日本ヒートアイランド学会論文集 Vol.3 (2008)
Journal of Heat Island Institute International Vol.3 (2008)

学術論文

COOLING THE METROPOLIS: AN ECONOMICANALYSIS TO ALLEVIATE URBAN HEAT ISLAND
Kimitoshi SATO
Graduate School of Economics, Rikkyo University
E-mail address: satokt@rikkyo.ac.jp
ABSTRACTThis paper shows the necessary conditions to efficiently produce nongaseous Gorman-Lancasterian characteristics
embodied in goods, and to optimally emit heat, and trace gases as gaseous attributes combined in the urban atmosphere.
Sen’s capability approach is used to define personal well-being, since the impacts of urban warming upon each resident
affect his/her functionings à la Sen. Any inhabitant consumes goods and emits heat and gases in the urban air. It
is demonstrated that any inhabitant maximizes his/her happiness function by consuming goods and emitting heat andgases in the ambient urban air. This paper introduces heat island integral to represent the magnitude of heat island
and an urban warming function. Also, producers and landscape gardeners in the metropolis provide goods or plant trees
in metropolitan residents’ gardens. A tax-subsidy scheme is proposed to cope with urban heat island, which aims tooptimally adjust heat in the urban atmosphere.
Key Words: Goods as a complex of Gorman-Lancasterian gaseous and nongaseous attributes, Heat as an intangible
attribute, Heat island integral, Urban heat island tax and subsidy scheme, Sen’s functionings and happiness function

1.

INTRODUCTION

1.1 The Issue of Urban Heat Island
Tokyo is now a gigantic Urban Heat Island with peaks
such as Otemachi and Shinjuku which are the central business districts of Tokyo. Whereas, the Imperial Palace, theMeiji Shrine, and the Shinjuku Gyoen National Garden
as Cool Islands are cooler, and the temperature of theseareas remains rather low compared to the hotter spots
mentioned above. By using a concept of Heat Island Integral, formal definitions of an urban heat island and a coolisland are given in Section 2.3.
In the 1930s there were only seven hot nights called
tropical nights in which the temperature in the nighttimerose over 25◦ C. Tokyo had 14 or 15 tropical nights per
year in the 1960s and this doubled in the 1990s. We experienced 67 tropical nights in 2000. In 2002, 53 days wereobserved in which the temperature rose over 30◦ C, andthis number greatly exceeded that of an ordinary year,
i.e., 38.4 days. The average number of tropical nights in
Hiroshima City was 4.9 in the 1930s and increased to 28.8
in the 1990s. Kumagaya City in the Saitama Prefecture
had ten times the amount of days over 30◦ C than it had
just a half-decade ago. The average temperature of Japan
hit a new high in November 2003.
The global mean temperature has risen 0.74◦ C in thepast hundred years. It is 0.9◦ C in Japan and 3◦ C in
Tokyo, thus, Tokyo has been warming more than fourtimes faster than the Earth. The area of Tokyo’s urban
district and its artificial emission became one hundred
times in the 20th century. These are direct causes of anurban heat island. Whereas, the increase in the temperature in Paris and New York were respectively about 0.9◦ C
and 1.6◦ C in the same period. Tokyo’s urban warming has
a remarkable value which is above the rising rate of thepast ten thousand years.
The Tokyo Metropolitan Research Institute for Environmental Protection estimated that the heat emitted
from cars and air conditioners has resulted in an increaseof 0.4◦ C and that from the loss of greenery by 1.4◦ C in one
hundred years. The green area was halved in Japan from1930 to 1990, thus, there has been an increase in radiantheat and a decrease in radiational cooling. It is therefore
a pressing need to manage the urban thermal pollution
due to an urban warming in large cities of Japan.
Urban agglomeration has been increasing the number ofhouses and condominiums, which entails a loss of greeneryand water. Factories have been decreasing, but houses
have been increasing: half of the 23 wards of Tokyo (60,000ha.) share housing which has augmented by 30%. Heatfrom houses and buildings has increased to three times
as much as three decades ago. It is now far ahead ofheat from cars. The diffusion rate of air conditioners was
30% thirty years ago, however, it is 120% now, i.e., many
households have more than two air conditioners.
Tokyo’s floor space of buildings and their energy consumption has considerably augmented in the past 100
years, so that high-rise buildings are limited to several
areas. However, due to the progress of urbanization andthe augmentation of energy intensity of office automation business machines, energy consumption has grown
rapidly. As the capacity intensity and floor area ratio-1-increase, the total energy released to the ambient atmosphere grows remarkably. Agreeability of urban life ismarkedly lost when the temperature exceeds 35◦ C, whena human would have serious problems of health. Tokyo
with horribly high humidity will experience such unpleasantness in the near future, as approaching year 2030.1
1.2 Proposal of an Urban Heat Island Model
Different from the results of Rouillon(2000) and (2001)
on irreversibility of global warming, urban warming would
be reversible within the not so distant future by employing some appropriate countermeasures against an urban
heat island. As there are some surgical operations for our
warming metropolises, some recovery may be expected.
We do not have to accept this as an incurable disease. Human beings are unable to have a “zero emission solution,”because they emit heat and CO2 even when breathing
and consuming goods. They are also by-products in numerous manufacturing procedures. Hence, we must aim
to find a “livable/viable”combination of socially optimallevels of the ambient air quality and heat emissions. The
environmental managers of many cities have been utilizing
the command and control as well as the tax and subsidy
systems. Efficiency of these systems, however, is likely to
be lost because they cannot be made cost-effective without exact knowledge of pollution emitted by each household or each firm. Furthermore, if it is impossible to directly observe the amount of each emission, the environmental managers cannot deal with the incentive problem,since relevant information is privately held and must be
elicited. Thus, very often, a pollution abatement scheme
can be neither informationally efficient nor incentive compatible.
However, heat emitted by each emitter is assumed to beobserved somehow or other, so this paper proposes a taxsubsidy system to deal with an urban heat island, which
can do without the exact knowledge of the heat emitted
by each economic agent. Moreover, since urban warming has become a socially significant issue in recent years,
an increased attention has been paid to scientific breakthroughs and powerful remedies that may be more immediate than economic instruments, such as greening methods and other technological developments to cool down
the urban atmosphere.
It has been so far a general case in public economic
theory that there is no public good at the beginning of
the model, and the issue is how to decide on an optimalquantity of the public good by using the private information, i.e., consumers’ preferences. Our case in the present
paper is, however, that the urban atmosphere exists asa public good at the outset, and the problem is how tochoose both an acceptable atmospheric quality and a sensory endurable heat emission. Hence, I propose a model
to provide an optimal amount of heat as an urban public
good. In order that this model be operational, necessity
compels us to devise a tax-subsidy scheme to determine
the sum of heat emissions released by metropolitans into
the urban atmosphere.
The urban environment should now be perceived to be
an intergenerational public good that we have to protect.
Metropolitan residents are to be involved in the problem
1 For simulations of future Tokyo, see, for example, Saitoh,
Shimada and Hoshi(1996), Saitoh and Yamada(1999), (2000)
and (2001).

of urban warming which is now confirmed to be causedby the emission of enormous heat, i.e., all the households
living in metropolises are polluters as well as victims of
the warming climate. The metropolitan government is incharge of controlling the total amount of heat emitted byresidents and producers to keep the urban climate not so
unpleasant in the very near future.
1.3 Some Incidents due to Urban Heat Island
The ecological system has undergone great changes inmetropolitan areas. The cherry-blossom front has changed,i.e., Tokyo could be the first city where the cherry trees
bloom in Japan irrespective of its latitude. Tokyo could
now be inhabitable for tropical parakeets and hemp palms
too. The cicadas have been observed to change from thosethat like the damp soil to those which are fond of the dry
soil, so that intermittent choruses of cicadas have beendiffering in recent summers of Tokyo. Green-banded swallowtails originally from the torrid zone are protected topropagate in the center of Tokyo with the urban heat island. Palm trees can tune themselves to life in Japan and
now winter in the Botanical Garden of Tohoku University in the Sendai city which lies north of Tokyo. Furthermore, something unusual has been happening in themarine ecosystem, i.e., some types of poisonous plankton
have increased due to an early outbreak of red tides, which
occured not in July nor August, but in May. The temperature of the surface of brine has increased these past
several years.
An incident of heavy rain occured in Nerima in Tokyoin September 3, 2003. It was 33.9◦ C in Nerima where
there was no sea wind. Whereas a sea breeze with a windvelocity of 3 meters per second blew to Otemachi in the
midst of Tokyo, where it was 32◦ C, and littoral Shinkiba with a breeze of 4m/s had 28.6◦ C. The mercury
touched 33.4◦ C in Tokyo, but 19mm per 10 minutes ofheavy rain from 18:30 to 18:40 decreased its temperaturefrom 31◦ C to 24.2◦ C, and it was accompanied by thunderand flashes of lightning. It was reported that Otemachi
had more rain than the twenty-one suburban areas: it was
30% more than the other areas at the maximum. The
Tokyo Metropolitan Government has appointed priority
areas where it endeavors to prepare rainpool improvement projects to raise the drainage capacity of the sewers.
There is also an increasing possibility that underground
shopping centers will be submerged by urban floods.
Large cities such as Tokyo, Osaka, Sendai, Hiroshima,
Fukuoka, Nagoya, and Sapporo consume an enormous
amount of energy by using cars and air conditioners, hence,urban warming has now become notable. Cars have been
multiplying unusually in Japan, and the 7 million cars
are in circulation every day, which end in huge energy
consumption. It follows that there is also an increase inemissions of heat from car air conditioners. One fifth ofcars are now RVs that drivers can buy cheaper than before the introduction of an auto tax. A few decades ago,
each family owned a car, but now it is more common foreach person to possess a car. Thus, the number of drivers
has augmented by 24% in this decade.
Urban heat island remarkably reveals itself in winters,
when the temperature does not go under the freezing
point. One of the main factors of urban heat island is thealteration of land usage, since the heat stored cruciallydepends upon what ground covering materials are used.

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The temperature is higher with concrete covering thanwith bare and turf grounds, as artificial covering takes inheat. Dryness has been advancing in supercities, because
the roads and grounds are paved with asphalt and concrete which give off heat during the day and take it in atnight. Also because of this paving, the water vaporized
by the ground has decreased and the reflection of heat issevere in the summertime.
It has been reported that urban heat island has been
accelerating air pollution, which has resulted in a dust
dome: dust pollution is shut in as if it forms a dome.
Photochemical smog occured for 23 (30, resp.) days inTokyo (in the Saitama Prefecture, resp.). However, theemission of NOx was 51,000 tons in Saitama, whereas it
was 65,000 tons in Tokyo. This inverse of phenomena
could be explained by urban heat island. Hence, countermeasures for tackling the issue of urban heat island havethe positive side effect of mitigating urban air pollution.
It is observed that urban heat island is closely related
to pollinosis or hay fever, and urban air pollution. The
pollen scattered much in 2005, which may be due to theprecedent summer’s heat wave, and thus an increase ofpollen allergy was worried in 2005.
1.4 The Approach of This Paper“Integral” and “exponential ”are two key adjectives
given to the present state of the large cities in peril ofurban heat island, e.g., population explosion, build up of
noxious nitrogen dioxides due to increasing cars, and decreasing natural green resources. All of these phenomenain the metropolitan areas, which have been induced byhuman activities, especially agglomeration due to urbanization and motorization share common integral and exponential features, particularly after the drastic economic
growth three decades ago. The atmosphere of the supercities has been revealing the similar integral and exponential trends most prominent in these past decades. In order
to alleviate the warming tendency in the metropolises, every economic agent may well pay as a user charge of theurban atmospheric air, i.e., a “heat island tax, ”introduced below.2
Temperature revealed an exponential trend most prominently in recent decades. The atmosphere is a composite
of gases such as nitrogen (N2 ), oxygen (O2 ), hydrogen(H2 ), water vapor(H2 O), and Greenhouse gases (GHGs)
such as carbon dioxide (CO2 ), methane (CH4 ), nitrous oxide (N2 O), and chlorofluorocarbons (CFCs). These gases
can be considered as gaseous attributes à la GormanLancaster, precisely defined below. The urban atmosphereis made up of 78% nitrogen, 21% oxygen, water vapor,trace gases supra, etc.
This paper proposes to consider urban atmosphere as
a complex of intangible attributes such as heat and gases,
and the goods combined by nongaseous attributes that are
produced by the firms and the green areas as a composite of biological attributes supplied by landscape gardeners. The paper also shows that each resident has his/herland planted by a landscape gardener and simultaneously
emits heat and trace gases. For this, I adopt an analytical

2 See, for example, Hourcade et al.(1997) for an international

carbon tax to moderate global warming. See also Henry(1989)
for a market of emission rights as an alternative to the tax
scheme.

framework of the New Consumer Theory by Gorman andLancaster, and Sen’s Capability Theory.
The paper proceeds as follows: Section 2 presents the
periurban model of regional warming in the attributes/
functionings framework à la Gorman-Lancaster-Sen andproposes the concept of Heat Island Integral. The model
involves residents, producers such as offices, hotels, manufacturers and farmhouses as producers, and landscapegardeners who play a very important role to cool down
the metropolis by roof-top gardening and tree-planting
activities, etc.
2.

URBAN HEAT ISLAND IN A PERIURBAN
CITY

2.1 Heat Island Intensity
Urban heat island is represented by a contour line on amap joining points of equal temperature in the inside andthe outside of a city. It is an analogy of a contour line of
an island in a sea. Heat Island Intensity is defined as thehighest height of an island.
Let τu and τr be the temperature of urban and periurban or urban areas of a city. Urban temperature can be
represented by ∆τ or (∆τ /τr ) × 100, where τu = τr + ∆τor ∆τ = τu − τr . The following notation is also used.
Ξ : heat capacity radiated from the surface to the atmosphere, i.e., sensible heat flux [W/m2 ]; Π : city size[m]; ∆T /∆m : the difference in temperature [∆T ] between the places vertically separated [◦ C]; Q : air specificheat [Wh/kg◦ C]; M : atmospheric density [kg/m3 ]; V :wind velocity [m/s].
Summers(1965) proposed a procedure to easily calculate heat island intensity, which was further simplified by
Takeo Oka as the formula:sΞΠ∆T /∆m.
∆τ =1800QM V
It has been confirmed that this equation can be adapted
to such cities as Lund in Sweden.
Oke(1973) also proposed the formula:∆τ = 1.93 log P − 4.76
where P is the population of a city. This function meansthat population and urban temperature are closely related. He believed that this formula would be applicable
to European and North American cities. Sundborg(1951)
was the first to introduce a traveling observation in Uppsala. He statistically verified the difference of temperature
among the midtown area and rural points as a functionof cloudiness, temperature, absolute humidity, and windvelocity. These factors can be considered to be attributesof the urban air. Duckworth and Sandberg(1954) leptsuddenly to fame by showing the urban high temperature in an aeriel photo of San Francisco. It was after
their paper that the phenomenon was named “urban heatisland ”in which there has been a growing interest sincethen. Park(1987) verified the correlation for Japanese andKorean cities, and Yamashita(1988) observed urban heatislands in Japanese cities such as Kawagoe City in the
Saitama Prefecture.
Heat Island Intensity for the center of Sendai City in
the Tohoku District in Japan will be about 10.5◦ C, since
the temperature is anticipated to increase by 5◦ C by 2030.

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The temperature rose 2.1◦ C in Sendai in the past hundredyears. It is the second highest record next to Tokyo. However, it is reported that the zelcova trees planted in themain streets in Sendai can make a decrease in the sensory
temperature of 5◦ C. In Tokyo, the center of urban heatisland is found in the Nakano or Suginami Wards in wintertime and heat island intensity is 3◦ C. It is 10∼15km
in diameter and its form is the same all day long.

Not to mention, every inhabitant, producer or landscapegardener resides in some block, so that an index β will
be omitted hereafter in almost all the cases, except for
describing some variables related to block β.
Different from von Thünen(1826), or usual urban economic theory, it is hypothesized that our periurban city
as follows:4

2.2

H2. The city is not necessarily circular and its center iscalled the Central Business District(CBD).

A Model of Urban Heat Island in a Periurban
City
This section introduces the Attributes/Functionings
model based on the Gorman-Lancasterian theory where
goods are regarded as a composition of characteristics.
For the sake of simplicity, the two terms, “attributes ”
and “characteristics,”are used interchangeably throughout this paper. All trace gases such as GHGs can beinterpreted as intangible gaseous attributes in our framework, since they partially compose the urban atmosphereas an urban public good. Characteristics of urban climate are in order: pollution substances, solar radiation,
cloudiness, precipitation, temperature, absolute and relative humidity, and wind velocity. These attributes compose the urban ambient atmosphere. This paper focusesupon heat as an attribute, since it is the main cause of urban warming. Before rushing into our theoretical model,
let me introduce some basic concepts of attributes.
Let us consider a monocentric periurban city which is
supposed to be a “climatically closed city,” i.e., there are
no climatic influences from and on the neighboring cities.
“Periurban”means the peripheral zone of the urban area,
where reside inhabitants, manufacturers, farmhouses andlandscape gardeners.3 For the sake of simplicity, assume
that our periurban city is composed of two regions, i.e.,
the urban area and the periurban area, which are divided
into many blocks, β ∈ ˛ = {1, ..., B} : the set of blocks. In
other words, our periurban city is represented by a meshof blocks, such as a grid of the center of Chicago. Assume
that whatever size and shape can be chosen for a block,as a smaller size in Tokyo or a larger one in Chicago.
Let there be N residents indexed by i ∈ N = {1, ..., N }:the set of inhabitants who live in the periurban belt. Eachindividual emits heat and trace gases when consuming
goods or services indexed by j and its set is J = {1, ..., J}.
For the sake of simplicity, it is assumed that each producer j supplies only one good j which is composed of C
characteristics indexed by c ∈ C = {1, ..., C}: the set of
attributes. Denote as qjc an amount of attribute c embodied in one unit of good j. There are producers: e.g.,
offices, hotels, manufacturers, and farmhouses.
Suppose that resident i chooses a landscape gardenernamed ` to have a part of his/her land planted with trees,and Λ = {1, ..., Λ} is the set of gardeners. Define q`s
as a biomass of species s in one square meter suppliedby gardener `, and S = {1, ..., S} as the set of species offlora and fauna as biological attributes. Let xij be resident i’s consumption of good j, and Ai` be his/her demand for the green area supplied by landscape gardener `,then, xi = (xi1 , ..., xiJ , Ai` ) is his/her consumption vector. There is also the metropolitan government whose
task is to reduce heat emission by making effective use ofan urban heat island tax/subsidy scheme defined below.
3 See Cavailhès et al.(2003) and (2004) for the concept of“periurban. ”

H1. The city is formed in a heterogeneous plain, where
the climate could differ among its areas.

H3. Its urban and periurban transportation systems areavailable in whatever direction.
H4. Land in the city is all owned by absentee landlords.
H5. All periurban residents commute to work for an officein the CBD, a gardener or in a farmhouse.
H6. All gardeners plant areas in a part of lands of residents in the periurban zone.
H7. Manufacturers produce goods and farmhouses makeagricultural products in the periurban region.
The urban atmosphere is regarded as a complex ofgaseous attributes including trace gases which are to bemainly generated by production and consumption activities. It is naturally assumed that the amount of gases suchas N2 , O2 , H2 are stationary, so I can focus upon heat andtrace gases as attributes in this paper. An index qjg is also
used hereafter to identify the gth gaseous attribute, andqjh is an amount emitted of heat as an attribute when
producing one unit of good j. Let G = {C + 1, ..., C + G}
be the set of trace gases which compose the urban atmosphere.
Taking urban warming into consideration, let us extend
and generalize the framework developed by Sato(2006a).
When producing one unit of good, each firm does notchoose but to jointly emit heat and trace gases as vexing
by-products, qjh ≥ 0, which is producer j’s unit emissionof heat. Thus, qjh xj is firm j’s amount emitted of heat
when it produces xj units of good j. Landscape gardeners
also emit heat and gases, q`h ≥ 0 and q`g ≥ 0, ∀g ∈ G,
which are gardener `’s unit emission of heat and any gas.
Hence, q`g A` and q`h A` are gardener `’s emitted quantityof a gas or heat in his/her production of A` units of greening service `. Inhabitants also emit heat and gases, qih ≥ 0
and qig ≥ 0, ∀g ∈ G, which is city dweller i’s unit emissionof heat and gas. Hence, qih Ai` and qig Ai` , ∀g ∈ G, is individual i’s emitted quantity of heat and a gas in his/her
consumption of Ai` units of greening service `.
Let zi0 be resident i’s available time as a numéraire
characteristic that he/she possesses, by which other attributes can be utilized. Amounts of each attribute embodied in the goods and the atmosphere which are consumed by resident i are given for any c ∈ C and for any
g∈G
X
zic =
qjc xij
(1)
j∈J

zg =

X

qjg

X

i∈N

xij +

i∈N

j∈J

+

X

qig

X
j∈J

xij +

X

qjg Aj +

j∈J

X X

X

q`g Ai``∈Λ

qig Ai`(2)`∈Λ i∈N`

4 See Samuelson(1983) and Huriot(1994) for the Thünenmodel.

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entire metropolis, as examplified in the Nerima incident
in the Introduction.

and
zh =

X

qjh

j∈J

+

X

X

xij +

i∈N

qih

i∈N

X
j∈J

xij +

X

qjh Aj +

j∈J

X X

X

q`h Ai``∈Λ

qih Ai` .

(3)`∈Λ i∈N`

In the above equation, zic means the consumption of
tangible nongaseous attributes which compose goods,
while zg and zh represent the total amount of a trace
gas and heat emitted by all residents, producers and gardeners. Note that the values of zg , ∀g ∈ G, and zh can bemeasured via ton or kilojoule. Heat and gases are generated both in the consumption and production of goods.
Every inhabitant is made to consume not only his/heremission but also the quantity emitted by the rest of thecity. When he/she uses goods, he/she emits heat andgases, which were already released when the goods weremade by producers.
Both of the above equations may be interpreted as characteristics availability functions, which convert commodities into attributes. The amount of any characteristic ineach good can be regarded as a parameter that is objective
and common to all consumers, i.e., it has a public-goodproperty. Thus, the inhabitants as consumers must behave as “quality takers”, since they can only change theirconsumption of zic , zih and zig , via the choice of xij andAi` . Producers and landscape gardeners can choose thecomposition of attributes embedded in their goods andtheir greening service.
A part, ηg zg , 0 < ηg < 1, of an aggregate emission ofgases is observed to stay in the urban atmosphere andthe rest, (1 − ηg )zg , is perceived to disintegrate. Of thisamount, about 46% of CO2 emission is absorbed by theoceans and forests as carbon sinks. An integration rateor an inverse of a lifetime of each trace gas is denoted
as λg , with 0 < λg < ηg , ∀g ∈ G, so that the mass ofthe gth gas that stays in the urban air is represented by(ηg − λg ) zg , ∀g ∈ G.
Let θg be a conversion parameter from mass(ton/year)
to concentration(ppmv), and zg (γ) be an amount emitted
at time γ of the gth trace gas. Then an amount of trace
gas accumulated from time t0 to time t, which is converted
into a concentration is given by
Z t
ζgt =
θg (ηgt − λtg )zg (γ) dγ, ∀g ∈ G.
(4)
t0

This can be written as
Z t
Θtg zg (γ) dγ, ∀g ∈ G
ζgt =

(5)

2.3 Heat Island Integral
The problem of how to represent the heat in the city
was analyzed in Sato(2006a) who introduced the conceptof “Heat Island Integral.”5 In effect, there must be differences in the temperature of building surface, back alleys,
rooftops, streets, and green tracts of land, which are directly exposed to the solar radiation. However, these differences of the surface temperature of the ground coveragecan be measured by utilizing the thermography.
Let Fβ be an area of a block β and Sβ be its level surface
projection, i.e., the area which could absorb the solar radiation. More precisely, it is the sum of developable areasof the ground coverage, e.g., the streets, the tree crowns,
the rooftops and walls of the buildings, which exist in Fβ .
Denote πβ (u, w) as a function of the surface temperatureof Sβ and ηβ (u, w) as a function of atmospheric temperature over Fβ , where u and w are the plane coordinates.
Then, the Riemann sum led me to propose a concept ofHeat Island Integral which reads, ∀β ∈ ˛ :
(Z Z
)
ZZ
1
ηβ (u, w)dudw .
πβ (u, w)dudw −
Υβ =
Sβ
Fβ
Sβ
Needless to say, the existence condition of this multiple
integral is that the functions, πβ (u, w) and ηβ (u, w), arecontinuous and compact in the domains Sβ and Fβ , and
it is easily seen that this condition is satisfied.
Remark 1. Υβ is not the result of heat emission on thepart of the economic agents, but it stems from the groundcoverage of the metropolis. Υβ measures the force of urban heat island in each block β, since the temperatureof asphalt and concrete is very often higher than that of
the atmosphere. Asphalt and concrete as meterials areobserved to absorb approximately 90% of the solar radiation, hence, the walls of buildings made of concrete absorb
the same amount of heat, which result in urban warming.
Let me give numerical examples of the values in Υβ . The
mercury stood at 27.5 degrees at an observation point inShinjuku at 14:00 on June 6, 2002, where the wall of abuilding made of concrete was observed to have the heatof 38◦ C and a street of asphalt had 45◦ C. The central
area of Shinjuku (Fβ = 4km2 ) must be extended to its
plane of projection (Sβ = 9.3km2 ), which is exposed tothe solar radiation. Also, heat is emitted from the underground shopping malls in Shinjuku. By thermography,
πβ (·) and ηβ (·) can be constructed.

t0

where Θtg ≡ θg (ηgt − λtg ) is a climatical parameter related
to the gth trace gas. An argument of time t is omitted
hereafter, unless necessary. Therefore one observes
Z = (ζ1 , ..., ζG )(6)

which affects all residents, producers and landscape gardeners in the periurban city.
Another equation is proposed for heat as a flow in thenext subsection. It is the microclimate in the block, whichmost influences any economic agent who resides or works
in β. However, climatical incidents depend upon not only
the concentrations in each block, but also those in the

The following formula is physically supposed: Sensible
Heat Flux of Block β(Ξβ ) = Heat Island Integral (Υβ )×Convective Heat Conductivity(κ), ∀β ∈ ˛, where κ varies
according to the temperature. Sensible heat flux from theground coverage of the metropolis therefore is representedbyXX
Ξβ (A, W ) = κΥβ (A, W ) .
(7)
β∈˛

β∈˛

Note that this function depends on the planted areas andthe amount of water. Both of them can diminish the
5 “Heat Island Potentail”developed by Iino and Hoyano(1995) incited me to propose this concept.

-5-value of Υβ by cooling down the ground coverage of thecity.
Let εi , εj and ε` be heat emitted by resident i, firm j,and landscape gardener ` in the metropolis. The definitions of εi , εj , and ε` are implicit in Eq.(3). Total heatemission E which affects all economic agents in the largecity is thus represented as
X
X
X
X
E(A, W ) =
εi +
εj +
ε` +
Ξβ
(8)
i∈N

j∈J

`∈Λ

β∈˛

The first three terms represent artificial emissions of heatand the last term is the sensible heat flux generated fromthe ground coverage, all of which are assumed here to depend upon the amounts of vegetation and water. Notethat heat contributes to urban warming, hence, a differentiability assumption is posed on the above equation.
Assumption 1. (i) E is convex and twice continuously
differentiable with ∂E/∂A < 0 and ∂E/∂W < 0. (ii)εi , εj , ε` and Ξβ are convex and twice continuously differentiable with ∂εe /∂A < 0 and ∂εe /∂W < 0, for any
e = i, j, `, ∂Ξβ /∂A < 0 and ∂Ξβ /∂W < 0.
Vegetation, Water and the Urban Warming
Function
Let Li , Lj and L` be the land leased by resident i, producer j, and landscape gardener `. Also let Ai ≡ ρi Li
and Aj ≡ ρj Lj be greening areas of their land required to
be planted with trees, where ρi and ρj are greening rates
for city dweller i and producer j, which are legally determined, e.g., at least 20% of the rooftop on the houses andbuildings, when they are newly built, enlarged or reconstructed in Tokyo. Suppose that landscape gardeners are
exempted from the obligation of greening a part of theirown lands in our model.
As I assumed that inhabitant i chooses one landscapegardener ` to green a part of his/her land Ai with plants,
at a cost of σ` /m2 . Then, each person is to have his/hergreen area planted with plants, insects and microorganisms in the soil. These living things are regarded as biological attributes that may be offered by gardener ` as
by-products, and let Ai` be his/her area Ai planted bygardener `. City dweller i therefore consumes biologicalattributes by possessing the green area in his/her land as
represented bycities to and fro. These green areas are regarded as amenities, which may be clubs or excludable public goods.
As I mentioned in the Introduction, the quantity of water existing in the metropolis is crucially important to coolsupercities, so underdrained rivers must be resuscitated.
Let W be the total amount of water in the metropolis
represented by
W = W0 + WP
(11)where W0 is the quantity of water that already exists and
Wp is the amount which is to be revived by public enterprises with a cost Ω(Wp ), e.g., resuscitating culverts.
By physics, it is easily concluded that the more abundant the amounts of A and W , the cooler the metropolis,i.e., the more, the better. Notice that A and W have apublic-good property, since their cooling effects can prevail all the blocks of the metropolis as beneficial externalities. This paper focuses on greening as one of theimportant natural factors which plays a significant role incooling down the supercity. Modeling the water management in more detail is presented in Sato(2006b).
Total heat emission defines the Urban Warming Function which represents the urban temperature:
U = U (E) .

2.4

zis = q`s Ai` , ∀s ∈ S, ∀` ∈ Λ.

(9)

Resident i can enjoy seeing creatures in his/her green area:
they may be trees, flowers, insects, birds, or minute animals.
Suppose that each producer has an environmental
branch which is in charge of planting trees
Pin a part of its
lot size Aj , with its cost σj . Let AP ≡ β∈˛ Aβ be thearea to be planted with street trees and tree lawns undertaken by public works, and Γ(Ap ) is its cost. Summingthese green areas yields
XX
X
X
A=
Ai` +
Aj +
Aβ .
(10)
i∈N `∈Λj∈Jβ∈˛Incidentally, passers-by can see hedges of others’ houses,
and can have beautiful views of parks. Children canamuse themselves in small parks which would stud the(12)

The following assumption is needed.
Assumption 2. U is convex and twice continuously
differentiable with dU/dE > 0.
Remark 2. Urban warming is a typical example of apublic good which is both nonrival and nonexcludable.
However, its impacts on each resident differ among regions, which can be treated as regional public goods. Urban heat island with 33.5degc occured in some blocks of
the Nerima Ward, Tokyo on July 21, 1999, which was
not necessarily due to the consumptions and productionsof goods in that area, but due to those of districts inTokyo, as well as the climate conditions of the block atthat time. Consequently, the worst that could happen to
any block where some meteorological conditions are satisfied at some point in time, as in the above unforeseen
incident.
Let λββ 0 be the difference of the temperature between
one block β and the other block β 0 ∈ ˛. Discussions
so far eventuate in the following conditions. Urban heatisland could occur if there are at least three conditions,
i.e., i) Υβ > 0 and Υβ − Υβ 0 > 0, ∀β, β 0 ∈ ˛, ii) λββ 0 >0, ∀β, β 0 ∈ ˛, and iii) some meteorological incidents happen at some time in a day on rare occasions. Whereas,
Cool Island is present if Υβ ≤ 0 and Υβ −Υβ 0 > 0, ∀β, β 0 ∈˛ and if λββ 0 ≤ 0, ∀β, β 0 ∈ ˛. As was mentioned in the
Introduction, the Imperial Palace, the Meiji Shrine, andthe Shinjuku Gyoen National Garden as cool islands arecooler than the center of Shinjuku, i.e., the sensible heat
flux of the latter was observed to be about 300W/m2 ,whereas it was less than 200W/m2 in the Shinjuku Gyoen National Garden.
2.5

Beings and Functionings of Residents Menaced by Urban Heat Island
The city itself is a complex of an infinite number of attributes. The Gorman-Lancasterian characteristics theory is the most suitable to analyze goods and the urban-6-

atmosphere which is perfectly divisible and decomposable
as gaseous attributes. The characteristic availability functions can be applied to any resident whose utilization differs from person to person. Consequently, each inhabitant’s functionings should be introduced as one of theimportant concepts à la Sen(1985) to fully appraise thevalue of goods or characteristics. Each resident’s physicaland climatical situations differ, so I must introduce thefunctionings which are represented below.
Metropolitan consumers use personal computers, printers, drive cars and cook meals. They also watch televisions, listen to music, and make photocopies. All ofthese behaviors may be considered as functionings, thus,
city dwellers emit heat and gases when they use theirfunctionings. Much heat is emitted from air-conditioners,
and large-sized refrigerators in houses and supermarkets
to store many things such as fruits, vegetables, fish, meat,
poultry, dairy produces and frozen foods.
Different from Sen, an inhabitant’s beings can be represented as a vector of functionings. Let fi be a vectorof resident i’s functionings. The set of person i’s functionings is denoted as Ki . Ki includes resident i’s physical ability which heavily depends upon its climate in theblock where he/she lives or works, e.g., the temperatureof more than 40◦ C in the daytime would bring criticalsituations for human bodies: an increase of body temperature, heat exhaustion, bad circulation, etc. Not to mention, the number of functionings Ki differs among persons.
If he/she chooses a vector of functionings, then his/herbeings are generated by his/her functionings,fik , ∀k =1, ..., Ki ∈ Ki . As for heat emission, there are two subNE
N
sets, KEi and Ki (Ki = Ki ∪ Ki ), i.e., the functionings
are
those
that
emit
heat, whereas, thosebelonging to KEiin KN
i do not. Different from the above examples of urban
warming, some functionings can cool the cities. Drawing
water from wells to sprinkle on the rooftops also cools thehouses. Generally, 20∼30% of heat enters from the windows and 10% of it flees from them. So we could makeefficient use of air conditioning by employing blinds andcurtains, since pulling down blinds provides excellent protection against the sun. Window glass type may well bechanged to one which does not easily accept heat. Theseare examples for some functionings of metropolitan residents. One liter of water per hour could be discharged
from the body when the thermometer shows above 30◦ C;
some people could die of dehydration. Also indicatedis that victims of heatstroke will exponentially increase
when the mercury exceeds more than 34◦ C. City dwellers
will want to have more water and cold drinks, to swim ina pool or in the sea, to see tropical fish in an aquarium.
When the temperature approaches 40◦ C, the number ofsunstroke victims will increase. Urban heat island willcause many residents’ deaths in metropolises. The number of people conveyed by ambulance was more than four
hundreds in 2003, which was the maximum in the five
years, since the observation was started. Many people
will be sick because they stay in rooms extremely aircooled for a long time. Heat-sensitive persons will haveto pay medical expenses when they suffer from heatstrokeor other diseases. Electricity consumption can be reducedby raising the set point of the temperature of air conditioners in the rooms.6 Medical methods can be developed
6 The power saving policies are enforced in the ward offices
in Tokyo and the municipal institutions where the temperature

to help alleviate the possible spread of tropical contagious
diseases carried by vermin. These are examples of functionings, too.
Remark 3. I would like to point out that the following
fact is very important. Residents emit heat by utilizing
some of their functionings, and at the same time, they
suffer from the heat they emit. That is, some functionings fik ∈ KE
i of a person could inflict pain upon his/herother functionings fik ∈ KN
i , by prioritizing too much theconvenience and the comfortableness as immediate profits in his or her everyday life. Humans are apt to think
only of the present and take a short view of things, andthis mentality may be one of the main causes which havebrought urban heat island in the metropolises.
Denote ziC = (zi1 , ..., ziC ) , ziG = (ziC+1 , ..., ziC+G ),and ziS = (ziC+G+1 , ..., ziC+G+S ). Amounts of each characteristic embodied in the goods, the plants in the green
area, the atmosphere and the greenery consumed by resident i in the periurban metropolis is given by“”zi = zi0 , ziC , ziG , ziS , zih .
(13)
The sum of green area A is composed by plants andother living things as biological attributes and the total
amount of water is also regarded as a composite of characteristics, which may be called generically as waterness
as above. A and W not only cool down cities, but also
give residents a tasteful life instead of a prosaic life, sothat they enter in city dweller i’s consumption vector of
attributes as public goods.
Personal beings of inhabitant i may be representableas:bi = (fi1 (zi , U ) , ..., fiKi (zi , U ))(14)
Hence, the following assumption is needed.
Assumption 3. For any i ∈ N and for any fik ∈ Ki ,fik is concave and twice continuously differentiable on theclosed convex consumption set Mi .
Remark 4. It is considered that a change in the numéraire attribute zi0 can vary resident i’s functionings. For
example, consuming a huge amount of goods in highconsumption societies in developed countries can be supported from combusting enormous quantities of fossil fuels, which can be acquired by utilizing the numéraire characteristic zi0 and functionings. The signs of ∂fik /∂zic and2∂ 2 fik /∂zic
depend upon what characteristic c is, i.e., it
can take a sign {+, 0, −} according to attribute c which isgood, irrelevant, or bad, respectively, for resident i’s wellbeing. If c is heat, then the sign may now be minus formany metropolitan residents in summertime because ofhealth problems due to an increasing level of urban heatisland. Assumption 3 means that the more heat emitted into the urban air, the less functionings that residents
can utilize to enjoy their metropolitan lives. They feelquite aggravated by the sweltering heat during the day,and they may suffer from heatstroke. Summer nights withmore than 25◦ C often keep them from sleep, so that they
have to endure sleepless nights.
of the air conditioners is set at 28◦ C. The same is true for many
companies.

-7-

2.6

Resident’s Happiness Function and Valuing
Personal Well-Being
Let any inhabitant i have his/her Happiness Function
which is assumed to depend upon his/her being, thus, one
observes
Hi = Hi (bi ) .
(15)
Any person’s use of functionings can vary his/her personal happiness. In order to obtain our desired results, I
need another differentiability assumption.
Assumption 4. For any i ∈ N, Hi is strictly concave
and twice continuously differentiable with (∂Hi /∂fik ) ·(∂fik /∂zi0 ) 6= 0 for at least one k ∈ Ki .
Denote resident i’s hedonic shadow price or hedonic
marginal willingness-to-pay (HMW) of any gaseous andnongaseous characteristic c, and of any biological attributes in greenery as an attribute:Pk∈Ki (∂Hi /∂fik ) (∂fik /∂zic )πic = P, ∀c ∈ C
k∈Ki (∂Hi /∂fik ) (∂fik /∂zi0 )Pk∈Kiπis = P(∂Hi /∂fik ) (∂fik /∂zis )k∈Ki (∂Hi /∂fik ) (∂fik /∂zi0 )Pk∈Kiπig = P(∂Hi /∂fik ) (∂fik /∂zg )k∈Ki (∂Hi /∂fik ) (∂fik /∂zi0 ), ∀s ∈ S, ∀g ∈ G

k∈Ki (∂Hi /∂fik ) (∂fik /∂U )(dU/dE)

P

The set of feasible functionings vectors for any personis the personal capability set, i.e., opportunities to achieve
personal well-being. Residents live in some block, andmany of them work in other block. When they move insummertime, they may suffer from the sweltering heat.
Hence, urban warming exerts an influence on residents,
wherever they live in large cities.
Let xi be resident i’s entitlements of goods, and Xi
be its set. Given xi and U, one can represent the setof feasible beings vector, or the personal capability set of
inhabitant i under urban warming as:
Bi (xi , U ) = {bi |bi = (fi1 (zi , U ) , ..., fiKi (zi , U )) ,f or some ki in Ki and f or some xi in Xi }.

P
πih =

emit heat as a vexing by-product. Consequently, πih < 0holds for many people, who feel displeased by the scorching heat due to urban
warming. However, they wish toP
avoid this, thus, c∈C∪G πic qjc À 0 still prevails for j if
it is an air conditioner. It cannot be helped, since everybody wants to keep his/her room cool, especially undera burning sun. It is not only heat but also the resulting
high humidity, and an actual and sensary high temperature due to an increase of heat, that could lower somefunctionings of residents. This fact can be represented by(∂fik /∂U ) (dU/dE) < 0, ∀i ∈ N, ∀fik ∈ Ki .

k∈Ki (∂Hi /∂fik ) (∂fik /∂zi0 ).

Remark 5. (i) πic , πis , πig or πih is a marginal contribution of each attribute to resident i’s marginal happiness through his/her functionings in terms of the numéraire characteristic zi0 . It may correspond to a “marginal
rate of substitution(MRS)”between each attribute andthe numéraire characteristic zi0 in the utility theoreticalcontext.
(ii) Note that our “MRS” is totally different from Drèze
and Hagen(1978), since it involves the concepts of functionings and a happiness function à la Sen, and moreover,it can be applied to detrimental attributes too. Hence,in Sato(2000) I had to replace a happiness function for autility function. Note also that the analysis of Drèze andHagen allows for negative MRSsP
of some consumers, withan additional assumption thati πic > 0 as expressedin our notation. This means that even if some person(s)put(s) negative marginal evaluation πic < 0 on some characteristic c, the aggregate value of MRSs over residents
can still be positive, i.e., they admit this attribute.
(iii) Whether inhabitant i considers a commodity ora service as good, irrelevant, or bad toP his or her functionings is confirmed
by
the
sign
of
c∈C∪G πic qjc forPeach good j and c∈B∪G πic q`c for any service `. Moreover, whether a commodity or a service is socially good ornotP may also be examined by summing
P over individuals of
π
q
for
any
good
j
and
ic
jc
c∈C∪G
c∈B∪G πic q`c for any
service `. Now, to keep the room open to the cool air, an
air conditioner is one of the necessities for our comfortable urban life under urban heat island, even if it does

Limiting our analysis to the persons’ risky aspects ofliving which are menaced by urban warming due to human activities, one may interpret bi as a “personal wellness index,” since bi , ceteris paribus, corresponds to somepersonal being. In our context an inhabitant enjoys hisor her life by using his/her functionings, which enhance
his/her happiness and well-being. The health of the population in the metropolis is not only an ultimate objective, but also a means which permits people to experienceagreeable urban lifestyles. The crucial problem is thatmany residents’ personal capability sets would shrink due
to the accelerating urban warming in the near future, i.e.,
Bt+1
(xt+1
, U t+1 ) ⊂ Bti (xti , U t ).
ii
In Sato(2000) I characterized the conditions for a socially optimal consumption of goods as well as the global
ambient air in terms of characteristics including gaseous
attributes. I added “climatical constraints” to confirm
an optimal composition of the global atmosphere to maximize a personal happiness function, which depends upon
his/her functionings. Section 3 presents the main results,
i.e., a derivation of optimality conditions for the urban climate and a design of heat island tax/subsidy scheme.
3.

HEDONIC OPTIMALITY CONDITIONSFOR METROPOLITANS UNDER URBAN
HEAT ISLAND

3.1 Periurban Residents
Consider that city dwellers know the risks of man-madefuture urban climate changes, and that they have an incentive to optimize the composition of the urban atmosphere in order to aim at achieving their personal bestbeing in the metropolis. Let δ be a distance from thecenter of the CBD. Denote ri (δ) as the rent of one square
meter of land, and each inhabitant’s rent is ri (δ)Li , when
he/she leases a lot Li from an absentee landlord. ξi (δ)
is the transportation cost to go to whatever place the-8-resident i likes, by using his/her private car(s), and public transportation systems such as buses, subways, trains,and streetcars. For example, with his/her family, he/she
visits museums or concert halls in the CBD or naturalparks in the periurban zone. It is assumed that his/hercommuting cost is payed by the firm in which he/sheworks, and that it is involved in zi0 .7
Denote ε−i (α−i ) = (ε1 (α1 ), . . . , εi−1 (αi−1 ), εi+1 (αi+1 ),. . . , εN +J+Λ (αN +J+Λ )). Let νi (εi (αi ), ε−i (α−i )) be resident i’s heat abatement cost which depends upon his/heremission of heat as well as those of others. A parameter
αi > 0 is associated with his/her effort to reduce heatemission. It is assumed that αi is known only to him/her.
Functionings to cool cities are, for example, economizing
in power by reducing the use of electrical fittings and byusing a bicycle or public transportations instead of a car.
As urban heat island accelerates, city dwellers will wantto buy newly developed, more efficient air conditioners,
which result in reduction of heat emission. These are examples of functionings to explain αi . Resident i has topay ti (αi ) as a heat island tax, in order for the urban atmosphere not to be warmed so much as to be unendurable
to live in.8
As is well-known, greening lands and rooftops are oneof the most effective ways to cool the metropolis. Let
the metropolitan government determine 0 < ϕ < 1 as
the refund rate for greening the area Ai` that resident
i requests landscape gardener `. Assume that the valueof ϕ is decided by the scientific data about tree-planting.
As defined, σ` is landscape gardener `’s greening cost per
square meter, hence, σ` Ai` is resident i’s greening cost,and ϕσ` Ai` is i’s refund for the effort to plant trees.
The set J includes all the goods and services such as
electricity and water but land and transportation, and pj
is a unit price of good j, then each inhabitant’s budget
constraint is given byXzi0 =pj xij + σ` Ai` − ϕσ` Ai` + ti (αi )j∈J+ νi (εi (αi ), ε−i (α−i )) + ri (δ)Li + ξi (δ).
The left-hand side of the above equation signifies the valueof numéraire attribute whose price is normalized to beone.
Assumption 5. For any i ∈ N, ri , νi , εi and ti areconvex and ξi is concave. Moreover, they are continuously
differentiable, with dri /dδ < 0, ∂νi /∂εi < 0, dεi /dαi < 0
dti /dαi < 0 and dξi /dδ > 0.
This assumption means that ri (δ) has a distance decay curve, and that ξi (δ) has an increasing curve of distance. The more heat is emitted, the more the urban air
warms, and the more costly life becomes. This signifies
that the more damages due to urban heat island augment,
the more residents have to pay for cool air, the more tax
they have to pay for alleviating the urban warming. Each
resident’s effort can diminish his/her tax payment.
7 Usually in the urban economics, a commuting cost is not
implicitly included in the income, but explicitly defined insteadof our traffic cost. Our model is somewhat different from thatin an ordinary urban economic theory.
8 See Sandmo(2000) for some related issues of taxes and alternatives to improve the environment.

The maximand is the personal happiness function.Each
periurban resident solves the following optimization problem:
M ax Hi = Hi (bi )Xs.t. zi0 =pj xij + σ` Ai` − ϕσ` Ai` + ti (αi )j∈J+ νi (εi (αi ), ε−i (α−i )) + ri (δ)Li + ξi (δ).
Now the first result is presented.
Lemma 1. For any periurban resident i ∈ N, an individually optimal consumption of goods jointly composed by
Gorman-Lancasterian gaseous and nongaseous attributes,
and of the green area as a complex of biological attributes
is characterized as: ∀j ∈ J, ∀` ∈ Λ
01XX
πic qjc ≤ pj , @πic qjc − pj A xij = 0
c∈C∪G∪{h}X

c∈C∪G∪{h}

πic q`c ≤ σ` (1 − ϕ),c∈S∪G∪{h}

8
<
0=

X

:

c∈S∪G∪{h}∂νi dεi
dti
≤−
,
dαi
∂εi dαiLi ≤ −

dξi /dδ,dri /dδ9=
πic q`c − σ` (1 − ϕ) Ai`
;

„

«
dti
∂νi dεi+αi = 0dαi∂εi dαi„«dξi /dδLi +δ=0dri /dδri ≤ −ρi σ` (1 − ϕ),{ri + ρi σ` (1 − ϕ)} Li = 0.

Remark 6. (i) The conditions presented are not only
necessary but also sufficient from the assumptions on thefunctions. In the first equations, πic signifies a hedonic
shadow price of an attribute acquired by utilizing inhabitant i’s numéraire characteristic through his/her functionings. The left-hand side of the first equation is thesum of resident i’s marginal evaluations of the nongaseous
attributes embodied in one unit of a good, as well as of thetangible attributes released when the good is produced.
Notice that the first formulae verify that any resident considers heat and gases as gaseous characteristics emitted
when consuming one unit of good j. The first conditions
mean that the unit price of the good is equal to the sumof marginal contributions of attributes to his/her happiness through his/her functionings. The conditions assurea Pareto optimality for a quantity of each good, and givea basis upon which goods household i chooses to buy.
(ii) The second conditions signify that individual i’s
marginal evaluation of species as biological attributes by
having the area Ai` of his/her land Li , planted by landscape gardener ` is equal to his/her greening cost per
square meter minus the refunded cost. The term, ϕσ`may be called a “heat island (alleviating) subsidy,”since
it represents the reward according to the person’s effort toplant trees in a part of his/her land. Many residents sufferfrom the fierce heat in summertime, if he/she feels very
displeased by the boiling weather due to urban warming.
(iii) The conditions about ti show that city dweller i’s
marginal evaluation of emitting a unit of heat is a matchfor his/her marginal heat island tax. The next conditions
let us know the lot size that the resident decides to lease-9-

from an absentee landlord. These are the equilibrium conditions of residential location. It is easily seen that Li isdetermined by the distance, the rent and the traffic costthat inhabitant i has to pay. The last two formulae determine the equilibrium rent in the metropolis. Note thatthe more rent increases, the more the area is greened, andthe less heat is emitted. Greening a rooftop or a garden as
a part ρi of his/her lot size, any resident can cool his/herhouse, which results in saving power. Thus, an incentive
is given to an inhabitant to have a part of his/her land
planted with trees and flowers from motives of selfishness.
3.2

Offices, Manufacturers and Farmhouses as
Producers
A framework that I earlier employed in Sato(2000) is
used to involve the phenomenon of urban warming due toheat released by the producers. I present the optimizationby profit maximizing producers to supply one good with
an optimal product quality to consumers. Let producer j(landscape gardener `) produce good j (service `) by using
Lj and xj0 (L` and x`0 ) as inputs,
price of xj0
P and the P(xP`0 ) is normalized to be one, with j∈J xj0 + `∈Λ x`0 ≤i∈N zi0 . Then, xj0 (x`0 ) is a numéraire attribute that
producer j (landscape gardener `) uses as an input.
Let yj = (xj0 , Lj , xj , qj1 , ..., qjC , qjC+1 , ..., qjC+G , qjh )be producer j’s input-outout vector, then it produces agood j as an output to maximize its profit subject to the
production function
ψj = ψj (xj0 , Lj , xj , qj1 , ..., qjC ,qjC+1 , ..., qjC+G , qjh ) ≤ 0(16)

where qjc , ∀c ∈ G, is an amount of a gas and qjh is aquantity of heat emitted in the urban atmosphere when
it produces one unit of good j, and (qjC+1 , ..., qC+G ) is avector of gaseous attributes. Meanwhile, (qj1 , ..., qjC ) is avector of nongaseous characteristics embodied in one unitof good j. Any producer therefore jointly produces heatand gaseous attributes. The production function may not
be convex, but the difficulties arising from nonconvexities
are not treated here, so I must make an assumption.
Assumption 6. For any j ∈ J, ψj is convex and twice
continuously differentiable on the closed convex production set Yj , with ∂xj0 /∂qjc > 0, ∀c ∈ C ∪ G, c 6= j 0 .
Furthermore, xj > 0 implies xj0 > 0 and Lj > 0, and∀∆ ∈ R+ , {yj |ψj (yj ) ≤ 0, xj0 ≤ ∆, Lj ≤ ∆} is compact.
It is assumed that πic and πig are truthful, since all
goods are private goods or publicly provided goods suchas gas, water and electricity, for which the residents haveto pay public utility charges. Hence, they cannot have forfree.
PWhen xij > 0, pj could be computed as c∈C∪G πic qjc
from Lemma 1, then the profit maximization problem forthe producers are given by99 To construct the producer’s profit function, I followedDrèze and Hagen(1978, p.510) who wrote that ”the implicit
price could be computed... and they would in equilibrium be
the same for all consumers. So we do not have to make pricedifferentiation among consumers.”This fact follows from theirassumption of nonsingularity of the technological matrix. If
the matrix qj and q` are nonsingular, then they have an inverse matrix. In our model qj and q` are not nonsingular,
so we assume the truthful revelation of πic , ∀c ∈ C ∪ G, andπis , ∀s ∈ S, ∀i ∈ N.

M ax Pj =XX

πic qjc xij − xj0 − rj (δ)Lji∈N c∈C∪G∪{h}− µj δxj − tj (αj ) − νj (εj (αj ), ε−j (α−j ))− χσj Aj + χϕσj Aj + (1 − χ)ωLjF
χ = 0 if j ∈ JF and χ = 1 if j ∈/ JF .
The first term of the R.H.S. is revenue. xj0 is the amountof labor used as an input, and rj (δ)Lj is the rent of land,
where Lj is a variable as an input. Denote ε−j (α−j ) =(ε1 (α1 ), . . . , εj−1 (αj−1 ), εj+1 (αj+1 ), . . . ,εN +J+Λ (αN +J+Λ )). Let νj (εj (αj ), ε−j (α−j )) be j’s heat
abatement cost, where αj > 0 is a parameter only known
to j to reduce heat emission. Note that νj depends on
the heat emissions of the rest of the city. It may be interpreted as an external cost to buy, for example, more
efficient power saving air conditioners. Heat island tax
tj is a function of αj which represents firm j’s effort todecrease an emissionof heat when producing and transP
porting xj = i∈N xij units of good j. More examples
to explain αj are in order: producer j encourages his/her
staff to commute by bicycle, to use hybrid or fuel cellcars as delivery vans, and to choose energy-saving type of
personal computers and printers in the office.
Let µj > 0 be a unit cost of transportation. It is necessary to include µj δxj as transportation cost in the above
equation, since delivery of goods result in emitting heat,
GHGs such as CO2 and CH4 , and noxious gases such as
SO2 and NOx .10 It is generally accepted that midtown
hotels and office buildings use more computers and air
conditioners than residents and landscape gardeners in
the periurban belt. The former would emit more heatand gases into the urban atmosphere so as to cool hotels
and offices, as urban heat island accelerates in the near
future. As producers, offices offer services, manufacturers
supply products, and farmhouses make agricultural products.
σj Aj is firm j’s greening cost, and ϕσj Aj is the refund
to firm j. Denote LjF as j’s farmland. Let ωLjF (0 <
ω < 1) be a reward given by the metropolitan government
to farmhouse j which has not turned its farmland into
housing. Let also JF be the set of farmhouses that are
exempted from the obligation to green a part of their own
lands.
Here I need another assumption.
Assumption 7. For any j ∈ J, rj , νj and tj are convex
and εj is concave. Moreover, they are twice continuously
differentiable with drj /dδ < 0, ∂νj /∂εj < 0, dtj /dαj < 0and dεj /dαj < 0.
In the presence of gaseous and nongaseous attributes,
I have the following result.
Lemma 2. For any periurban producer j ∈ J, necessary conditions for Pareto optimal product quality interms of gaseous and nongaseous attributes are: ∀c ∈ C,
c 6= j 0 , ∀g ∈ G, g 6= j 0 , h 6= j 0
Xi∈N

πic xij ≤

∂xj0
∂Lj
∂xj
+ rj
+ µj δ
∂qjc
∂qjc
∂qjc

10 CO emitted from the transportation branch is 20% in
2Japan.

-10-

0=

X

πic xij −

i∈N

−µj δ
X

πig xij ≤

i∈N

0=

∂xj
∂qjc

∂xj0
∂Lj
− rj
∂qjc
∂qjc

«
qjc

∂xj0
∂Lj
∂xj
+ rj
+ µj δ
∂qjg
∂qjg
∂qjg
X

πjg xij −

i∈N

∂xj0
∂Lj
− rj
∂qjg
∂qjg

«
∂xj
−µj δ
qjg
∂qjg
X
∂xj0
∂νj
∂xj
∂Lj
πih xij ≤+ rj+
xj + µj δ
∂q
∂q
∂ε
∂q
j
jh
jh
jh
i∈N
0=

X

πih xij −

i∈N

−µj δ

∂xj
∂qjh

∂νj
∂xj0
∂Lj
− rj
−
xj
∂qjh
∂qjh
∂εj

«
qjh

«
dtj
∂νj dεj+αj = 0dαj∂εj dαj
„
«
µj xj
µj xj
,
Lj +
δ=0
Lj ≤ −
drj /dδ
drj /dδ∂xj
rj ≤ −χρj σj (1 − ϕ) − µj δ
− (1 − χ)ωΠj
∂Lj

ff
∂xj
0 = rj + χρj σj (1 − ϕ) + µj δ
+ (1 − χ)ωΠj Lj∂Lj

dtj∂νj dεj≤−,dαj∂εj dαj„of producer j ∈/ JF . Whereas it equals to the terms of
MTC and the reward given to farmhouse j ∈ JF . The
amount of ωπj is given as a subsidy to farmhouse j forpreserving its farmland in the periurban area, which could
be expected to somewhat mitigate urban heat island.
3.3 Periurban Landscape Gardeners
Next, efficiency conditions for periurban landscape gardeners are derived. Their job is to plant trees in areas oflands that residents or producers possess, so they are exempted from the obligation of greening a part of their own
lands in our model. They supply plants as biological, andtherefore nongaseous attributes. Let r` (δ) L` be the land
rent, where r` (δ) is the rent per square meter and L` isthe land leased from the absentee landlord.
Denote ε−` (α−` ) = (ε1 (α1 ), . . . , ε`−1 (α`−1 ), ε`+1 (α`+1 ),. . . , εN +J+Λ (αN +J+Λ )). Let ν` (ε` (α` ) , ε−` (α−` )) be gardener `’s abatement cost, where α` > 0 is a parameter
only known to gardener `. Landscape gardeners can attempt to serve green areas to residents by using a newtechnology which does not emit too much heat. Let µ` δA`
be their transportation cost,
P where µ` is a unit cost oftransportation and A` =Thus, they emiti∈N Ai` .
trace gases such as CO2 , CO, NOx and SO2 into the air,when they use fertilizers, serve and transport their garden
plants. Any gardener serves a green area to residents tomaximize his/her profit subject to the production functionψ` = ψ` (x`0 , L` , A` , q`C+1 , ..., q`C+G , q`C+G+1 , ...,
q`C+G+S , q`h ).

χ = 0 if j ∈ JF and χ = 1 if j ∈/ JF .

Remark 7. (i) The first equations establish a Pareto optimality for an amount of each attribute and determine avector of optimal nongaseous characteristics embodied in
the goods supplied by producer j. The L.H.S. of the firstequation is the marginal revenue which is the aggregateof the residents’ marginal evaluations of an infinitesimalchange in an attribute embedded in xj . Its R.H.S. is themarginal cost in terms of the numéraire characteristic andthe land toP
produce qjc , as well as j’s marginal transportation cost.
i πic xij is the marginal social value of good j,which is the sum of the personal evaluations of a change
in an attribute when the quantity of good xj is produced.
(ii) The second two formulae verify a Pareto optimal
amount of of each gas as a gaseous attribute. The L.H.S.
is the social value of any gas, and the R.H.S. consists ofthree terms: the first two terms are the marginal cost interms of xj0 and Lj , and the last term is the marginaltransportation cost when producing xj units of good j.
(iii) The next two equations show a Pareto optimalquantity of heat as an intangible attribute. The L.H.S.
is the social value of heat, and the R.H.S. consists of fourterms: the first two terms are the marginal cost in termsof xj0 and Lj , the third term means the marginal damage
to emit heat, and the last term is the marginal transportation cost when producing xj units of good j.
(iv) As in Lemma 1, tj is the heat island tax of producer j, and the formulae of Lj determine the equilibrium
lot size. The last two equations determine the equilibrium
land rent, which is equal to the sum of the terms of greening cost σj /m2 and marginal transportation cost(MTC)

Hence, each gardener solves the optimization problem:
XX
M ax P` =πic q`c Ai` − {x`0 + µ` δA`i∈N c∈S∪G∪{h}+ t` (α` ) + ν` (ε` (α` ), ε−` (α−` )) + r` (δ)L` }.
Two assumptions are needed for this maximization
problem.
Assumption 8. For any ` ∈ Λ, ψ` is convex and twice
continuously differentiable on the closed convex production set Y` , with ∂x`0 /∂q`c > 0, ∀c ∈ S ∪ G, c 6= `0 .
Furthermore, A` > 0 implies x`0 > 0 and L` > 0, and∀∆ ∈ R+ , {y` |ψ` (y` ) ≤ 0, x`0 ≤ ∆, L` ≤ ∆} is compact.
Assumption 9. For any ` ∈ Λ, r` , ν and t` are convex
and ε` is concave. Moreover, they are twice continuously
differentiable with dr` /dδ < 0, ∂ν` /∂ε` < 0, dε` /dα` < 0and dt` /dα` < 0.
We state the following result.
Lemma 3.
For any periurban landscape gardener` ∈ Λ, necessary conditions f or P areto optimal qualityof planted green area as a complex of biological attributes
are: ∀s ∈ S, s 6= `0 , ∀g ∈ G, g 6= `0
X
i∈N

-11-

πis Ai` ≤

∂x`0
∂L`
∂A`
+ r`
+ µ` δ
∂q`s
∂q`s
∂q`s

0=

X

πis Ai` −

i∈N

−µ` δ
X

πig Ai` ≤

i∈N

0=

∂A`
∂q`s

∂x`0
∂L`
− r`
∂q`s
∂q`s

«
q`s

∂x`0
∂L`
∂A`
+ r`
+ µ` δ
∂q`g
∂q`g
∂q`g
X

πig Ai` −

i∈N

∂x`0
∂L`
− r`
∂q`g
∂q`g

«
∂A`
−µ` δ
q`g
∂q`g
X
∂L`
∂A`
∂x`0
πih Ai` ≤+ r`+ µ` δ
∂q
∂q
∂q
`h
`h
`h
i∈N
0=

X

πih Ai` −

i∈N

−µ` δ

∂A`
∂q`h

∂Lj
∂x`0
− r`
∂q`h
∂q`h

M in {Φ + Ψ + Γ + Ω}
where Γ(AP ) is the cost to plant the area AP with trees,and Ω(WP ) is the cost to revive the amount of water, WP ,in our periurban metropolis.
Assumption 10. Dβ , Γ, and Ω are concave and twice
continuously differentiable with dDβ /dU > 0, ∀β ∈ ˛,dΓ/dAP > 0 and dΩ/dWP > 0.
Since I seek for socially optimal quantities of greenery,water and heat emission, the first order conditions are:

«

q`h
„
«
dt`
∂ν` dε`dt`∂ν` dε`≤−,+α` = 0dα`∂ε` dα`dα`∂ε` dα`
„
«
µ` A`
µ` A`
, L` +
δ=0L` ≤ −
dr` /dδdr` /dδ
„
«
∂A`
∂A`
,
r` + µ` δL` = 0.
r` ≤ − µ` δ
∂L`
∂L`
Remark 8. As in Lemmata 1 and 2, some of the equations are reminiscent of Samuelson’s Conditions, since attributes, including biological ones are public goods. Gardeners make their products and transport them, so they
have to pay for transportation, µ` δA` , where µ` is a unit
cost of transportation. They plant trees on the lands ofprivate houses. The equations establish a Pareto optimality for an amount of each attribute and determine a vectorof optimal quality characteristics that any gardener can
supply its biological product to residents. The R.H.S. of
the first equation is the marginal cost composed of threeterms: the first is the marginal cost of the numéraire attribute and the second is the marginal cost of land to
supply one unit of plant as a biological attribute, and thethird is the marginal transportation cost. The first two
equations signify the marginal social value in terms of thenuméraire attribute, where the marginal social value is thesum of the personal evaluations of a change in each biological characteristic. The next eight equations have thesimilar implications as for the producers. As in the above
propostions, the last two formulae define the equilibrium
rent, which is equal to gardener `’s MTC.
3.4

waves.11 Also, Ψ includes the costs of cooling installations in stockyards and of increasing water consumption
due to cooling them. The social damage due to heat island therefore is Φ + Ψ. Hence, our problem of social costto be minimized is

The Metropolitan Government with an Urban Heat Island Tax/Subsidy Scheme
Here I explain a heat island tax/subsidy scheme. For
that purpose, define the monetary damages due to urban
heat island as
P the sum of costs to deal with public damages: Φ ≡β∈˛ Dβ (U (E)) and the sum of abatement
costs
to
cope
with
private damages: Ψ ≡
P
ν
(ε
(α
e
e
e ), ε−e (α−e )), where e is a generic ine∈N∪J∪Λ
dex for any heat emitter in the metropolis. Examples ofpublic damages and private damages are the inundationof a subway station, houses flooded above floor level orup to the floorboards. Φ embraces economic losses in thedeaths of people and domesticated animals due to heat

X dDβ (U ) dU ∂E

dΓ (AP )
=0
dAP

(17)

X dDβ (U ) dU ∂E
dΩ (WP )=0+dU
dE ∂WP
dWP

(18)

dU

β∈˛

dE ∂AP+β∈˛andX dDβ (U ) dU ∂E
β∈˛

+

dU

dE ∂εeX∂νe (εe (αe ) , ε−e (α−e ))= 0.
∂εee∈N∪J∪Λ(19)

The first term of the formula (17) signifies the marginal
damage of losing one square meter of green area, and thesecond term is the marginal cost of greening an area withan additional one square meter of the area. Similarly,
the first term of Eq.(18) is the marginal damage of losing
one cube meter of water, and the second term means themarginal cost to revive one cube meter of water in themetropolis.
Here presented is the main result.
Theorem. (i) Urban heat island tax is of the form à
la Groves represented by, ∀e ∈ N ∪ J ∪ Λ :Z αe∂νe (εe (ae ), ε−e (α−e )) dεedae + Qe (a−e )te (αe ) =
∂εe
dae
0
where Qe (a−e ) is a constant of integration independent ofae .
(ii) Heat island subsidies are given by
X
XXS=
ϕσ` Ai` +ϕσj Aj +ωπj Lj .
`∈Λ

j∈J\JF

j∈JF

Proof: By adopting the Laffont’s (1982) differential
method, one observes from Lemmata 1∼3 thatff
X  ∂νe (εe (αe ), ε−e (α−e )) dεedte (αe )
+
= 0.
∂εe
dαe
dαe
e∈N∪J∪Λ(20)
11 In 2003 Europe was overcome by abnormal weather withrecord fierce heat and fatalities numbered at more than 13,000
in France. Paris especially experienced violent heat wave withmore than 40◦ C. A heavy death toll was recorded because
many people did not have an air-conditioning in not so hotsummers in Paris.

-12-Integrating this yieldsX
te (αe ) =e∈N∪J∪Λ

−

X

Z αe

∂νe (εe (ae ) , ε−e (α−e )) dεe
dae
∂εe
dae
e∈N∪J∪Λo+ Qe (a−e ) .
(21)0Suppose that there exists a unique solution to the above
problem of social cost. As we aim to find the heat island
tax corresponding to the social optimum, integrating andsubstituting Eq.(19) into Eq.(21) givesXte (αe ) =e∈N∪J∪ΛX Z αe dDβ (U ) dU ∂E dεe
β∈˛

0

dU

dE ∂εe daedae + Q

(22)
where Q is a constant of integration.
Consequently, the sum of urban heat island taxes is represented byXX
te (αe ) =Dβ (U (E)) + R(23)e∈N∪J∪Λβ∈˛where R is a constant of integration.
It can be refunded to the collaborators of tree planting
plus the rewards to farmhouses as:
X
X
XS=
ϕσ` Ai` +ϕσj Aj +ωπj Lj .
(24)`∈Λj∈J\JF

j∈JF

Hence, subsidies are given to residents and producers
as cooperation for their efforts to have trees and flowers planted in the required percentage of their land lots.
The third term is the sum of rewards given to farmhouses
to preserve their farmlands. S is the sum of heat island
subsidies to cool down the metropolis. An amount of
R − S is used to undertake public works at the cost ofΓ(AP ) and Ω(WP ). Q.E.D.
Remark 9. What is important is that the R.H.S. of
the formula (23) does not depend upon any unobservableparameter αe , thus, it is immune to whatever manipulation on the part of the residents, producers and gardeners. The problem of preference revelation problem under incomplete information need not be discussed here.
As for monitoring, εe can be partially measured by theamounts of gas and electricity used by each agent. Then,
our tax/subsidy scheme is said to be strategy proof. Another important feature is that the tax levied can fully
cover the social damage, and the part of the tax overpaid
R can be redistributed to the contributors in order to cool
the metropolis as shown above.
Allow me to enumerate some issues for further research.
First of all, urban economic theoretical analysis must be
made to realistically involve urban heat island and land
use with an exact knowledge of this phenomenon. Secondly, a model incorporating both greening and water
management in more detail will be proposed in future
research. Finally, whether an optimal urban atmosphericquality can be implemented in a dynamic context will be
one of the main concerns of a consequent paper. It willshow that the composition of characteristics in the urban

atmosphere can be intertemporally optimized by adjusting each component. Thus, we have to design a dynamic
process to implement our urban hedonic optimality conditions.12
Now is the time when our civilization and urbanization
are changing the urban climate, which cannot but entail modifications of our lifestyles. The situation that westand in is very different from that of Pre-Industrial Revolution.13 I have shown efficiency conditions, but have notcharacterized the climatical optimality. I do not at all insist that we can artificially control the urban atmosphericcompound of heat and gases as intangible characteristics.
What I have done is just to theoretically derive the hedonic optimality conditions in an economic sense.
Acknowledgement
The previous version of this paper was presented at theRegional Science Workshop in Sendai held at the Graduate School of Information and Sciences, Tohoku University, September 10, 2003. I gratefully acknowledge to
Komei Sasaki, Kei Fukuyama and the participants at theworkshop for valuable comments and suggestions, which
substantially improved the earlier version of the paper.
The revised version was delivered at the annual meetingof the Society for Environmental Economics and Policy
Studies held at University of Tokyo, September 27, 2003.
The further revised paper was presented at the annual
meeting of the Japanese Economic Association held at
Meiji University, October 13, 2003. Some revisions weremade thereafter. It was also presented at the seminar of
the Department of Economics held at McGill University,
Montreal in October 12, 2007. I am appreciative of thestimulating discussion with Christopher Green and thepraticipants and their useful comments and suggestions
at that meeting. I bless him for his hospitality and kindness during my stay. This paper was also presented at theseminar held at Keio University, in December 10, 2007.
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Received June 21, 2007, Accepted February 1, 2008.

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