, T., Joshi, K.K., Kato, T., Yokoi, T., 2012. Optimal regulation on building size and city boundary: an effective second-best remedy for traffic congestion externality. Reg. Sci. Urban Econ. 42 (4), 619–630. Kraus, M., 1989. The welfare gains from pricing road congestion using automatic vehicle identification and on-vehicle meters. J. Urban Econ. 25 (3), 261–281. Li, Z.-C., Wang, Y.-D., Lam, W.H.K., Sumalee, A., Choi, K., 2014. Design of sustainable cordon toll pricing schemes in a monocentric city. Netw. Spat. Econ. 14 (2), 133–158. Lindsey, R., Verhoef, E., 2001. Traffic congestion and congestion pricing. In: Button, K.J., Hensher, D.A. (Eds.), Handbook of Transport Systems and Traffic Control. Elsevier Science, Pergamon, pp. 77–105. Mun, S., Konishi, K., Yoshikawa, K., 2003. Optimal cordon pricing. J. Urban Econ. 54 (1), 21–38. Pines, D., Kono, T., 2012. FAR regulations and unpriced transport congestion. Reg. Sci. Urban Econ. 42 (6), 931–937. Sullivan, A.M., 1983. The general equilibrium effects of congestion externalities. J. Urban Econ. 14 (1), 80–104. Tikoudis, I., Verhoef, E.T., van Ommeren, J.N., 2018. Second-best urban tolls in a monocentric city with housing market regulations. Transp. Res. B Methodol. 117, 342–359. Part A. Verhoef, E.T., 2005. Second-best congestion pricing schemes in the monocentric city. J. Urban Econ. 58 (3), 367–388.

Further reading Kono, T., Notoya, H., 2012. Is mandatory project evaluation always appropriate? Dynamic inconsistencies of irreversible and reversible projects. J. Benefit Cost Anal. 3 (1), 1–31. Kono, T., Kitamura, N., Yamasaki, K., Iwakami, K., 2013. Quantitative analysis of dynamic inconsistencies in disaster prevention infrastructure improvement: an example of coastal levee improvement in the city of Rikuzentakata. Environ. Plan. B Urban Anal. City Sci. 43 (2), 401–418.

CHAPTER SEVEN

Road investment evaluation under land use regulation

7.1 Introduction Up to the preceding chapter, we explored how to design optimal density and zonal regulations for efficient traffic flow in cities. This chapter focuses on road investment.a In congested cities, not only land use policy but also road investment can reduce congestion. But road investment requires a large area of land, which could otherwise be used for residential use. Hochman (1975) studies how much of the land should be allocated for roads in cities with congestion tolls. Akai et al. (1988) show that selfish, privately owned, profit-maximizing commuting companies end up charging the price exactly equivalent to the optimal toll and that land can be efficiently allocated for commuting. However, introducing the first-best congestion tolls is practically difficult, if not impossible. This chapter therefore explores how much of the land should be allocated for roads in a city without congestion tolls. The optimal land use for roads has been studied by Kanemoto (1977) and Arnott (1979). Kanemoto (1977) examines the cost-benefit analysis in a second-best situation where congestion is unpriced. Kanemoto (1977) concludes that the naı¨ve cost-benefit analysis based on market land rent tends to support overinvestment in roads at the center of the city because of the distorted market prices due to traffic congestion. Whether this tendency holds at the boundary or not depends on the price elasticity. If the compensated demand for housing has a price elasticity less than 1, there will be an underinvestment in roads near the boundary. To mitigate congestion, local governments around the world impose density regulations that restrict lot size, building size or height, and/or city a

Some part of this chapter is based on studies with Takayuki Kaneko and Kei Matsushita.

Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00007-8

© 2019 Elsevier Inc. All rights reserved.

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size, as discussed in the preceding chapters, in addition to road investment. Against traffic congestion, Pines and Sadka (1985) and Wheaton (1998) explore lot size (LS) regulation and conclude that regulating lot size achieves a result similar to congestion pricing, that is, a first-best policy. However, LS regulation, which controls only the size of each lot, is not always an effective tool because even a small lot can have a high-rise building that accommodates several households (e.g., condominium). In particular, congestion is caused by high-rise buildings, which are regulated by floor area ratio (FAR) regulation rather than LS regulation. Moreover, as the preceding chapters have demonstrated, although LS regulation and FAR regulation both attempt to mitigate population density-related externalities, they have distinct characteristics. In particular, FAR regulation is at most a second-best policy, whereas, as stated earlier, optimal LS regulation in a city composed only of detached houses can be a first-best policy. FAR regulation mitigates congestion externality but instead generates deadweight losses in the floor area market. In addition to such density regulations, policymakers invest in roads. So, we have to explore a method of cost-benefit analysis for road investment in a city with FAR regulation because FAR regulation is at most the second best. We will focus on the FAR regulation and road investment in this chapter because density regulation and road investment are important policies to mitigate congestion.b We need to examine cities with and without FAR regulations. In cities without FAR regulation, FAR is determined by market equilibrium. In summary, this chapter aims to find how to evaluate road investments in the following three regimes: (1) cities with optimal FAR regulation, (2) cities with fixed FAR, and (3) cities with FAR determined in a competitive market. The remainder of this chapter is organized as follows. Section 7.2 develops a monocentric city model with high-rise buildings. Section 7.3 sets the welfare function and three regimes. Section 7.4 examines the optimal FAR regulation based on the welfare function. Section 7.5 finds changes in FAR according to road investment in each regime. Section 7.6 explores a method of cost-benefit analysis for road investment in a closed city, and Section 7.7 explores the same in an open city. Section 7.8 concludes this chapter followed by Technical Appendix.

b

FAR regulation can act as a UGB regulation by reducing building size to zero.

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169

7.2 The model 7.2.1 The city We adopt the standard model of a congested monocentric city, which is adopted by the preceding chapters. But we suppose a city composed of multiple discrete zones, rather than a continuous city for two reasons. First, road investment is usually carried out in a discrete way, and second, it is easy to see how we should consider price distortions in our analysis. The city is circular and is symmetric along any radial axis. The residential area expands from x ¼ 1 at the CBD edge to x ¼ xb at the city boundary. The city is divided into several concentric rings of equal width; the total floor supply per unit area of land at x, that is, on xth ring, is denoted F(x), which represents FAR at x.c At each distance a constant fraction ρ of the land is used for roadd; the share of land available for housing at x from the CBD edge is given by 2π[1  ρ]x. The model has four agents: residents, city government, land owners, and developers. The city government can impose FAR regulation on all buildings to control traffic congestion externality. The developers rent land from landowners and build dwelling units in the residential area under the regulated FAR. The real urban policy imposes maximum FAR (and minimum FAR in some cities). In this chapter the government sets the FAR per building (total floor area) as an equality constraint such that if the optimum FAR is less (resp. greater) than the market FAR, then the government should impose maximum (resp. minimum) FAR regulation.e

7.2.2 Household behavior The city is inhabited by N identical households. For simplicity, population is identified as the number of households. Although a household can choose any location within the city to reside, only one residence is chosen. Residents have a quasilinear function V ¼ u( f ) + z, where subutility u( f ) is a function of f which expresses housing square footage, and numeraire composite goods z that include all consumer goods except floor space. c

For a sufficiently large city radius in comparison with the lot size of a building, the building size (or FAR) can be treated as a continuous function of distance. d This assumption follows Pines and Sadka (1985), Wheaton (1998), and Brueckner (2007). e A detailed discussion will follow later in this chapter using Fig. 7.1.

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Each household earns income y per period and pays tax G, defined as Pb G ¼ N1 xx¼1 2πxρðxÞRðxÞ, to the city government, where R(x) is land rent. Tax is used to buy land for roads from the landowners. A household rents floor space from the developers at the price of r per square foot of floor space per period. The commuting cost to and from the CBD for a household residing at x is denoted by T(x). The income constraint is, therefore, expressed as y ¼ r(x)f(x) + z(x) + T(x) + G. Because many households can reside in a building, floor rent r equals maximum floor rent bid by a household as a result of competition among residents. Maximizing r with respect to f gives the optimal dwelling size. The bid floor rent function at x, denoted r(V(x), T(x)), which comprises utility and commuting costs, is given by   1 r ðV ðxÞ, T ðxÞÞ  max ½y + uðf ðxÞÞ  T ðxÞ  V ðxÞ  G (7.1) f ðxÞ f ðxÞ

7.2.3 Landowners’ benefit Landowners rent land either to the city government to make roads or to the developers to build residential buildings, whoever offers them the highest price. The landowners’ profit is expressed by Xxb Xxb Xxb Ω¼ 2πxρ ð x ÞR ð x Þ + 2πx ½ 1  ρ ð x Þ R ð x Þ ¼ 2πxRðxÞ x¼1 x¼1 x¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼

Xxb

¼NG

2πx½F ðxÞr ðxÞ  SðF ðxÞÞ x¼1

(7.2)

The last equality is explained in the next section.

7.2.4 Developers’ behavior Developers combine housing capital (building materials) and land to produce residential buildings under prevailing FAR regulations. Developers are assumed to be perfectly competitive and are, therefore, price-takers.f Floor space supply per unit of land at each location x is denoted by F(x), which also expresses FAR at that location. The net profit from floor supply is given by Fr  S(F), where S(F) expresses construction cost of F floor space. f

As a result, they can be treated as an aggregate developer who maximizes profit from building construction.

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171

Thereby, Π in Eq. (7.3) sums developers’ net profit from total floor space supply in the city: Xxb 2πx½1  ρðxÞ½F ðxÞr ðxÞ  SðF ðxÞÞ  RðxÞ: (7.3) Π¼ x¼1 The profit of the developers will be zero because of perfect competition. Therefore, R(x) can be derived from Eq. (7.3) as Eq. (7.2).

7.2.5 Commuting cost: The external factor For simplicity, we assume that automobiles are the only mode of commuting and that only one radial route extends across the residential zone between the CBD and the urban boundary. Also, commuting cost is incurred only when commuting to and from the CBD boundary.g Because vehicles from more distant locations increasingly join traffic while moving towards the CBD, traffic volume is not uniform across all locations but increases towards the central locations. So, under congestion conditions, the unit cost of commuting for a commuter at x depends on the traffic flow ^ ðxÞ denote the number of households residing beyond across ring x.h Let N Pb ^ ðxÞ  xm¼x ring x, which is given as N 2πmð1  ρðmÞÞnðmÞ. Then, assuming ^ ðxÞ that one member of each household commutes to the CBD to work, N represents the number of commuters passing through ring x. We adopt the following commonly used functional form of the commuting cost to cross ring x as used by Brueckner (2007):   ^ ðxÞ=½2πxρ γ , where η, δ, γ are positive parameters. Tunit ðxÞ ¼ η + δ N   ^ ðxÞ=½2πxρ γ denotes congestion externality, where The term δ N ^ ðxÞ=½2πxρ is the “traffic volume/road capacity” ratio at ring x. The comN muting cost for residents at x is, then, given by Xx Xx   ^ ðmÞ=½2πρm γ T ðxÞ ¼ T ð m Þ ¼ ηx + δ N (7.4) unit m¼1 m¼1 When an additional commuter joins traffic flow at x, the resultant change in ^ ðxÞ. Multiplying this by N ^ ðxÞ gives congestion cost is given by dT unit ðxÞ=d N the total externality caused by unpriced congestion, which is expressed by g h

Commuting costs within the CBD are assumed to be unchanged by regulations in the residential zone. This setting on commuting cost is identical to that adopted by Wheaton (1998) or Brueckner (2007). A different setting on commuting cost is possible. However, results denoted in propositions, which appear later in this chapter, would not be fundamentally different.

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  ^ ðxÞ dT unit ðxÞ ¼ γδ N ^ ðxÞ=½2πxρ γ  τðxÞ N ^ ðxÞ dN

(7.5)

where τ(x) equals congestion toll at band x, which fully internalizes congestion externality.

7.2.6 Market equilibrium The market equilibrium conditions are as follows. First the total population N is accommodated within the urban boundary, yielding Eq. (7.6). Next, because households can choose a residential location costlessly, the household utility is the same across all locations, which is denoted by V in Eq. (7.7). Finally, as shown in Eq. (7.8), in a unit land area, the total floor space supplied by developers at a location (left side) equals the total floor space consumed by households at that location (right side). Xxb 2πx½1  ρðxÞnðxÞ ¼ N (7.6) x¼1 V ðxÞ ¼ V

(7.7)

F ðxÞ ¼ nðxÞf ðxÞ

(7.8)

7.3 Welfare functions of FAR and UGB regulations and city regimes We adopt a Benthamite social welfare function, which sums up the total household utility and developers’ net profit in a closed city, as given in Eq. (7.9), where W is total social welfare. The welfare level W depends on the FAR and the road investment, which is expressed as a function of floor area ratio from 1 to xb, and the fraction of road ρ as given by Xxb W 2πx½1  ρðxÞnðxÞV + Ω (7.9) x¼1 The social welfare depends on FAR regulations and road investment. So the function can be expressed as   W ¼W F , ρ (7.10) |{z} |{z} FAR Road regulation investment

where F ¼ F ðxÞ 8x 2 ð1, xb Þ, ρ ¼ ρðxÞ 8x 2 ð1,xb Þ.

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Road investment evaluation under land use regulation

We will next discuss the cost-benefit analysis of road investment in a closed city and an open city.i As defined in Chapter 3, a closed city is one whose population does not change in response to the population density regulation. An example is Silicon Valley, which mostly attracts skilled ICT workers, and as such the total population is not affected much by the density regulations. Examples of open cities are towns primarily consisting of residences for workers employed elsewhere, such as “bed towns” in Japan as discussed in Chapter 3. We examine three regimes of FAR regulation: Regime 1—cities with optimal FAR regulation. Regime 2—cities with fixed FAR. Regime 3—cities with no FAR regulation. In Regime 1, the city planner changes the FAR regulation according to the change in road investment, and the regulation is always at optimal FAR. In Regime 2, the city planner first sets FAR at some value (any value including the optimal level at the initial stage) and does not change the regulation in spite of change in road investment. In Regime 3, there is no FAR regulation, and the FAR is decided by market equilibrium. These three regimes represent typical practical situations of cities. These three regimes are graphically shown in three dimensions in Fig. 7.1 in which one axis represents road widening, the second axis represents FAR, and the axis perpendicular to the paper plane represents the social welfare. This graph only shows the conceptual relationship between them. Needless to say, we cannot represent FAR regulations or road widening as one axis because they are a set of variables. But using this graph the three FAR F Optimal regulation Regime 3 Regime 1 Regime 2

Welfare

Road project

Road widening,

Fig. 7.1 Three-dimensional conceptual representation of the three regimes. i

See Chapter 3 for further discussion on the closed and open city models.

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Traffic congestion and land use regulations

regimes are concisely explained as follows. Given a road width, the optimal FAR can be defined to maximize the social welfare. In Regime 1, the change according to the change in road width is along the optimal FAR path. In Regime 2, the FAR is fixed even if the road width changes. In Regime 3, market mechanisms determine the FAR, so probably in most cases the FAR increases greater than that in Regime 1.

7.4 Optimal FAR The optimal FAR regulation without UGB regulation is defined as follows. The optimal FAR regulation at x 2 (1, xb) requires FAR F∗(x) such that F ∗ ðxÞ ¼ arg max W ðF, ρÞ s.t. (7.3)–(7.10), where F ¼ F ðxÞ 8x 2 ð1,xb Þ, ρ ¼ ρðxÞ 8x 2 ð1, xb Þ. The absence of UGB regulation implies that xb is determined endogenously in the market.j Based on the definition earlier, the optimal FAR regulation (i.e., the optimal floor space) satisfies the conventional first-order condition: dW ðF,ρÞ=dF ðxÞ ¼ 0 at any x. This optimal condition implies that if the FAR (or regulated floor space) of buildings at all locations is optimal, any infinitesimal deviation from the regulated FAR at each location x, that is, F(x), will not change the social welfare W. Applying the first-order condition yields the following optimal condition: dW ðF, ρÞ ¼ dF ðxÞ

E ðF ðxÞÞ |fflfflfflffl{zfflfflfflffl}

Change in total congestion externality

+

DðxÞ |ffl{zffl}

¼ 0 8x 2 ð1xb Þ

(7.11)

Change in deadweight loss in floor space market at x

Pb where E(F(x))  xm¼1 2πm[1  ρ(m)]n(m)[dT(m)/dF(x)] and DðxÞ  2πx ½1  ρðxÞnðxÞ½r ðxÞ  ∂SðF ðxÞÞ=∂F ðxÞ, where ∂SðF ðxÞÞ=∂F ðxÞ denotes marginal construction cost of floor space. See Appendix 7(1) for the derivation of Eq. (7.11). Using Eqs. (7.4), (7.5), the effect of a marginal increase in F(x) on the commuting cost T(m) at any location m is expressed by ^ ðνÞ Xm τðνÞ d N ^ ðνÞ dT ðmÞ Xm dT unit ðνÞ dN ¼ ¼ ν¼1 d N ν¼1 N ^ ðνÞ dF ðxÞ ^ ðνÞ dF ðxÞ dF ðxÞ j

(7.12)

This definition implies that the regulation targets only one city, not the whole system of cities. In most countries (e.g., the United States, EU countries, and Japan), urban planning, which is carried out by local governments, targets only their own city. Hence, we adopt this definition. However, if urban planning is carried out from the whole nation’s viewpoint, we should target the system of cities.

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Road investment evaluation under land use regulation

^ ðνÞ is the congestion externality per comwhere recalling Eq. (7.4), τðνÞ=N muter at location ν. Therefore E(F(x)) in Eq. (7.11) denotes “change in the total congestion externality,” which is the same as population-weighted total of the change in commuting costs across the cityk induced by a change in FAR at ring x. On the other hand, D(x) represents total marginal deadweight loss (which is the difference between rent and marginal cost of floor space) in the floor space market at ring x, resulting from the FAR regulation.l The optimal condition of FAR regulation is expressed by Eqs. (7.11) and (7.12). Accordingly, even under the optimal FAR regulation, deadweight loss exists in the regulated FAR market because the change in congestion externality according to the regulation is not zero as long as congestion exists. The existence of deadweight loss implies that FAR regulation is a second-best policy because of the indirect adjustment of population density. This is because FAR regulation can only control total floor space of a building without controlling the per capita floor space consumption directly.

7.5 Change in FAR according to road investment in each regime In our context the welfare depends on FAR regulation and road investment, expressed as F and ρ, respectively. Optimal FAR regulation F is a function of ρ in any of the three regimes. The reason is that, when the road investment changes, the traffic condition changes. In Regime 1, if the FAR regulation is set to be optimal, the FAR regulation must change due to change in traffic congestion. In Regime 2, even though optimal FAR changes according to the change in ρ, the FAR is fixed. In Regime 3, because there is no FAR regulation, the developers will change the FAR according to the change in ρ because they can obtain more profits. However, we can regard FAR regulation F as a function of ρ in Regimes 1–3. When F is a function of ρ, the effect of the road investment on the social welfare can be expressed as dW ðF, ρÞ Xxb ∂W dF ðmÞ ∂W ¼ + m¼1 ∂F ðmÞ dρðxÞ dρðxÞ ∂ρðxÞ k

8x 2 ð1,xb Þ

(7.13)

E() total commuting cost”; the latter is expressed by Pxin Pxb is not to be confused with “the change b ν¼12π[d(n(ν)T(ν))/dF(x)]ν ¼ E(F(x)) + ν¼12πT(ν)[dn(ν)/dF(x)]ν. l A condition similar to Eq. (7.12) is shown in a two-zone model in Chapter 2 that considers intrazonal population externality, whereas this chapter analyzes congestion externality, which has interzonal effect.

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Traffic congestion and land use regulations

The first term on the right-hand side, [∂ W/∂ F(m)][dF(m)/dρ(x)], expresses partial change in welfare with respect to change in FAR caused by road investment at x. The second term expresses partial change in welfare with respect to change in the road investment at ring x. Eq. (7.13) holds in Regimes 1–3, but the following conditions apply in the corresponding regimes: In Regime 1, ∂ W/∂ F(m) ¼ 0 because FAR is optimal. In Regime 2, ∂ F(m)/∂ ρ(x) ¼ 0 because FAR is fixed. In Regime 3, r(x)  ∂ S(F(x))/∂ F(x) ¼ 0 because the FAR is not regulated and the developers maximize their profit to change the FAR.

7.6 Cost-benefit analysis of road investment in a closed city 7.6.1 Social welfare This section focuses on a closed city. In a closed city the welfare is expressed by the residents’ utility and the landowners’ profit: Xxb W¼ 2πx½1  ρðxÞnðxÞV ðxÞ + Ω x¼1 The effect on the welfare of a road investment can be expressed as follows (see Appendix 7(2) for derivation): Xxb dW ðF, ρÞ ∂T ðmÞ ∂F ðmÞ ¼  m¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂F ðmÞ ∂ρðxÞ

Xxb ∂SðF ðxÞÞ ∂F ðmÞ 2πx½1  ρðxÞ r ðxÞ  + m¼1 ∂F ðxÞ ∂ρðxÞ Xxb ∂T ðmÞ 2πm½1  ρðmÞnðmÞ  2πx½F ðxÞr ðxÞ  m¼1 ∂ρðxÞ

(7.14)

 SðF ðxÞÞ 8x 2 ð1,xb Þ The first term expresses the change in total commuting cost due to change in FAR caused by the change in road investment at x. The second term expresses the effect of the change in road investment on deadweight loss in the floor space market. The third term expresses the change in total commuting cost caused by the change in road investment at x. The last term expresses the opportunity cost of land used in road investment. The road expansion and the change in FAR change the endogenous utility level. However, Eq. (7.14) does not include the terms related

Road investment evaluation under land use regulation

177

to the utility. Actually the term expressing the residents’ utility is cancelled out by the term in the landowners’ profit. The reason is, as the utility increases (resp. decreases), the profit of the landowners decreases (resp. increases). These effects in both sides are cancelled out. See Eqs. (A7.9), (A7.15) in Appendix 7(2) for a detailed derivation.

7.6.2 Closed city with optimal FAR regulation The first-order condition of the optimal FAR is ∂W/∂ F ¼ 0. Then, using Eq. (7.14), we can express the effect on the social welfare induced by the road investment as in Proposition 7.1, which shows that the formula of cost-benefit analysis can use the market land rent. Proposition 7.1 (Cost-benefit analysis in a closed city with optimal FAR regulation) The cost-benefit analysis in Regime 1 in a closed city is expressed by Eq. (7.15): Xxb dW ðF,ρÞ ∂T ðmÞ ¼  m¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in road investment

 2πx½F ðxÞr ðxÞ  SðF ðxÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(7.15)

Opportunity cost of land used for road widening

Eq. (7.15) shows that the welfare change associated with road investment is composed of two terms. The first term expresses the change in total commuting cost caused by the change in road investment at ring x. The second term expresses opportunity cost of land used for road widening. Unlike Kanemoto (1977), this formula implies that the cost-benefit analysis based on the market land rent has no bias even in the existence of congestion. In Kanemoto (1977) the cost-benefit formula cannot use the market price of land rent because of the existence of distortion caused by congestion externality. However, in Regime 1, the change in distortion caused by change in total congestion externality is cancelled out by the change in deadweight loss caused by FAR. This cancelling-out holds regardless of any changes in road investment. This is why we can use the market land rent in the cost-benefit analysis in a closed city with optimal FAR regulation.

178

Traffic congestion and land use regulations

7.6.3 Closed city with fixed FAR The FAR being fixed means it will not be affected by the change in road investment, that is, ∂ F/∂ ρ ¼ 0. Then, using Eq. (7.14), we can rewrite the effect on social welfare induced by road investment as in the succeeding text, which shows that the formula of cost-benefit analysis can use the market land rent. Proposition 7.2 (Cost-benefit analysis in a closed city with fixed FAR) The cost-benefit analysis in Regime 2 in a closed city is expressed by Eq. (7.16): Xxb dW ðF,ρÞ ∂T ðmÞ ¼  x¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in road investment

 2πx½F ðxÞr ðxÞ  SðF ðxÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(7.16)

Opportunity cost of land used for road widening

This equation is exactly the same as Eq. (7.15). However, there is an economic reason why we obtain an identical formula. In Eq. (7.16) the change in road investment does not affect the FAR, which means it does not affect the deadweight loss caused by FAR regulation. This can be explained by the Harberger welfare formula, which is expressed as dW/dΞ ¼ direct affect P + i[ri  ∂ Si/∂ Xi][∂ Xi/∂Ξ] where [ri  ∂ Si/∂ Xi] is the distortion (e.g., the price minus the marginal cost) in market i; X is the output in market i and Ξ is the policy variable.m In Eq. (7.16) the distortion [ri  ∂ Si/∂ Xi] is the deadweight loss in the floor area market. The policy Ξ implies change in road investment. The change in output associated with the policy change, ∂Xi/∂Ξ, is the change in FAR due to change in road investment, that is, ∂ F/∂ ρ. This is the reason why we can use the market land rent in the cost-benefit analysis. In addition, although Eq. (7.16) is exactly the same as Eq. (7.15), the value of each term is different because the situations of FAR differ so that traffic volumes differ between the two regimes.

7.6.4 Closed city with no FAR regulation Under no FAR regulation, developers maximize profit per unit of land with respect to F(x). Solving the relevant first-order condition for F(x) provides r  ∂ S/∂ F ¼ 0. Using this equation and Eq. (7.14), we can rewrite the effect on the m

See supplementary note in Chapter 2 for more details.

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Road investment evaluation under land use regulation

social welfare of road investment as in the succeeding text, which shows that the formula of cost-benefit analysis can use the market land rent. Proposition 7.3 (Cost-benefit analysis in a closed city with no FAR regulation) The cost-benefit analysis in Regime 3 in closed city is expressed by Eq. (7.18): Xxb dW ðF,ρÞ ∂T ðmÞ ∂F ðmÞ 2πm½1  ρðmÞnðmÞ ¼ x¼1 dρðxÞ ∂F ðmÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in FAR caused by road investment

Xxb ∂T ðmÞ  x¼1 2πm½1  ρðmÞnðmÞ  2πx½F ðxÞr ðxÞ  SðF ðxÞÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in road investment

Opportunity cost of land used for road widening

(7.17) The first term expresses the change in total commuting cost due to change in FAR at ring x.The others are the same as in Eqs. (7.15), (7.16). This is also expressed by the market land rent. Without FAR the distortion does not exist because there is no regulation on FAR and the developers take the best price for their benefit. This is why we can use the market land rent in a closed city with no FAR regulation. However, the range of the demand forecast differs from Eq. (7.16) because of the change in total commuting cost due to change in FAR at ring x caused by road investment.

7.6.5 Comparing the three regimes of the closed city The cost-benefit formulae in the three regimes are summarized as follows: Optimal FAR regulation: Xxb dW ðF,ρÞ ∂T ðmÞ ¼  x¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in road investment

 2πx½F ðxÞr ðxÞ  SðF ðxÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Opportunity cost of land used for road widening

(7.18)

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Traffic congestion and land use regulations

Fixed FAR: Xxb dW ðF,ρÞ ∂T ðmÞ ¼  m¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in road investment

 2πx½F ðxÞr ðxÞ  SðF ðxÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(7.19)

Opportunity cost of land used for road widening

No FAR regulation: Xxb dW ðF,ρÞ ∂T ðmÞ ∂F ðmÞ ¼  x¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂F ðmÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in FAR caused by road investment



Xxb

∂T ðmÞ 2πm ½ 1  ρ ð m Þ n ð m Þ  2πx½F ðxÞr ðxÞ  SðF ðxÞÞ x¼1 ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in road investment

Opportunity cost of land used for road widening

(7.20) In Regime 1, deadweight loss exists in the regulated FAR market because the change in congestion externality according to the regulation is not zero as long as congestion exists (let F∗ in Fig. 7.2 denote the optimal FAR). However, dW/dF ¼ 0 means that the deadweight loss is offset by the change in total commuting cost due to change in FAR at that location. This means that we do not have to think about the distortion and only have to see the direct effect on the social welfare by the partial change in road investment. In Regime 2, the FAR is fixed, which means that the change in road investment does not affect the FAR, that is, ∂F/∂ρ ¼ 0. This means that there will be no need to consider the effect on the welfare caused by the change in FAR due to road investment (let F f in Fig. 7.2 denote the fixed FAR; r  ∂S/∂F will not change even when ρ changes). This also means that we only have to see the direct effect on the social welfare caused by the change in road investment.

Road investment evaluation under land use regulation

181

Fig. 7.2 Floor rent and marginal cost curve.

In Regime 3, r  ∂S/∂F ¼ 0 means that there is no distortion in the floor market (let FM in Fig. 7.2 denote the market equilibrium FAR). This means we have to see the effect on the total commuting cost caused by the change in FAR due to road investment. Note that all the cost-benefit formulae are expressed by using the market land rent.

7.7 Cost-benefit analysis of road investment in an open city 7.7.1 Social welfare This section explores the cost-benefit analysis of road investment in an open city. In an open city the social welfare is expressed by the landowners’ profit: W ¼Ω which is different from the social welfare in a closed city. The effect of road investment on the social welfare can be expressed as in the succeeding text: Xxb dW ðF,ρÞ ∂T ðmÞ ∂F ðmÞ ¼  m¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂F ðmÞ ∂ρð xÞ Xxb ∂SðF ðxÞÞ ∂F ðmÞ 2πx½1  ρðxÞ r ðxÞ  + m¼1 ∂F ðxÞ ∂ρðxÞ Xxb ∂T ðmÞ 2πm½1  ρðmÞnðmÞ  m¼1 ∂ρðxÞ  2πx½F ðxÞr ðxÞ  SðF ðxÞÞ 8x 2 ð1,xb Þ

(7.21)

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This equation is exactly the same as Eq. (7.14). The reason why the equations are the same is that the term expressing the effect on residents’ utility is cancelled out in Eq. (7.14). This means that the equation will not be affected even if the utility is fixed when the city is small and open.

7.7.2 Open city with optimal FAR regulation The basic step for deriving the cost-benefit analysis method is the same as in Section 7.6. We use the equation dW/dF ¼ 0. By using this Eq. (7.21), we derive the equation in the succeeding text, which shows that the formula of cost-benefit analysis can use the market land rent. Proposition 7.4 (Cost-benefit analysis in an open city with optimal FAR regulation) The cost-benefit analysis in Regime 1 in an open city is expressed by Eq. (7.22). This is exactly the same as Proposition 7.1: Xxb dW ðF,ρÞ ∂T ðmÞ ¼  m¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in road investment

 2πx½F ðxÞr ðxÞ  SðF ðxÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(7.22)

Opportunity cost of land used for road widening

The equation and the meaning of each term are the same as Eq. (7.15).

7.7.3 Open city with fixed FAR We adopt the same process as in Section 7.6.3. Using the equation ∂ F/∂ ρ ¼ 0 and Eq. (7.21), we can rewrite the effect on the social welfare induced by road investment as in the succeeding text, which shows that the formula of cost-benefit analysis can use the market land rent. Proposition 7.5 (Cost-benefit analysis in an open city with fixed FAR) The cost-benefit analysis in Regime 2 in an open city is expressed by Eq. (7.23). This is exactly the same as Proposition 7.2:

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Xxb dW ðF,ρÞ ∂T ðmÞ ¼  x¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in road investment

 2πx½F ðxÞr ðxÞ  SðF ðxÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(7.23)

Opportunity cost of land used for road widening

The equation and the meaning of each term are the same as Eq. (7.16).

7.7.4 Open city with no FAR regulation Under no FAR regulation the first-order condition r ∂S/∂F ¼ 0 holds as shown in Section 7.6.4. Using this equation and Eq. (7.19), we can rewrite the effect on the social welfare of the road investment as in the succeeding text, which shows that the formula of cost-benefit analysis can use the market land rent. Proposition 7.6 (Cost-benefit analysis in an open city with no FAR regulation) The cost-benefit analysis in Regime 3 in an open city is expressed by Eq. (7.24). This is exactly the same as Proposition 7.3: Xxb dW ðF, ρÞ ∂T ðmÞ ∂F ðmÞ ¼  x¼1 2πm½1  ρðmÞnðmÞ dρðxÞ ∂F ðmÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Change in total commuting cost due to change in FAR caused by road investment



Xxb

∂T ðmÞ 2πm½1  ρðmÞnðmÞ  2πx½F ðxÞr ðxÞ  SðF ðxÞÞ ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} x¼1

Change in total commuting cost due to change in road investment

Opportunity cost of land used for road widening

(7.24) The equation and the meaning of each term are the same as Eq. (7.17).

7.8 Conclusion This chapter examines the cost-benefit analysis in a closed and open city by studying the effect of road investment on social welfare. In a city with

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FAR regulation, we can use market land rent in the cost-benefit analysis. In Kanemoto (1977) the market land rent differs from the shadow price of land because there is commuting congestion externality. In contrast, in the current study, commuting congestion externality does exist, but the change in the externality caused by the expansion of road is cancelled out by the change in distortion in the floor area market. This means the cost-benefit formula does not have to take account of the distortion in the land rent market. However, the distortions in the floor area market and the commuting market still exist, which is why the economy is not the first best. The effect of road investment on social welfare is expressed by using market land rent. This means the cost-benefit analysis based on market land rent expresses the true value without distortion. In addition, the cost-benefit formulae are the same whether the city is open or closed. We have examined the cost-benefit analysis in a monocentric city and come to the conclusion earlier. However, this conclusion holds even if we extend the model in any of the following three ways: (1) combining lot size zoning and FAR regulation, (2) making the city nonmonocentric, and (3) adding other cities. For (1), it is correct to use market land rent under lot size-regulated cities because the economy can achieve the first best, and the current study shows that it is correct to use market land rent under FARregulated cities, too. This means we can use the market land rent in the cost-benefit analysis in cities where both regulations are combined. For (2) and (3) the optimal FAR regulation will change if we extend the model in these ways. However, we have shown that we can use the market land rent in the cost-benefit analysis in a FAR-regulated city, no matter whether it is optimal or not. This means we can still use the market land rent in the cost-benefit analysis even if we extend the model in terms of (2) and (3). Future study can explore a stepwise policy of the combination of FAR regulation and road investment. In Regime 1 of this chapter, we suppose that FAR regulation follows road investment. But in the real world, this order is not necessarily typical. Besides, FAR regulation and road investment may not be optimal in most cases. In this situation, how shall we approach the optimal policy combination? Fig. 7.3 shows the welfare function of road policies and FAR regulations, taking road policy as one of the two horizontal axes; FAR regulation as the other horizontal axis and the welfare as the vertical axis. This graph is conceptually depicted for simple understanding, although road stock or FAR regulation cannot be represented as one axis. Generally the present status

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Road investment evaluation under land use regulation

Social welfare Optimal

Present status Road stock

FAR regulation

Fig. 7.3 Combination of road investment and FAR regulation.

of the FAR regulation and the road stock is not optimal. So, we have to approach the optimal point efficiently. While there are many approaches, we would like to present one way proposed by Miyakoshi et al. (2010a, b). Many studies have addressed how the components of fiscal spending affect economic growth but have not explicitly inquired into how to adjust the components to achieve the efficient point starting from the present status. Miyakoshi et al. (2010a,b) investigate how to determine the optimal adjustment by introducing a gradient method, which explicitly takes account for the adjustment cost. The use of this method is a practical way to approach the efficient point.

Technical Appendix 7 (1) Derivation of optimal FAR The term ∂W(F, ρ)/∂F(x) in Eq. (7.11) is derived as follows: Eq. (7.9) yieldsn ∂W ðF, ρÞ Xxb ∂V ðmÞ ∂Ω 2πm½1  ρðmÞnðmÞ ¼ + 8x 2 ð1, xb Þ (A7.1) m¼1 ∂F ðxÞ ∂F ðxÞ ∂F ðxÞ The first term on the right side of Eq. (A7.1) denotes the effect of a marginal increase in the floor supply at x on the total household utility; the second n

Note that V(m) ¼ V and

Pxb

m¼12πm[1  ρ(m)]∂n(m)/∂ F(x) ¼ ∂ N/∂ F(x)

(¼ 0 in a closed city).

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term ∂Ω/∂F(x) represents the effect of the same on the landowners’ total net profit, which, recalling Eq. (7.2), is given by ∂ΩðF, ρÞ Xxb ∂r ðmÞ ¼ m¼1 2πmF ðmÞ ∂F ðxÞ ∂F ðxÞ

∂SðF ðxÞÞ + 2πx r ðxÞ  8x 2 ð1, xb Þ: ∂F ðxÞ

(A7.2)

Recalling that ∂r/∂V ¼ ∂r/∂T ¼ ∂G/∂T ¼  1/f and F ¼ nf from Eq. (7.1) and Eq. (7.8), respectively, differentiation of the bid rent function Eq. (7.1) with respect to F(m) yields

∂r ðmÞ 1 ∂T ðmÞ ∂V ðmÞ ∂G ¼ + + (A7.3) ∂F ðxÞ f ðmÞ ∂F ðxÞ ∂F ðxÞ ∂F ðxÞ This can be arranged into

∂V ðmÞ ∂r ðmÞ ∂T ðmÞ ∂G ¼ f ðmÞ  + ∂F ðxÞ ∂F ðxÞ ∂F ðxÞ ∂F ðxÞ

(A7.4)

Plugging (A7.2) and (A7.4) into (A7.1) and noting that F(m) ¼ n(m)f(m) yields

Xxb ∂W ðF, ρÞ ∂r ðmÞ ∂T ðmÞ ∂G ¼ + + 2πm ½ 1  ρ ð m Þ n ð m Þ f ð m Þ m¼1 ∂F ðxÞ ∂F ðxÞ ∂F ðxÞ ∂F ðxÞ

Xxb ∂r ðmÞ ∂SðF ðxÞÞ ð Þ ð Þ + 2πx r x  2πmF m + m¼1 ∂F ðxÞ ∂F ðxÞ ¼

Xxb

∂r ðmÞ Xxb ∂T ðmÞ  2πm½1  ρðmÞnðmÞ m¼1 ∂F ðxÞ ∂F ðxÞ

Xxb ∂G ∂SðF ðxÞÞ  2πm ½ 1  ρ ð m Þ n ð m Þ + 2πx r ð x Þ  m¼1 ∂F ðxÞ ∂F ðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2πmρðmÞF ðmÞ m¼1

¼N ∂G=∂F ðxÞ

Pxb

(A7.5)

Note that m¼12πm[1  ρ(m)]n(m)∂G/∂F(x) ¼ N∂G/∂F(x) because ∂G/ Pb ∂F(x) is identical for all residents. Then, using G ¼ N1 xx¼1 2πxρðxÞRðxÞ and R(x) ¼ F(x)r(x)  S(F(x)), ∂G/∂F(x) can be derived as

Road investment evaluation under land use regulation



∂G 1 ∂SðF ðxÞÞ ¼ 2πxρðxÞ r ðxÞ  ∂F ðxÞ N ∂F ðxÞ X 1 xb ∂r ðmÞ 2πmρðmÞF ðmÞ 8x 2 ð1, xb Þ: + m¼1 N ∂F ðxÞ

187

(A7.6)

Plugging (A7.6) into (A7.5), ∂W(F, ρ)/∂F(x) is given as Xxb ∂W ðF, ρÞ ∂T ðmÞ ¼  m¼1 2πm½1  ρðmÞnðmÞ ∂F ðxÞ ∂F ðxÞ

∂SðF ðxÞÞ 8x 2 ð1, xb Þ: +2πx½1  ρðxÞ r ðxÞ  ∂F ðxÞ

(A7.7)

The optimal FAR regulation implies ∂W(F, ρ)/∂F(x) = 0, leading to Eq. (7.11). (2) Derivation of cost-benefit analysis ∂W/∂ρ(x) is derived as follows: dW ðFðρðxÞÞ, ρÞ Xxb ∂W ∂F ðmÞ ∂W ¼ + 8x 2 ð1, xb Þ: x¼1 dρðxÞ ∂F ðmÞ ∂ρðxÞ ∂ρðxÞ

(A7.8)

The first term on the right-hand side, [∂W/∂F(m)][∂F(m)/∂ρ(x)], expresses partial change in welfare with respect to change in the market equilibrium FAR caused by road investment at x. The second term expresses partial change in welfare with respect to change in the road investment at x. ∂W/∂F(m) is given by Eq. (A7.7). ∂W/∂ρ(x) can be expressed as follows: ∂W ðF, ρÞ Xxb ∂V ðmÞ ∂Ω 2πm½1  ρðmÞnðmÞ ¼ + 8x 2 ð1, xb Þ: (A7.9) m¼1 ∂ρðxÞ ∂ρðxÞ ∂ρðxÞ The first term on the right side of Eq. (A7.8) denotes the effect of a marginal increase in the road investment at x on the total household utility; the second term ∂ Ω/∂ ρ(x) represents the effect of the same on the landowners’ profit, which, recalling Eq. (7.2), is given by Xxb ∂Ω ∂r ðmÞ 2πmF ð m Þ ¼ 8x 2 ð1, xb Þ: x¼1 ∂ρðxÞ ∂ρðxÞ

(A7.10)

Recalling that ∂r/∂V ¼ ∂r/∂T ¼ ∂r/∂G ¼  1/f and F ¼ nf from Eqs. (7.1), (7.8), respectively, differentiation of the bid rent function Eq. (7.1) with respect to ρ(x) yields

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∂r ðmÞ 1 ∂T ðmÞ ∂V ðmÞ ∂G ¼ + + ∂ρðxÞ f ðmÞ ∂ρðxÞ ∂ρðxÞ ∂ρðxÞ

(A7.11)

This can be arranged into

∂V ðmÞ ∂r ðmÞ ∂T ðmÞ ∂G ¼ f ðmÞ  + ∂ρðxÞ ∂ρðxÞ ∂ρðxÞ ∂ρðxÞ

(A7.12)

Plugging Eq. (A7.10) and Eq. (A7.12) into Eq. (A7.9) and noting that F(m) ¼ n(m)f(m) yields

Xxb ∂W ðF, ρÞ ∂r ðmÞ ∂T ðmÞ ∂G ¼ + + 2πm½1  ρðmÞnðmÞ f ðmÞ m¼1 ∂ρðxÞ ∂ρðxÞ ∂ρðxÞ ∂ρðxÞ Xxb ∂r ðmÞ 2πmF ðmÞ + x¼1 ∂ρðxÞ Xxb ∂r ðmÞ Xxb ∂T ðmÞ  2πm½1  ρðmÞnðmÞ ¼ m¼1 2πmρðmÞF ðmÞ m¼1 ∂ρðxÞ ∂ρðxÞ Xxb ∂G 2πm½1  ρðmÞnðmÞ  m¼1 ∂ρðxÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼N ∂G=∂ρðxÞ

Pxb

(A7.13)

Note that m¼12πm[1  ρ(m)]n(m)∂G/∂ρ(x) ¼ N∂G/∂ρ(x) because ∂G/ Pb 2πxρðxÞRðxÞ ∂ρ(x) is identical for all residents. Then, using G ¼ N1 xx¼1 and R(x) ¼ F(x)r(x)  S(F(x)), the effect of the marginal increase in the road investment at x on the tax can be derived as ∂G 1 ¼ 2πx½F ðxÞr ðxÞ  SðF ðxÞÞ ∂ρðxÞ N 1 Xxb ∂r ðmÞ + 2πmρðmÞF ðmÞ 8x 2 ð1, xb Þ: m¼1 N ∂ρðxÞ

(A7.14)

Plugging (A7.14) into (A7.3) gives the partial effect on the welfare by the road investment expressed as Xxb ∂W ðF, ρÞ ∂T ðmÞ ¼  m¼1 2πm½1  ρðmÞnðmÞ ∂ρðxÞ ∂ρðxÞ  2πx½F ðxÞr ðxÞ  SðF ðxÞÞ

(A7.15)

Substituting Eq. (A7.11) and Eq. (7.6) into Eq. (A7.7), we obtain the effect of road investment on social welfare as expressed in Eq. (7.14).

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189

References Akai, N., Fukushima, T., Hatta, T., 1988. Optimality of a competitive equilibrium in a small open city with congestion. J. Urban Econ. 43, 181–198. Arnott, R.J., 1979. Unpriced transport congestion. J. Econ. Theory 21, 294–316. Brueckner, J.K., 2007. Urban growth boundaries: an effective second-best remedy for unpriced traffic congestion? J. Hous. Econ. 16, 263–273. Hochman, O., 1975. Market equilibrium versus optimum in a model with congestion: note. Am. Econ. Rev. 65, 992–996. Kanemoto, Y., 1977. Cost-benefit analysis and the second best land use for transportation. J. Urban Econ. 4, 483–503. Miyakoshi, T., Kono, T., Terasawa, K., 2010a. Optimal adjustment of the composition of public expenditure in developing countries. Pac. Econ. Rev. 15 (5), 577–595. Miyakoshi, T., Tsukuda, Y., Kono, T., Koyanagi, M., 2010b. Economic growth and public expenditure composition: optimal adjustment using the gradient method. Jpn. Econ. Rev. 61 (3), 320–340. Pines, D., Sadka, E., 1985. Zoning, first-best, second-best, and third-best criteria for allocation land for roads. J. Urban Econ. 17, 167–183. Wheaton, W.C., 1998. Land use and density in cities with congestion. J. Urban Econ. 43, 258–272.

CHAPTER EIGHT

Changes with future ICT technologies and future studies This book has addressed land use regulations, disentangling complex mechanisms of the dependency between spatial externalities and land use regulations theoretically, and derives practical ways for optimizing them. Each chapter explores a different situation in a city to illustrate the mechanisms. Chapter 2, assuming a closed two discrete-zone city model, shows the necessity for minimum and maximum FAR regulations to maximize social welfare. Chapter 3 explores the difference in optimal regulation between an open city and a closed city. Chapter 4 targets a growing city with congestion. Chapter 5 addresses a city with agglomeration economies and traffic congestion. Chapter 6 introduces cordon pricing in a city with land use regulation. These chapters show that a second-best situation has different regulations because distortions can depend on land use regulations differently. Up to Chapter 6, we discuss how to design optimal density and zonal regulations in congested cities. In contrast, Chapter 7 focuses on road investment, exploring how much of the land should be allocated for roads in a city without congestion tolls. This final chapter first discusses a different situation that can arise in the future. Information and communication technology (ICT) has drastically changed our life. By virtue of the fast-improving rate of ICT, at this rate, our society will soon change to a totally different society. ICT is expected to drastically reduce the supply-side cost in many situations and to produce convenient life. One thing we should note, however, is that ICT cannot exclude externalities completely without economic policies, even if the latest ICT is implemented. For example, an automatic transportation system can replace our current inefficient driving behavior. In other words, ICT may enable our limited resources to be used more efficiently. However, as long as people want to commute at the same time, say in the morning, the traffic volume will concentrate at such time. As a result, there remain congestion externalities. Still, we have to address such congestion externalities via some economic policies. Traffic Congestion and Land Use Regulations https://doi.org/10.1016/B978-0-12-817020-5.00008-X

© 2019 Elsevier Inc. All rights reserved.

191

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ICT can reduce the cost of implementing economic policies including the first-best policies. For example, with some sensing gadgets, any car driving can be fully traced; so the first-best toll can be imposed at each location at each time. Nevertheless, in many cases, we cannot use the first-best policies because of several reasons. For example, firms would not like to reveal their trading networks completely to conceal their production technologies. In this situation, locational externalities in business zones, which are discussed in Chapter 5, cannot be internalized. Next, some politicians might not like to implement some first-best policies because the related agents may react against the policies, as discussed by many previous papers shown in Table 1.4 in Chapter 1. Furthermore, costs for implementing the first-rate sensing gadget might not be low even with the most state-of-the-art ICT. In situations where the first-best policies cannot be implemented to internalize all the externalities, we need to consider second-best policies, which are explored in this book. However, as this book has demonstrated, it is not so easy to design second-best policies properly partly because our society is spatial. In the future also, we will need a rigorous derivation of second-best policies, considering advanced ICT and relevant political and economic constraints. In this situation, because the city must be composed of heterogeneous zones, optimal control theory taking account of multiple different zones, adopted in Chapters 5 and 6, is useful. In addition, in such a rapidly changing society, the income distribution will change drastically. So, it is important to check who gains and who loses and not only how much the society gains or loses.a Many economists argue that the first-best and second-best policies themselves should be implemented only from the viewpoint of efficiency, based on Richard Musgrave’s (1959) book The Theory of Public Finance, which classifies policies into the allocation of goods and the distribution of income. However, total policies may change the income distribution drastically. So, at least, we should check this income distribution. Besides such future settings, we have to explore some other situations from the current book. First, commuting cost function in this book (i.e., the Bureau of Public Roads or BPR function) is a traditional static function in which the timing of commuting is not a choice variable and where traffic flows and speeds at each location are constant over time. This is a common a

With the existence of heterogeneous agents in a spatial economy, Allais surplus cannot be consistent with compensation tests, whereas the surplus is consistent with compensation tests in a point economy. This is demonstrated by Kono and Kishi (2018). They show what kind of a surplus measure is appropriate in a spatial economy.

Changes with future ICT technologies and future studies

193

way of describing traffic congestion in the urban areas. But the bottleneck model, which was introduced by Vickrey in 1969, also expresses congestion externality. The bottleneck model is the dynamic model of traffic congestion in which the choice of departure times is endogenous and where dynamic patterns of travel delays are key features. However, most papers exploring bottleneck congestion do not consider urban spaces in the city. Recently, some economic papers have modeled dynamic congestion in urban space. For instance, Fosgerau and de Palma (2012) and Takayama and Kuwahara (2017) consider a continuous city with a central bottleneck, exploring welfare improvements of heterogeneous residents by an optimal toll regime. These papers mainly focus on the effects of bottleneck congestion and an optimal time-varying congestion toll on the spatial structure of cities. In addition, we should consider different modes of transportation such as rails and buses. In that situation, we have to take into account all relevant distortions caused by congestion, land use regulation, and rail fare. Past studies have explored these ingredients in isolation, but no paper has considered them simultaneously with land use regulations. There are many papers exploring a two-transport mode system. For example, Tabuchi (1993), Arnott and Yan (2000), and Danielis and Marcucci (2002) explore a railway system in a model with congested roads. However, since these papers do not consider space, they cannot examine land use regulation. Haring et al. (1976), Anas and Moses (1979), and Sasaki (1998) suppose urban space with two-transport modes but do not study transportation policies or land use regulation. Buyukeren and Hiramatsu (2016) explore transportation policies and optimal UGB regulation by using a simple model of only two discrete zones. We have to consider that if commuters have access to an alternative mode in an urban area, mitigating congestion externality may cause urban sprawl. In addition, we should consider the end point of a rail line inside the city, where residents living before the end point can choose cars or railway and residents living beyond the end point can choose only cars. In this situation also the optimal control theory taking account of heterogeneous zones, which is shown in Chapters 5 and 6, is useful. Next, this book considers only a monocentric city. Many large cities, which have large population and are terribly congested with traffic, have sub-business district (SBD). For a nonmonocentric city, by setting mixed-use zones in a monocentric and nonmonocentric city having residents with idiosyncratic tastes and in a system of cities with homogeneous

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residents, respectively, Anas and Rhee (2006, 2007) and Anas and Pines (2008) show that an expansive UGB may be necessary, and nonmonocentric city may require a different land use regulation from a monocentric city. That implies that the setting of geography plays an important role in determining the properties of optimal land use regulations. Although the results in a monocentric case should be reasonably valid for any modifications on the model as long as the fundamental relationship between residential density and commuting costs is preserved (Brueckner, 2007; Sridhar, 2007), we have to extend our analyses to nonmonocentric cities. Next, in real cities, there are many heterogeneous people living in a city. So, even if the regulations increase the social welfare, all the agents will not necessarily increase their respective welfare. When implementing the regulations, we need to consider how each agent is affected. Policies including land use regulations should consider the distribution of wealth among agents and the efficiency of regulations. Indeed, policies are politically acceptable if they can be designed in such a way that a large majority of the population will benefit. The distribution of wealth in the case of urban policies has been analyzed by many papers (e.g., Dubin et al., 1992; Arnott et al., 1994; Buyukeren and Hiramatsu, 2016). Similarly, the distribution of wealth in the case of land use regulations has been analyzed. For instance, investigating urban growth controls, Brueckner (1995) shows that the growth controls harm consumers while benefiting landowners. This result clearly shows that we should differentiate residents from landowners for welfare analysis. The geographical difference in changes in land rent implies that the benefits of landowners depend on the location of the land they own and that the location of households also matters for changes in the welfare of households if heterogeneous residents reside separately. In particular, income heterogeneity yields a residential segregation pattern in which population groups are sorted into different zones (Fujita, 1989). However, past theoretical studies on land use regulation have not considered heterogeneous people. Land use regulations are quantitative regulations. Actually, land use regulations can be replaced by equivalent property tax policies, as shown in Pines and Kono (2012) where homogeneous residents are considered. However, if there are heterogeneous agents in a city, both are not the same. For example, a certain type of industry needs rather large spaces so that a large building is appropriate from an efficiency viewpoint. In this situation a common regulation on the heterogeneous industries leads to an inefficient outcome. Property tax depending on the location is preferable. Like this, we need to explore spatially variable property tax in addition to land use

Changes with future ICT technologies and future studies

195

regulations. Assuming homogeneous residents, Kono et al. (2019) explore spatially variable property tax in addition to other tax availabilities, but we need other settings including heterogeneous residents. Finally, urban economists have basically focused on cities themselves. However, cities are surrounded by agricultural areas, where farmers work, and forest, where natural resources including ecosystem services are provided. For residents to coexist with agriculture and natural resources beneficially, land use policies should consider their benefits. Several studies investigate how large the natural habitat must be to sustain the ecosystem services, focusing on land use competition between humans and the ecosystem (Walker, 2001; Eppink et al., 2004; Eichner and Pethig, 2006, 2009). Eichner and Pethig (2006) integrate the microfounded ecosystem model with an economic model to show that the optimal city size is smaller than the equilibrium city size because of external benefits caused by an increase in the size of natural habitats to enrich the quality of ecosystem services. On the other hand, unlike the two-zone setting by Eichner and Pethig, introducing a continuous distance dimension and the human-wildlife conflict, Yoshida and Kono (2018) derived some different land use policies. Likewise, we should consider land uses other than urbanized use to explore land use regulation. In summary, our book provides some novel results and some useful analytical methods, but there are many factors we have to take into account for designing optimal land use regulations in various urban situations. To design second-best policies, not only targeted externalities but also distortions produced by policies should be considered. In addition, spatial factor adds complexity. However, we need to systematize different results according to different situations to provide practical methods for urban land use regulations. This remains for future work.

Acknowledgments Most work related to this book started when Kirti Joshi and Tatsuhito Kono met in 2004 in Tohoku University, Japan. We would like to thank many professors, groups, students, and our families for supporting us. Without their kind assistance and guidance, we cannot even continue our research lives. Above all, we would like to express our deep appreciation for the late Prof. Hisa Morisugi, our common mentor, for his kind and enthusiastic guidance.

References Anas, A., Moses, L.N., 1979. Mode choice, transport structure and urban land use. JTEP 6 (2), 228–246. Anas, A., Pines, D., 2008. Anti-sprawl policies in a system of congested cities. Reg. Sci. Urban Econ. 38 (5), 408–423. Anas, A., Rhee, H.-J., 2006. Curbing excess sprawl with congestion tolls and urban boundaries. Reg. Sci.Urban Econ. 36 (4), 510–541.

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Index

Note: Page numbers followed by f indicate figures, t indicate tables, and np indicate footnotes.

A Agglomeration economies, 4, 21, 110 Agricultural land rent, 38–39, 62–63, 74, 86 Alonso models business zone cities, 11, 12f congested roads, 6, 7f extensions of, 9t, 12 high-rise buildings and roads, 10f leaps from, 10t Automatic transportation system, 191

B Bed towns, 1–3, 58, 58np, 173 Benthamite social welfare function, 89, 172 Bid rent, 109, 186 Bottleneck model, 192–193 Bureau of Public Roads (BPR) functions, 111, 122 Business zones, 105 commuting cost, 111 developers’ behavior, 110 FAR regulation, 132 firms’ behavior, 107–110 market clearing conditions, 112–113 model city, 106–107 numerical results, 124–127 optimal FAR regulations, 115–119 simulation model, 120–127 social welfare, 113–115 zonal boundaries, 119–120, 136

C Central business district (CBD), 6. See also Business zones cordon pricing, 144, 147, 151–152 road investment analysis, 169, 171 City government, 34, 59, 145–146, 169–170

Closed city model commuting cost, 61–62 cost-benefit analysis of road investment, 176–181 developers’ behavior under FAR regulation, 60–61 with no FAR regulation, 60 household behavior, 59 market equilibrium conditions, 62–63 optimal FAR regulation and urban boundary, 63–69, 74 Communication trips, 4, 108, 111, 122–123, 165 Commuting cost, 192–193 business zones, 111 open and closed city model, 61–62 road investment, 171–172 Competitive market, 54, 168 Complementary condition, 81np Condominium, 8–10, 15, 105, 127, 139–140 Condominium zones, 105 FAR regulation, 117, 119, 133 LS regulation, 117, 119 Congestion externality, 3–4, 23, 33, 39, 42–43, 42–46f, 48, 99–101, 111, 163, 171, 174–175, 191–193 pricing, 4–5, 11, 143–144, 167–168 tolls, 23, 36, 39, 46–47, 122–123, 167 Constraint qualification, 79, 149np Construction cost, 86–88, 91, 110, 170–171, 174 Cordon line, 1–3, 145, 147–149, 151, 152f, 153, 161, 165 Cordon pricing, 15, 143 city configuration, 144–145, 145f developers’ behavior under FAR regulation, 146

197

198 Cordon pricing (Continued ) FAR regulation characterization, 151–154, 164 distortions and congestion, 163 simulation model, 154–157, 157f household behavior, 145–146 market equilibrium conditions, 148–149 optimal location of cordon line, 149–151, 161–162 traffic flow and commuting cost, 147 UGB regulation characterization, 151–154, 164 simulation model, 154–157, 157f Cost-benefit analysis, 167, 184 closed city, road investment in, 168, 173, 176–181 with fixed FAR, 178, 180 with no FAR regulation, 179–180 with optimal FAR regulation, 177, 179 open city, road investment in with fixed FAR, 182–183 with no FAR regulation, 183 with optimal FAR regulation, 182

D Deadweight loss, 115, 116f, 133 Demand-side cost, 61, 111, 147 Density gradients, population Moscow and Paris, 1–3, 2f Tokyo (e.g., Machida), 1–3, 2f Density regulations, 73f, 92–93, 105–106, 167–168, 173 Detached housing zone, 105–106, 135 Developer-cum-landowner, 84 Developers, 12, 23–24, 33–35, 60, 85, 110, 146, 169–171 Developers’ behavior business zones, 110 cordon pricing, under FAR regulation, 146 growing city, 85–89, 102 open and closed city model under FAR regulation, 60–61 with no FAR regulation, 60 road investment analysis, 170–171 two-zone city model, 25

Index

Distortion, 65–66, 115, 143 price, 5, 15, 53–54, 53–54f, 169 Dollar-equivalent welfare gain, 44np

E Employment elasticity of productivity, 4, 4np

F FAR regulation. See Floor area ratio (FAR) regulation Fixed FAR, 178, 182–183 Floor area market, 29–32, 68–69, 150–151, 163–165, 168 Floor area ratio (FAR) regulation, 1, 1np, 10, 22. See also Optimal FAR regulation in business zone, 132 combined road investment, 184–185, 185f in condominium zone, 133 cordon pricing, 151–157 in growing city, 86, 86–87f, 92f road investment, 172–174 three-zone city model, 106f Floor production function, 37, 121, 124 Floor rent, 22, 27–28, 34, 50, 59, 66f, 85, 92f, 96–97, 124, 146, 170, 181f

G Gradient method, 185 Growing city developers’ behavior, 85–89, 102 household behavior, 84–85 market clearing conditions, 89 model city, 84 optimal FAR regulation, 90–97, 95t, 103 exceptional cases, 97–98 static city, 97 traffic congestion externality, 99–101

H Hamiltonian functions, 80–81 equivalency with Lagrangian, 81 Harberger welfare function, 26–27, 53–55, 67np, 68 Heterogeneous residents, 194

199

Index

High-externality zone, 23, 83–84 High-rise buildings, 8–11, 10f, 168 Homogeneous residents, 194–195 Household behavior, 24–25, 59, 84–85, 112, 145–146, 169–170 Household utility, 24, 26, 62–63, 72, 84–85, 112, 148, 172 Human-wildlife conflict, 194–195

I Income constraint, 34, 59, 112, 146, 170 Income distribution effects, 13, 14t Information and communication technology (ICT), 191 automatic transportation system, 191 cost reduction, 192

J Japan bed towns, 58, 58np employment elasticity of productivity, 4, 4np

K Knowledge spillover, 4 Kuhn-Tucker condition, 78–79, 147

L Labor population, 113 Lagrangian function, 78 business zones, optimal regulations, 114, 128 equivalency with Hamiltonian, 81 optimal cordon pricing, 149–150, 159 optimal FAR in closed city, 63–64 in open city, 70 Lagrangian method, 78–81 Lagrange multiplier, 60, 147, 148np Laissez-faire, 38–43, 40–41t, 122, 125t Landowners’ profit, 187, 170 Land rent, 46–47, 47f Land use regulations, 1 distribution of wealth, 194 efficiency, 6, 7t

income distribution effects, 13, 14t spatial mechanisms, 5–6 Le Chatelier’s principle, 32 Locational externality, 109, 110f, 124 Lot housing zone, 106, 119, 135, 139 Lot size (LS) regulation, 1, 8, 22 open and closed city, 69 three-zone city model, 106f Lot size zoning, 167–168 Low-externality zone, 23, 83–84

M Machida, density gradient, 1–3, 2f Market clearing conditions business zones, 112–113 growing city, 89 two-zone city model, 25–26 Market equilibrium cordon pricing, 148–149 open and closed city model, 62–63 road investment analysis, 172 Market equilibrium FAR, 27–33 Maximization of social welfare, 90 Maximum FAR regulation, 11, 21–23 business zones, 118 city with point CBD, 7t, 9t open and closed city, 117 Microfounded ecosystem model, 194–195 Minimum FAR regulation, 11, 21 business zones, 118 land rent profiles, 46–47, 47f monocentric city, 33–47 optimal conditions (see Optimal FAR regulation) two-zone city model city configuration, 23–24, 24f developer’s profit, 25 household utility, 24–25 market clearing conditions, 25–26 Minkowski-Farkas lemma, 79–80 Monocentric city model minimum FAR regulation, 33–47 road investment analysis, 169 Moscow, density gradient, 1–3, 2f

200

N Negative externalities, 21 No FAR regulation open and closed city, 60 road investment analysis closed city, 178–179 open city, 183 Numeraire composite goods, 34, 59, 145–146, 169

O Open city model commuting cost, 61–62 cost-benefit analysis of road investment, 181–183 developers’ behavior under FAR regulation, 60–61 with no FAR regulation, 60 household behavior, 59 market equilibrium conditions, 62–63 optimal FAR regulation and urban boundary, 70–72, 75 population, 57 Optimal control theory, 15–16, 80, 105–106 Optimal FAR regulation, 46f, 63 closed city, 63–69, 74 closed vs. open-city, 72–73 at cordon line, 153, 165 growing cities, 89–98, 95t, 103 Harberger welfare function, 26–27 interpretation of, 31–32, 31f vs. market FAR, 27–33 open city, 70–72, 75 for residential areas, 48 road investment analysis, 174–175, 185 closed city, 177 open city, 182

P Paris, density gradient, 1–3, 2f Perfect competition, 25, 32, 35, 55, 60, 134 Pigouvian tax, 4–5, 124 Point CBD, 8–10, 9t Pontryagin’s maximum principle, 128 Population density, 35, 37–38 and commuting cost, 62, 62f open and closed city model, 60

Index

Positive externalities, 48, 102, 105 Price distortions, 53–54, 53–54f Price elasticity, 167 Production function, 25, 107 Property tax, 194–195 Public land ownership, 107

Q Quasi-linear utility function, 34, 59

R Regularity condition, 79 Rent bidding, 109 Residents, 1, 13, 22, 24, 26, 34, 59, 107, 145–146, 169, 193–195 Road capacity, 111, 122, 147, 171 Road investment closed city, cost-benefit analysis, 176 with fixed FAR, 178, 180 with no FAR regulation, 178–180 with optimal FAR, 177, 179 combined FAR regulation, 184–185, 185f commuting cost, 171–172 congested monocentric city, 169 developers’ behavior, 170–171 FAR regulations, 172–176, 173f floor rent and marginal cost, 180, 181f household behavior, 169–170 landowners’ profit, 170 market equilibrium conditions, 172 open city, cost-benefit analysis, 181 with fixed FAR, 182–183 with no FAR regulation, 183 with optimal FAR, 182 optimal FAR, 174–175, 185 UGB regulations, 172–174

S Second-best policy, 15–16, 22, 29–30, 30–31f, 33f, 51f, 143–144, 168, 175, 192, 195 Shadow prices, 114–115, 116f Silicon Valley, 58, 173 Social welfare, 26–27, 49, 89 business zones, 113–115 marginal change, 90 maximization of, 90 road investment and FAR regulations, 172

201

Index

Static city, 97 Sub-business district (SBD), 193–194 Supply-side cost, 61, 108–109, 111, 147, 191

T Tax, 4, 188, 124, 126–127, 170, 194–195 Technological externality, 62–63 The Theory of Public Finance, 192 Three-zone city model, 106, 106f. See also Zoning Tokyo, density gradient, 1–3, 2f Toll-regime, 122, 155, 156t Total commuting cost, 35–36, 61 with cordon pricing, 154–155 Total marginal deadweight loss, 175 Traffic congestion, 3–4 bottleneck model, 192–193 FAR-regulated growing city, 99–101 Traffic volume, 35–36, 61, 111, 146–147, 171, 178, 191 Transversality condition, 81 Two-transport mode, 193 Two-zone city model city configuration, 23–24, 24f developer’s profit, 25 household utility, 24–25 market clearing conditions, 25–26

U Unit-distance commuting cost, 154, 111, 147 Unpriced congestion, 35–36, 171–172 Urban growth boundary (UGB) regulation, 1, 8, 23 cordon pricing, 151–157 monocentric city, 38 open and closed city model, 72, 73f road investment, 172–174 Urban growth pattern, closed monocentric city, 100, 100f Urban planning, 174np Utility, 24, 37

W Wage, 24–25, 94, 108, 110, 112, 123 Welfare, 44, 45f

Z Zero-profit condition, 110, 35 Zoning, 1, 8, 119–120 business zone, 136, 138 condominium zone, 137–138 lot housing zone, 137, 139