日本ヒートアイランド学会論文集 Vol.1 (2006)
Journal of Heat Island Institute International Vol.1 (2006)

学術論文

URBAN HEAT ISLAND: AN ENVIRONMENTAL
ECONOMIC MODELING
Kimitoshi SATO
Graduate School of Economics, Rikkyo University
E-mail address: satokt@rikkyo.ac.jp
ABSTRACTThis paper presents an environmental economic modeling, in which I propose heat island integral to represent the forceof heat island and urban heat island function. Also presented is a heat island tax/subsidy scheme; the subsidy based in
the numerical data of tree carbon fixation is given to contributers to cooling the city by providing vegetation in the city.
In our model any metropolitan resident consumes goods and have landscape gardeners plant trees in his/her gardens,
and producers supply goods. Every emitter emits heat and gases in the atmosphere, which results in urban heat island
and an anormous amount of cost, so that it is a recent economic problem of great importance for large cities such as
Tokyo. This paper tries to give an economic theoretical solution to this prolbem.

1.

INTRODUCTION

The Tokyo Metropolis is now observed to have two“suns”; one is the sun in the sky, the other is beneaththe surface. The latter may rival solar heat, and Tokyo is
now a gigantic Heat Island with peaks such as Otemachi(the central business district of Tokyo) and Shinjuku. It
has been reported that the temperature for these areas at
18:00 in July 31, 2031 is respectively anticipated as 45.5
[See Saitoh, Ito, Yamada(2006)]. Nevertheless, the Imperial Palace, the Meiji Shrine, and the Shinjuku Gyoen
National Garden as Cool Islands are cooler, and the temperature of these areas remains rather low compared tothe hotter spots mentioned above. Formal definitions ofheat island and cool island are given in the next section.
Recently, megacities have been struck by localized torrential downpours of more than 100mm per hour in thesummertime, e.g., an area in the Nerima Ward in Tokyo
experienced 33.5◦ C and 131mm/h in July 21, 1999. There
were ascending currents and a huge cumulonimbus which
was ten times larger than a usual one. Nonetheless, only a
few millimeters of raindrops were observed in Musashino
City and West Tokyo City, which are not so distant fromthe Nerima Ward. This is a very distinctive phenomenonof urban heat island. It is considered that breezes fromthe Tokyo Bay, the Sagami Bay and the Sea of Kashima
converged to the area in the Nerima Ward, which madethe atmosphere unstable and resulted in localized heavy
rains. They were harder than the deluge of rain in the
Tokai District in 2000 where 589mm was recorded, and the
maximum precipitation per hour was 114mm. With thehelp of sandbags and water bags, everyone should prepare
against the inundation caused by unforeseen urban localized torrential downpours originally due to urban heatisland. Tokyo had 15 incidents of heavy rain in the lasttwo decades. The rainfall was 57.5mm/h in October 13,

2003, which was the second highest record in the historyof meteorological observation. The power failure due totelephone polls broken by a downpour tangled traffic on
the highway and the railroad.
This paper proposes to consider urban atmosphere as acomplex of gaseous attributes or characteristics, and thegoods or commodities combined by nongaseous attributes
that are produced by the firms and the green areas as acomposite of biological attributes supplied by landscapegardeners. The paper proposes the model where eachresident has his/her land planted by a gardener and simultaneously emits heat and trace gases. For this, I adopt
an analytical framework of the New Consumer Theory by
Gorman and Lancaster Theory. Section 2 presents the
periurban model of urban warming in the framework à
la Gorman-Lancaster and introduces the concept of Heat
Island Integral. Our model involves residents, offices,manufacturers and farmhouses such as producers, andgardeners who play a very important role to cool down
a metropolis by roof-top and wall gardening and treeplanting activities, etc.
2.

2.1

URBAN HEAT ISLAND IN A PERIURBANMETROPOLISA Model of Urban Heat Island in a Periurban
Metropolis
The Characteristics/Functionings model is introducedin this subsection.
In the Gorman-Lancasterian New
Consumer Theory, goods are regarded as a compositionof characteristics. For the sake of simplicity, the twoterms, “attributes” and “characteristics”, are used interchangeably throughout this paper. All trace gases suchas Greenhouse gases(GHGs) can be interpreted as gaseous
attributes in our framework, since they partially compose
the urban atmosphere as an urban public good. This pa--41-per focuses upon emitted heat as a intangible attribute,since it is the main cause of urban warming.
It was Luke Howard, an English meteorologist, who
first noticed that the urban temperature was higher than
the rural one by making observations of climate data in
London in 1807∼1816. He already calculated that it was
3.7◦ C higher at nights in the urban area by comparing theobserved values of the inside and the outside of London.
Consider a monocentric periurban metropolis 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 and landscapegardeners. For the sake of simplicity, assume that our
periurban metropolis is composed of two regions, i.e., theurban area and the periurban area, which are divided into
many blocks, β ∈ ˛ = {1, . . . , B} : the set of blocks. In
other words, our periurban metropolis is represented bya mesh of 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.
Each individual 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 thateach 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 three
categories of producers: i.e., officies, manufacturers andfarmhouses.
Suppose that resident i chooses a gardener named `to have a part of his/her land planted with trees, andΛ = {1, . . . , Λ} is the set of gardeners. Define q`s as
a biomass of species s in one square meter supplied bygardener `, and S = {1, . . . , S} is 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 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 of aheat island tax/subsidy scheme defined below. Not tomention, every inhabitant, producer or gardener resides
in some block, so that an index β will be omitted hereafter in almost all the cases, except for describing somevariables related to block β.
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
such as N2 and O2 are stationary, so we can focus uponheat and trace gases as attributes in our paper. An index qjg is also used hereafter to identify the gth gaseous
attribute, and qjH is an amount emitted of heat as an
attribute when producing one unit of good j. Let G ={C + 1, . . . , C + G, H} be the set of intangible attributes
such as trace gases and heat in the urban atmosphere.
When producing one unit of good, each producer does
not choose but to jointly emit heat and trace gases as vexing by-products, qjH ≥ 0 and qjg ≥ 0, ∀g ∈ G, which areproducer j’s unit emission of heat and each gas. Thus,
qjH xj (resp. qjg xj ) is firm j’s amount emitted of heat

(resp. gas) when it produces xj units of good j. Landscape gardeners also emit heat and gases, q`H ≥ 0 andq`g ≥ 0, ∀g ∈ G, which are gardener `’s unit emission ofgas and heat. Hence, q`H A` (resp. q`g A` ) is gardener `’semitted quantity of a heat(resp. gas) in his/her production of A` units of greening service `. Inhabitants also
emit heat and gases, qiH ≥ 0 and qig ≥ 0, ∀g ∈ G, which
are city dweller i’s unit emission of gas and heat. Hence,qiH Ai` and qig Ai` are individual i’s emitted quantity ofa heat and gas in his/her consumption of Ai` units ofgreening service `.
Let zi0 be resident i’s numéraire characteristic thathe/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 byXzic =qjc xij , ∀c ∈ C
(1)
j∈J

and
zg =

X

qjg

j∈J

+

X

xij +

i∈N

XX

`∈Λ i∈N

qig

X
j∈J

X

qjg Aj +

j∈J

xij +

X

q`g Ai``∈ΛX X

qig Ai` , ∀g ∈ G.(2)`∈Λ i∈N`

In the above equation, zic means the consumption of
nongaseous attributes which compose goods, while zg(resp. zH ) represents the total amount of a trace gas (resp.
heat) emitted by all residents, producers and gardeners.
Note that the value of zg , ∀g ∈ G, can be measured via
ton or kilojoule, as in the Introduction. Heat and gases
are generated both in the consumption and production of
goods. Every inhabitant is made to consume not only
his/her emission but also the quantity emitted by therest of the metropolis. When he/she uses goods, he/sheemits heat and gases, which were already released when
the goods were made by producers. Both of the above
equations may be interpreted as characteristic availability functions, which convert commodities into attributes.
The amount of any characteristic in each good can be regarded as a parameter that is objective and common to
all consumers, i.e., it has a public-good property. Thus,
the inhabitants as consumers must behave as “quality takers”, since they can only change their consumption of zic
and zig , via the choice of xij . Producers and gardeners
can choose the composition of attributes embedded in the
goods and the services.
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 43% of CO2 emission is absorbed by theoceans and forests as carbon sinks. An integration rate oran inverse of a lifetime of each trace gas is denoted as λg ,with 0 < λg < ηg , ∀g ∈ G, so that the mass of the gth gasthat stays in the urban air is (ηg − λg ) zg , ∀g ∈ G. Allowing θg be a conversion parameter from mass(ton/year) toconcentration(ppmv), and zg (γ) be an amount emitted attime 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

ζgt =

Z t
t0

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θgt (ηgt − λtg )zgt (γ) dγ, ∀g ∈ G\{H}.

(3)This can be written as
Z t
ζgt =
Θtg zgt (γ) dγ, ∀g ∈ G\{H}

(4)

t0

where Θtg ≡ θgt (ηgt − λtg ) is a climatical parameter related
to the gth trace gas. An argument of time t is omitted
hereafter. Another equation is proposed for heat as aflow in the next subsection.
It is the local climate in the block, which most influences any economic agent who resides or works in β.
However, climatical incidents depend upon not only theconcentrations in each block, but also those in the entire
metropolis, as exampled in the Nerima incident in the
Introduction. Therefore one observes
Z = (ζ1 , . . . , ζG )(5)which affects all residents, producers and gardeners in theperiurban metropolis.
2.2 Heat Island Integral
The next issue is how to represent the heat in our
metropolis. In effect, there must be differences in thetemperature of building surface, back alleys, rooftops,
streets, and green tracts of land, which are directly exposed to the solar radiation. However, these differencesof the surface temperature of the ground coverage can bemeasured by utilizing infrared cameras or remote sensing.
Let Aβ be an area of 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 Aβ .
Denote τβ (u, w) as a function of the surface temperatureof Sβ and αβ (u, w) as a function of atmospheric temperature in Aβ , where u and w are the plane coordinates.
Then, the Riemann sum led me to propose a concept ofHeat Island Integral which reads(Z Z
1τβ (u, w)dudw
Υβ =
Sβ
Sβ
)
ZZ
αβ (u, w)dudw , ∀β ∈ ˛.
(6)−radiation. Also, heat is emitted from the underground
shopping center in Shinjuku. By remote sensing, τβ (·)
and αβ (·) can be constructed, thus, one can calculate Υβ
for each block.
The following formula is physically supposed: Sensible
Heat Flux (εβ ) = Heat Island Integral (Υβ )× Convective
Heat Conductivity(κ), ∀β ∈ ˛, where κ varies according
to the temperature. Block β is called a Heat Island if
Υβ > 0, ∀β ∈ ˛, and a Cool Island if Υβ ≤ 0, ∀β ∈ ˛. As
was mentioned in the Introduction, the Imperial Palace,the Meiji Shrine, and the Shinjuku Gyoen National Garden as cool islands are cooler than the center of Shinjuku, i.e., the heat capacity of the latter was observed to
be about 300W/m2 , whereas it was less than 200W/m2
in the Shinjuku Gyoen National Garden by the measurement of the Ministry of Environment. Furthermore, by
its observation, the Arakawa Rivers in Tokyo was muchcooler than the above areas, because they nearly absorb
all of the solar radiation.
2.3

Vegetation, Water, Heat, and the Urban
Warming Function
Let Li , Lj and L` be the land owned by resident i,
producer j, and gardener `. Also let Ai ≡ ρi Li andAj ≡ ρj Lj be greening areas of their land required to beplanted with trees, where ρi and ρj are greening rates forcity dweller i and firm j, which are legally determined,e.g., at least 20% of the rooftop on the houses and buildings, when they are newly built, enlarged or reconstructedin Tokyo. Suppose that gardeners and farmhouses are exempted from the obligation of greening a part of their own
lands in our model.
As I assumed that inhabitant i chooses one gardener `to green a part of his/her land Ai with plants, at a costof σ` /m2 . Then, each person is to have his/her green
area planted with plants, insects and microorganisms inthe 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 by gardener `. Denote S as the set of species. City dweller i therefore consumes biological attributes by possessing the green area
in his/her land as represented byzis = q`s Ai` , ∀s ∈ S, ∀` ∈ Λ.

AβNeedless to say, the existence condition of this multiple
integral is that the functions, τβ (u, w) and αβ (u, w), arecontinuous and compact in the domains Sβ and Aβ , and
it is easily seen that this condition is satisfied.
Υβ is not the result of heat emission on the part ofthe economic agents, but it stems from the ground coverage of the metropolis. Υβ measures the force of urban
heat island in each block β, since the temperature of asphalt and concrete is very often higher than that of theatmosphere. It is observed that asphalt and concrete absorb about 90% of the solar radiation, so that the wallsof buildings made of concrete absorb approximately thesame amount of heat, which result in urban heat island.
Let me give some numerical examples of the values in Υβ .
The mercury stood at 27.5 degrees at an observation pointin Shinjuku at 14:00 on June 6, 2002, where a wall of abuilding made of concrete was observed to have the heat of38◦ C and a street of asphalt had 45◦ C. The central areaof Shinjuku (Aβ = 4km2 ) must be extended to its plane ofprojection (Sβ = 9.3km2 ), which is exposed to the solar(7)

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 P
in a part ofits lot size Aj , with its cost σj . Let AP ≡ β∈˛ Aβ be
the area to be planted with street trees and tree lawns
undertaken by public works, and Γ(Ap ) is its cost. Summing these green areas yields
XX
X
X
A=
Ai` +
Aj +
Aβ .
(8)
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 thecities to and fro.
The quantity of water is crucially important to coolsupercities, so underdrained rivers must be resuscitated.

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Let W be the total amount of water in the metropolis
represented by
W = W0 + WP
(9)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.
Sensible heat f lux from the ground coverage of themetropolis therefore is represented byXX
εβ (A, W, Z) = κΥβ (A, W, Z) .
(10)
β∈˛

β∈˛

Note that this function depends on the planted areas andthe amount of water. Both of them can diminish thevalue of Υβ by cooling down the ground coverage of thecity. However, the above equation is also dependent upon
the vector of concentration of trace gases.
Let εi , εj and ε` be heat emitted by resident i, firm j,
and gardener ` in the metropolis. Total heat emission E
which affects all economic agents in the metropolis is thus
represented as
X
X
X
X
E(A, W, Z) =
εi +
εj +
ε` +
εβ
(11)
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 todepend upon the amounts of vegetation, water, and thevector of concentration of gases.
2.4 Urban Warming Function
Together with the concentrations of GHGs as stocks,
total heat emission defines the Urban Warming Function:
U = U (E, Z) .

(12)

Urban heat island is a typical example of a public goodwhich is both nonrival and nonexcludable.
However,
its impact on each resident varies from region to region,
which can be treated as a regional public good. As was
mentioned in the Introduction, urban heat island with
33.5◦ C occured in some blocks of the Nerima Ward, Tokyo
on July 21, 1999, which was not necessarily due to theconsumptions and productions of goods in that area, but
due to those of districts in Tokyo, as well as the climate
conditions of the block at that time. Consequently, theworst that could happen to any block where some climatical conditions are satisfied at some point in time, as
in the above unforeseen incident. Let Dβ (U ) is a monetary damage of block β due to urban heat island. With
the differentiability of Dβ (U ), it is naturally assumed
that (∂Dβ /∂U )(∂U/∂E) > 0 and (∂Dβ /∂U )(∂U/∂Z) >0, ∀β ∈ ˛.
The urban heat island function U depends upon the
total heat emission E and the vector of concentrations Z,
both of which compose the urban atmosphere of the atmosphere. Urban heat island may now be regarded as anegative externality which has been directly provided by
the economic agents who emit heat and GHGs into the urban air. Its climate damages can be measured in physicalunits, which are, in order: i) an increase in heat strokes
due to temperature rises, ii) an increase in the occuranceof showers and localized torrential downpours, iii) spreadof infectious diseases carried, for example, by mosquitoes,iv) changes of biodiversity in an urban ecosystem, etc.
3.

DESIGNING A HEAT ISLAND TAX / SUBSIDY SCHEME

3.1 A Heat Island Tax
Let E = N∪J∪Λ be the set of emitters in the metropolis. For example, consider a driver who emits both heatby using a car with an air-conditioner and gases such as
CO2 . Let εe0 be an initial amount of heat emitted byemitter e and εe be his/her current emission. Assume
that the amount of heat emitted can be converted into
CO2 , so that it is possible to construct a cost function,with which each metropolitan emitter e is confronted.
Ce = αe (εe0 − εe ) + tεe , ∀e ∈ E(13)where t is a heat island tax rate which may change annually. Then, tεe is an amount of tax payed by emittere. The objective for each emitter is to minimize the costwhich is related to urban heat island.
In order to obtain our desired result, I need another
differentiability assumption.
Assumption. For any e ∈ E, Ce is strictly quasi-concave
and twice continuously differentiable with Ce (0) = 0.
Simple calculation of cost minimization readst=dαe, ∀e ∈ Edεe(14)which means that the tax rate equals to the marginal
abatement cost(MAT) of any emitter. It is concluded
therefore that every MAT is same, irrespective of the curvature of cost function of any emitter. Then the sum of
tax is
X
X dαe
εe
(15)
T =
tεe =
dae
e∈E
e∈E
which may cover the damages due to urban heat islandand the rest can be refunded as subsidies for the collaboraters to cool down the city. Thus, one observesXT =Dβ (U ) + S.
(16)
β∈˛

3.2

Heat Island Subsidies Based on the Numerical Data of Tree Carbon Fixation
Let the set of trees be T, and let zτ g be an amount of
CO2 anticipated to be fixed by tree τ. Denote dτ as tree
τ ’s diameter at the height of a chest and ρτ as a coefficient
related to tree τ. Denote also %τ as a coefficient of tree τand υτ be the decision coefficient, then we have in generalthatzτ g = %τ d%τ τ , R2 = υτ , ∀τ ∈ T.
(17)
For example, one has zτ g = 0.005%2.496 , R2 = 0.957, forginkgo, in Fujiwara(2004). According to him, Toshima-44-

Ward in Tokyo has a room for vegetation for about 180
thousand of trees. If they will be planted in five years,
it is anticipated that 988t-CO2 will be fixed at the end of
the last year.
The CO2 fixation by any tree reads:zτ g hτ nτ = %τ d%τ τ hτ nτ , R2 = υτ , ∀τ ∈ T.

(18)

Let ni`τ be the number of trees that resident i commissions landscape gardener ` to plant his/her garden. And
njτ is assumed to be the number that producer j plants
in its site of firm or factory. Let also nβτ be the numberof trees planted by some public work in block β. If theamount of vegetation is denoted Vτ = hτ dτ nτ , ∀τ ∈ T,
then the CO2 fixation by all trees is given by the following formula:Xz τ g h τ nτ
τ ∈T

=

X

0
zτ g h τ @

τ ∈T

1
XX
`∈Λ i∈Nni`τ +

X

njτ +

j∈J

X

nβτ A . (19)β∈˛As greening lands and rooftops are one of the mosteffective ways to cool the metropolis, let the metropolitan
government determine 0 < ϕi < 1 as the refund rate forgreening the area Ai` that resident i requests landscapegardener `. Assume that the value of ϕi is decided by
the scientific data about tree-planting. As defined, σ` isgardener `’s greening cost per square meter, hence, σi` Ai`
is resident i’s total cost, and ϕi σi` Ai` is `’s refund for theeffort to plant trees. Candidates of subsidy sharing aredetermined by tree-planting density, i.e.
«„Pτ ∈T zτ g hτ nτ, ∀i ∈ N.
(20)
ϕ i = ϕi
Ai`
Similarly, for producers one observes„P«τ ∈T zτ g hτ nτ
ϕ j = ϕj, ∀j ∈ J
Aj
Heat island subsidies are therefore as follows:XX
XS=
ϕi σi` Ai` +ϕj σj Aj .
i∈N `∈Λ[2] Gorman, W.(1956/1980), “A Possible Procedure forAnalysing Quality Differentials in the Egg Market”,
Journal paper N. 3129, Iowa Agricultural Experiment
Station, 1956; published in Review of Economic Studies, 47, 1980, 843-856.
[3] Lancaster, K.(1966), “A New Approach to Consumer
Theory”, Journal of Political Economy, 74, 1966, 132157; reprinted in Lancaster, K., Modern Consumer
Theory, Edward Elgar, Hants, 1991, Chapter 2.
[4] Lancaster, K.(1971), Consumer Theory: A New Approach, Columbia University Press, New York, 1971.
[5] Saitoh, T. S., S. Ito, and N. Yamada(2006), “A Measurement of Urban Thermal Environment and a Simulation of Heat Island in an Urban Street Canyon,”
Heat Island Institute International Special Issue, 3042.
[6] Sato, K.(2006), “Beautifying and Reviving Urban
Heat Islands by means of Water, Vegetation and Technological Innovations”, presented at the annual meeting of the Japanese Economic Association held at Osaka Gakuin University, October 22.
[7] Sato, K.(2008), “Cooling the Metropolis: An Economic Analysis to Alleviate Urban Heat Island”, Journal of Heat Island Institute International.

(21)

(22)

j∈J

In the above equation, the first term is the subsidy given
to residents and the second term is the subsidy which goesto producers.
This paper has applied the economic way of thinking to an environmental problem of growing importanceand has proposed an environmental economic modelingof urban heat island. My discussion has proceeded on
the premise that the urban atmosphere is a composite ofgaseous characteristics as well as heat as an important
attribute. Sato(2006) also incorporates farmhouses andtechnological innovations to cool urban heat islands.
References[1] Fujiwara, N.(2004), “An Evaluation of the CO2 Reduction by Urban Greening–A Case Study in Toshima
Ward in Tokyo,” Environmental Information Sciences, 18, 241-245(in Japanese).

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