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Suminori Tokunaga
Landownership and
Residential Land Use
in Urban Economies
Existence and Uniqueness of
the Equilibrium
With 35 Figures
. Springer
Suminori Tokunaga, Ph.D.
Associate Professor of Economics
International School of Economics and Business Administration
Reitaku University
2-1-1 Hikarigaoka. Kashiwa.
Chiba, 277 Japan
ISBN-13: 978-4-431-68412-1 e-ISBN -13: 978-4-431-68410-7
DOl: 10.1007/978-4-431-68410-7
Printed on acid-free paper
© Springer-Verlag Tokyo 19<)6
Softcover reprint of the hardcover 1s t edition 1996
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Preface
This book examines the impact of landownership on residential land use
equilibria. The residential location behavior of the landowning household is
analyzed in a microeconomic framework, and the existence and uniqueness of the
residential land use equilibrium is shown. In addition, we examine how this
land use equilibrium will be altered with changes in landownership. The study
will develop three models of landownership by using the bid rent function
approach.
In the first model (Chapter 3) we propose a basic model of residential land
use, called an a-landownership model, in which a group of city residents is
assumed to own the circular area from the CBD up to a radius a. Our model
includes both the absentee landownership model and the public landownership
model as special cases. The existence and uniqueness of the land use
equilibrium will be shown. We will examine how the land use equilibrium will be
affected by changes in the landownership parameter a and other parameters such
as household incomes and populations. This model, however, should be viewed as
a simple prototype for introducing landownership into urban analysis. Thus, in
the second model (Chapter 4) we replace a-land with an arbitrary subset, say S,
of the city's land. In the context of this extended model, called an S
landownership model, we examine the existence and uniqueness of the residential
equilibrium and conduct comparative static analyses.
In the third model (Chapter 5) we extend the first model by introducing
crowding externalities and examine the impact of the minimum lot size
regulation on the land use equilibrium in order to remedy crowding
externalities. Furthermore, we examine how the utility levels and the total
land rent will be affected by changes in the landownership parameter.
This book is a revision of my Ph.D. dissertation Impact of Landownership on
Residential Land Use Equilibria, 1992, at the University of Pennsylvania. I am
indebted to many individuals who contributed to my academic development and
helped make a success of my dissertation.
First of all, lowe a special debt of gratitude to Professor Masahisa Fujita,
my dissertation supervisor at the University of Pennsylvania. Without his
continuous and persistent guidance and encouragement during my graduate years,
my dissertation and this book would never have been undertaken.
Preface
My profound thanks are due to Professor Tony E. Smith for many helpful
suggestions at various stages of my dissertation work. Professor Ronald E.
Miller contributed many comments and suggestions concerning this study, and I
am deeply grateful. I also express my thanks to Professors Lawrence R. Klein,
F. G. Adams, and Thomas W. Dunfee for their valuable suggestions and encoura
gement at the University of Pennsylvania.
In addition, Bettina Aten, Toni Horst, Kazuhito Ikeo, Ryoichi Ishii, Yoshio
ltaba, Dongju Kim, Chiaki Nakano, Koichi Ohno, Mitsuru Ota, Takashi Owaki,
Pornchai Rujiprapa, Shinichi Suda, Iwao Taka, Shuetsu Takahashi, Fumihiko
Tanabe, Hisashi Tanizaki, Kenichi Tsukahara, Akihiro Watabe, and Masakazu
Watanuki offered valuable suggestions and comments.
Furthermore, I wish to thank my former advisors at the University of Tsukuba,
Professors Takao Fukuchi, Shuntaro Shishido, Yoshihiro Maruyama, Yasuhiro
Sakai, Haruo Ohnishi, Yoshimi Kuroda, Yuji Kubo, and Noriyoshi Oguchi for their
valuable suggestions and encouragement. Especially, I acknowledge my deepest
appreciation to Professor Takao Fukuchi, who was my main advisor, for his moral
support and many helpful suggestions throughout the study. I am deeply indebted
to the late President Sentaro Hiroike, President Mototaka Hiroike, Vice
President Toshio Osawa, Dean of Department Masao Komatu, and Professor Yukimasa
Nagayasu of Reitaku University for their help and moral support. This book
would have been impossible without the financial support for publication from
the Reitaku University. I am also grateful to the editors of Springer-Verlag
Tokyo for their valuable suggestions and assistance. I am grateful to Akihiro
Kawakami and Mikio Matsui for their work in making the index. I am particularly
thankful to Yuko Fujita, Motoko Vining, and Dottie Dunfee for their
encouragement and hospitality during the stay of my family in the United
States.
Finally, [ am grateful to my parents, Sango and Kazuko, for their unfailing
support and understanding throughout the period of this study in the United
States, and my wife, Akiko, for her unfailing encouragement and patience
enabled me to complete this monograph.
Tokyo, Japan
October 1995
Suminori Tokunaga
Contents
Preface v
Chapter 1. Introduction
1.1. Objective of the book
1.2. Review of Related Research 2
1.3. Plan of the book 4
Chapter 2. Behavior of the Household with Land 7
2.1. Introduction 7
2.2. Basic Model of the Household with Land 7
2.3. Characteristics of Maximum Utility 8
2.4. Characteristics of the Optimal Lot Size and
the Autarkic Land Rent 14
2.5. Conclusion 17
Chapter 3. Impact of a-landownership on Residential Land Use
Equilibria 19
3.1. Introduction 19
3.2. The a-landownership Model 21
3.3. Definition of the Land Use Equilibrium 24
3.4. Existence and Uniqueness of the Land Use Equilibrium 26
3.4.1. Existence of the Equilibria 27
3.4.2. Uniqueness of the Equilibrium 32
3.5. Comparative Statics 33
3.5.1 Changes in a and y~ 33
3.5.2 Changes in yO, Nand N 35
1 1 2
3.6. Conclusion 43
Appendix 3.A: Proof of Lemma 3.6 44
Appendix 3.B: Proof of Proposition 3.2 47
Appendix 3.C: Proof of relation (3.31) 49
Appendix 3.0: Formal derivation of results for case
in Section 3.5.2 51
Contents
Appendix 3. E Formal derivation of results for case 2
in Section 3.5.2 55
Chapter 4. Impact of S-Iandownership on Residential Land Use
Equilibria 63
4. I. Introduction 63
4.2. The S-landownership Model 63
4.3. Existence and Uniqueness of the Equilibrium 66
4.4. Comparative Statics 71
4.5. Conclusion 72
Appendix 4.A: Proof of relation (4.22) 74
Appendix 4.B: Proof of Proposition 4.2 75
Appendix 4.C: Proof of relation (4.25) 77
Chapter 5. Landownership and Welfare Impacts of Zoning
Regulations 81
5.1. Introduction 81
5.2. A Crowding Externalities Model
without a Zoning Regulation 82
5.2.1. The Residential Choice Behavior 83
5.2.2. The Land Use Equilibrium 88
5.3. A Crowding Externalities Model
with a Zoning Regulation 91
5.3.1. The Residential Choice Behavior 91
5.3.2. The Land Use Equilibrium 93
5.4. Impact of Minimum Lot Size Regulation
on Residential Land Use Equilibria 98
5.5. Conclusion 107
Appendix 5.A: Case of a Large MLS Regulation 109
Chapter 6. Conclusion 113
References 119
Author index 123
Subject index 125
Chapter 1
Introduction
1.1. Objective of the Book
The objectives of this book are twofold. First, we propose three theoretical
models of urban residential land use, in which landownership is explicitly
considered. Second, we show that the land use equilibrium uniquely exists, and
examine how this land use equilibrium will be altered with changes in
landownersh ip.
In the first model, we propose a basic model of residential land use, called
an a-landownership model, in which a group of city residents is assumed to own
the circular area from the CBO up to a radius a. This first model includes both
the absentee landownership model and the public landownership model as special
cases. In other words, we propose a model of two types of households with and
without landownership. We show the existence and uniqueness of the land use
equilibrium for this model, and examine how this land use equilibrium will be
affected by changes in such exogenous parameters as the landownership parameter
a, non-land income and population size.
An a-landownership model in which we assume that a circular area is jointly
owned by a group of city residents should be viewed as a simple prototype for
introducing landownership into an urban residential choice model. Thus, we
replace the a-land with an arbitrary subset, S, of the city's land, and call
this extended model an S-landownership model. This S-landownership model is the
second model. By using the extended approach, we show that given any S, there
exists a unique land use equilibrium. We also conduct comparative statics of
the equilibrium in this general context.
The first and second models assume that land is allocated to the highest
bidder. In other words, they assume that the government is not involved in the
urban land market. In fact, most big cities have many problems. One of them is
crowding externalities. That is, as the density of households in a neighborhood
increases, the environmental quality diminishes partly because of an increase
in noise, littering, crimes and so on, and partly because of a decrease in the
open space and green areas in a neighborhood. Thus, in the third model we
2 Landownership and Residential Land Use in Urban Economics
extend an a-landownership model by introducing a crowding externalities
variable. That is, we introduce a zoning regulation that sets a minimum lot
size (MLS regulation) per household in order to remedy for crowding exter
nalities. We show the impact of the MLS regulation on the land use equilibrium
and furthermore examine how the utility level will be affected by changes in
the landownership parameter. Thus, this book will demonstrate that the size of
landownership plays an important role in residential land use equilibria.
1.2. Review of Related Research
During the last twenty five years a large literature, part of which is
sometimes called the new urban economics, has examined the equilibrium
configuration of an urban residential market. These models which assume that
utility-maximizing households consume the amount of composite consumer good and
land (the lot size of the house), and choose a location at which to live, are
called the basic model of residential choice. This model is based on the
pioneering work of Isard (1956), Beckmann (1957), Alonso (1964), Muth (1969),
and Mills (1972a). By solving the maximization problem, the bid rent function
is obtained. The bid rent function approach was first established by Alonso
(1964) who generalized the agricultural bid rent theory of von Thiinen (1826).
This approach is essentially the same as the indirect utility function approach
introduced into the urban land use model by Solow (1973). Many urban
economists, especially Schweizer, Varaiya, and Hartwick (1976) and Kanemoto
(1980), further developed this bid rent/indirect utility function approach. The
concept of relative steepness of bid rent functions was introduced by Fujita
(1985, 1989).
The land use equilibrium theory was developed by Alonso (1964), Muth (1969),
Casetti (1971), Mills (I 972a, b), Solow (1973), Beckmann (1969,1974), and
Montesano (1972), among others. There are two ways to prove the existence of
the land use equilibrium. First, the existence of land use equilibria has been
studied by Mackinnon (1974), Schweizer, Varaiya, and Hartwick (1976), King
(1980), Richter (1980), and Karmann (1982) by using fixed-point methods. On the
other hand, Fujita (1985,1989) has proven the existence and uniqueness of the
land use equilibrium using a boundary rent curve approach. A concept similar to
the boundary rent curve was introduced by Kanemoto (1980). Fuj ita (1985, 1989)
generalized it to the case of variable lot size and used it for the study of
the existence and uniqueness of the equilibrium and optimal land use. Without
the assumption of ordered bid rent functions, the existence of the land use
i. introduction 3
equilibrium was established by Fujita and Smith (1987), which provided a
general existence proof in the context of n-dimensional space. Since the latter
approach is mathematically simple and intuitively appealing, we use it in
Chapter 3 of this book. In particular, on the subject of the equilibrium of a
spatial model with different income level groups, there are many literatures,
for example, Beckmann (1969) and Montesano (1972) present a model in which
persons have a common utility function but income level varies according to the
Pareto distribution.
The definition of closed and open cities may be attributed to Wheaton (1974).
The public landownership model was introduced by Solow (1973) and further
studied by Kanemoto (1980). Fujita (1989) has classified these models into four
cases. That is, the closed-city model under absentee landownership (CCA model),
the open-city model under absentee landownership (OCA model), the closed-city
model under public landownership (CCP model), and the open-city model under
public landownership (OCP model). The comparative statics of land use equili
brium were first studied by Wheaton (1974) in the case of the single household
type and by Wheaton (1976), Hartwick, Schweizer, and Varaiya (1976), Arnott,
MacKinnon, and Wheaton (1978), and Miyao (1981) in the case of multiple income
classes. These model showed comparative statics results depend on whether the
city is assumed to be open or closed. Nevertheless, a model of residential land
use in which a group of city residents is assumed to own a part of the city's
land has not been studied in urban economic theory. As a consequence, we pro
pose a basic model of residential land use in Chapter 3, called an <x-land
ownership model, in which a group of city residents is assumed to own the cir
cular area from the CBD to a radius <x, and a general model of residential land
use in Chapter 4, called an S-landownership model, in which we replace the
<x-land with an arbitrary subset, say S, of the city's land.
Crowding externalities with space among households has been studied by
Richardson (1977), Grieson and Murray (1981), Tauchen (1981), Scotchmer (1982),
and Miyao and Kanemoto (1987). Richardson (1977) suggested that crowding may
result in a positive rent gradient. This interesting idea was proved unlikely
by Grieson and Murray (1981) and Tauchen (1981). Fujita (1989, Ch.7) proposed a
log-linear specification of the model by Richardson (1977). This externality
model proposed by Fujita (1989) is very useful not only for the analysis of the
efficient provision of the local public goods established by Samuelson (1954)
and Tiebout(1956), but also for that of neighborhood externalities such as cro
wding externalities, racial externalities, and traffic congestion. As a con
sequence, we extend an <x-landownership model by introducing crowding externali
ties variable in Chapter 5.
