An Intertemporal Computable General Equilibrium
Model of Sustainable Toyohashi City with
Green Energy and Smart Technology
Shamsunnahar KHANAM
Graduate School of Environmental and Life Sciences, Toyohashi University of Technology
andYuzuru MIYATA
Graduate School of Architecture and Civil Engineering, Toyohashi University of Technology
1. Introduction
Over the last few decades the negative impacts of transport depicting a growing concern for our environment (Van
Wee1)). In Japan automobiles are still increasing. If this continuous, CO2 emissions in this sector may increase throughthe first half of the 21st century. Consequently, a study of measures for reducing these CO2 emissions is essential. In thispaper, possible automotive technologies, introducing electric vehicle, improvements in fuel consumption and theintroduction of electric vehicle society are discussed. The main objective of our study is to introduce SustainableToyohashi City with green energies and smart grid technology are to mitigate consequent fossil oil dependencies, thus
reduction of GHG emissions in transportation sector, and making sure that the energy is used efficiently. In ourprevious study market penetration of introducing electric vehicle has evaluated already. However, this would require
the simultaneous implication of fuel-consumption improvements and the introduction of electric vehicles in the society.
Automobile consumers would be reluctant to accept these technologies, particularly electric vehicles, because of theirhigh purchase-price and low benefits in terms of operating economy. Acceptance will require financial and institutional
support from the public sector in introducing these automotive technologies into the Japanese transportation sector. The
study reported here has three aims: simulating the amount of CO2 emissions from Toyohashi’s automotive sector for 20
years from 2005; evaluating the potential of electric vehicle if the price of goods falls (like the cost of EVs drop as aconsequence of high production); and considering the individual dweller and stakeholders’ responses of implementing
the measures to introduce electric vehicle society; each household’s response as a seller and buyer because in our study
we assume that every household will have the electricity generation sources from solar.
2. The Model
2.1 Assumption of the Model
The main assumptions made in the model are as follows:(1) There are thirty four markets including thirty two commodity markets, and two factor (labor and capital) markets.
(2) Commodity, labor, and capital markets are perfectly competitive, and the real economic world is in equilibrium
at each time under the endogenously determined capital stock and(2) exogenously given labor endowment. (3) Each household has perfect foresight. (4) The simulation period is 20
years interval of 2005 to 2025.
2.2 Industries
Since the adjustment cost in industries’ investment is not considered in the model, the industries’ intertemporal
optimization behavior is reduced to the static optimization behavior, i.e. profit maximization. Industries have Leontief
technology with respect to intermediate composite and value added inputs, and CobbDouglas technology for bothintermediate inputs, and labor and capital inputs. Constant returns to scale are assumed in the technology. Due to linear-1-homogeneity in production technology, industries' behavior becomes cost minimization under the given commodity
demands and prices, which can be written as;
32

Min∑ p x + (1 + tp )(w ⋅ L + r ⋅ K )
i ij

j

j

(1)

( j = 1,⋅ ⋅ ⋅,31)

j

i =1

( xij , L j , K j )subject to
X j = Min {11f j ( L j , K j ),
a oj
A2 j
αj

31

∏x }

(2)

a ij
1j

i =1

(1−α j )f j ( L j , K j ) ≡ A1 j L j K j
31∑ a =1a oj , A2 j > 0,(3)

( a ij ≥ 0)

ij

(4)

i =1

where
pi : price of commodity I, xij : intermediate input of industry i's product in industry j , tpj : net indirect tax rate imposed on
industry j’s product (indirect tax rate - subsidy rate), w : wage rate, r : capital return rate, Lj : labor input in industry j, Kj :capital input in industry j, Xj : output in industry j, a0j : value added rate in industry j, aij : share parameter onintermediate input xij, Aij ,αij:technical parameters in industry
Cost minimization yields conditional demands for intermediate goods, labor, and capital in production process.
x ij = A 2 j

a ij

i

LD j = [KD j = [

32

pk

∏ (a
p
k =1

(1 − α j ) r

α jw

)

(5)

a kj

kj

]

α j

a oj X j

(6)

A1 j

α jw
( 1 − α ) a oj X j](1 − α j ) r
A1 j

(7)

j

where LDj : conditional demand for labor in industry j, KDj : conditional capital demand in industry j
Zero profit condition is realized in industries under perfect competition at each time.
profit = p j X j

31

(8)

∑ p x − (1 + tp )(w ⋅ LD + r ⋅ KD ) = 0
i ij

j

j

j

i =1

2.3 Household
Households in Toyohashi city are assumed to be identical. The utility function is specified as a CES type of composite
consumption and leisure. The representative household consumes composite goods and leisure time so as to maximize
the integration of discounted utility function over time. Then the composite good is divided into thirty two types ofcommodities maximizing a sub-utility function.
The budget constraint imposed on the household is as follows. First, the household income is composed of fullincome, which is defined as income obtained if the household supplied its entire labor endowment, post depreciation
capital income, current transfers from the government, labor income, property income, and current transfers from theexternal sector. A part of wage and capital incomes is transferred to the external sector. Direct taxes are subtracted fromthe household full income. Then the household allocate its disposable income on consumption and leisure. The balanceof income and expenditures is saved financing capital investment. Thus household behavior may be expressed asfollows:

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∞

σ

∫ σ - 1[{(1 − β ) C

Max

1/v

( v −1) / v

0

G ,H

+ β 1 / v F ( v −1) / v }v /( v −1) ](σ −1) / σ e −η ( t ) dt

•

(9)
•

KS = (1 − ty ) FI / p I − ( p / p I ) C − (1 − ty )(1 − l o )( w / p I ) F − ( E / E ) KS

(10)

FI ≡ (1 − l o ) w + (1 − k 0 )( r − p I δ ) KS + LI + KI + TrGH + TrOH

(11)

t

η (t ) ≡ ∫0 ξ ( s )ds

(12)

where
σ: intertemporal elasticity of substitution, β: share parameter, ν: elasticity of substitution, C : composite consumption
good, F : leisure, η(t) : subjective discount rate, KS : capital stock endowed per household, ty : direct tax rate••

Gross investment per household, Ip, for capital accumulation is written as Ip= KS +δKS + ( E/ E ) KS . Let us further
specify Ip as a Leontief function of commodities produced by industries

Ip = Min{Ip1 / b1 ,⋅ ⋅ ⋅, Ip 31 / b31 }31bi>0, ∑ b i = 1 .

where

(13)

It is assumed that the gross investment is made minimizing the investment

i =1

31

costs ∑ p i Ip i , thus it leads to Ipｉ= bi Ip (i =1,…, 31). Denoting the price of capital good it is derived as
i=1

pI =

31

∑

b i p i since pI Ip =

i =1

31

∑ p b Ip . Let us now introduce the current value Hamiltonian in order to solve the
i i

i =1

intertemporal optimization problem defined in equations (9) to (12).
σ[{(1 − β )1/v C (v −1) / v + β 1/ v F (v −1) / v }v /(v −1) ](σ −1) / σσ −1+ λ[(1 − ty)(1 − lo )(w / pI )(1 − F ) + (1 − ty){(1 − k0 )(r / pI − δ )KS +H (t ) =(14)•(LI + KI + TrGH + TrOH) pI } − p ⋅ C / pI − (E/ E)KS]

where, λ: costate variable associated with KS.
Necessary and sufficient conditions for maximizing the objective function (9) include;•・equation (10)・ λ = −∂H / ∂KS + ξλ ・C and F maximize the Hamiltonian at each time・the transversality
condition. These conditions are expressed in mathematical form as follows:••

KS = (1 − ty ) FI / pI − ( p / pI )C − (1 − ty )(1 − lo )(w / pI ) F − ( E/ E••

λ = λ [ ξ − (1 − ty )( 1 − k 0 )( r / p I − δ ) + E / E ]
•

(15)
(16)
(17)

C = ( 1 − β ) p − v Ω [( E / E ) p I / λ p ] σ•

F = β [( 1 − ty )( 1 − l o ) w ] − v Ω [( E / E ) p I / λ (1 − ty )( 1 − l o ) w ] σ(18)LS = 1 − F

(19)

lim
λ ⋅ KS ⋅ e − η ( t ) = 0
t→ ∞

(20)

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Ω ≡ [(1 − β ) p 1− v + β ((1 − ty )(1 − l o ) w)1− v ](σ − v ) /( v −1)(21)

Moreover, the composite good is divided into commodities produced by industries through maximizing CobbDouglas sub-utility function (20).
31

∏ C

Max

31

α j

(

j

∑ α

j

(22)

= 1)

j =1

j=1

subject to
31

∑ p

j

⋅C j = p ⋅C

j =1

where ,Cj : consumption of commodity j, pj : price of commodity j
Therefore we obtain the consumption demand for products of industries as;
C j = α j p ⋅C / p j
31

pj

j =1

j

∏ (α )

where p =(23)

( j = 1 , ⋅ ⋅ ⋅ , 31 )α j

2.4 The Government
The government obtains its income from direct and net indirect taxes, and current transfers from the external sector,
and then it expends the income on government consumption, current transfers to households, and current transfers to
the external sector. The difference between income and expenditures are saved. Nominal consumption expenditures oncommodities/services are assumed to be proportional to the government revenue with constant sectoral share. These areexpressed as the following balance of payments.
ty ⋅ Y +31∑ tp ( w ⋅ LDii

+ r ⋅ KD i ) + TrOG

(24)

=

i =1

31

∑ p CG + TrGH + WTC + TrGO + SG
i

i

i =1

where CGi : government consumption expenditures on commodity I, TrGH : current transfers to households, TrGO :current transfers to the external sector, SG : government saving, TrOG : current transfers from the external sector
2.4 The External Sector
The external sector gains its income from Japan's imports, current transfers from the government, labor income
transfers, and property income transfers. And then it expends the income on Japan's exports, current transfers tohouseholds and the government, labor and property income transfers to Japan. Quantity of exports of commodities are
exogenously given, and that of imports is assumed to be proportional to the domestic commodity demand. These arealso described as the following balance of payments.
31∑ p EMii+ TrGO+ KIO + LIO =i+ TrOH+ TrOG

(25)

i =1
31

∑ p EXi+ KI + LI + SO

i =1

where EXi : export of commodity I, EMi : import of commodity I, SO : savings of the external sector ( =-national current
surplus), LIO : labor income transfers to the external sector (= lｏ· w ·LS ), KIO : property income transfers to theexternal sector (= k0 · r · KS )2.5 Balance of Investment/Savings
In the model, private capital accumulation is internalized by maximizing the integration of the utility function.
In the real economy, however, there are other investments including public and housing investments etc. Here otherinvestments are supposed to be financed by savings of the government and the external sector.

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31

(26)

∑ p I = SG + SO
i

i

i =1

where
Ii : demand for commodity i by other investments, SG : savings of the government, SO : savings of the external sector
Similar to the private investment, demand for commodities by other investments is derived in the following way. Let
other investments, I , be a Leontief composite with technological parameters, bi’. Then the price of I , qＩ, is derived as;
31

∑ b p

qI =

i

(27)

31

(

i

i =1

∑ b = 1, b ≥ 0 )
i

i

i =1

Therefore other investments and their demand for commodities can be calculated as;

I = ( SG + SO ) / q I(28)

I i = bi I (i = 1,L,31)(29)

2.6 Prices
Wage and capital return rates, and commodity prices satisfy some price equation in temporal equilibrium because of
the zero profit condition. The relationship between the commodity price and production costs in industry i in temporalequilibrium is denoted as follows:31

31∑ p x + (1 + tp )(w ⋅ LD + r ⋅ KD )] / X = ∑ p a + (1 + tp )(w ⋅ ld + r ⋅ kd )pj = [

i ij

j

j

j

j

i=1

i ij

j

j

j

( j = 1,⋅ ⋅ ⋅,31)

(30)

i=1

in which aij≡xij /Xj , ldj≡LDj /Xj , and kdj≡KDj /Xj .
Given a wage and a capital return rates, we can formally calculate commodity prices as follows
P = ( I − A ' ) − 1 [( 1 + tp j )( w ⋅ ld j + r ⋅ kd j )](31)

Where, P : vector of commodity prices, A': transposed matrix of industries' input coefficients
2.7 Intertemporal Equilibrium Conditioncommodity market
X i + EM i =31∑ x + C + CG + Ip + I + EX
ij

i

i

i

i

i

(32)

(i = 1,⋅ ⋅ ⋅,31)

i =1

labor market
LS =

31

∑ LD

(33)

j

i =1

capital rental market
KS =

(34)

31

∑ KD

j

i =1

In numerical computation, iteration for the equilibrium prices is made on the capital return rate letting labor be thenumeraire (w=1 ).
3. Concluding Remarks
In Toyohashi city, GHG emissions are growing in transportation sector because of an increase in the number ofvehicles. Not only the growing concern on emission reduction, recently have we seen how growing demand rapidly
pushes prices on oil up. Normally, demand and supply finds a balance but with declining supply and increasing
demand the world could face an uncontrollable situation unless alternative energy sources are developed fast.
Thus, the main objective is to introduce a sustainable Toyohashi city with green energy and smart technology to
mitigate consequent fossil oil dependencies by developing alternative energy supplies and making sure that the energy
is used efficiently. In our study electric vehicles are the main subject. Some might ask how the use of EVs will-5-

influence our climate if the electricity should come from burning gasoline and diesel. Actually, EVs will be the moreenvironmentally friendly and alternative even if not charged with renewable energy such as wind or solar power.
Another important aspect is that all the power plants are strongly regulated by the government of Japan. However,energy generation from the renewable sources will truly be significant for reducing emissions and resource circulationin the society. Toyohashi city is considered one of the most potential for generating the renewable sources of energies,
especially wind and solar energies because of its location and weather. To the south of Toyohashi city is the Pacific
Ocean, and the city opens onto Mikawa Bay in the west. Thus, there is always a strong tendency of the wind. Moreover,the climate of this area is relatively warm and sunny days are more frequent (see Figure 1) round the year.

Figure 1. Sustainable Toyohashi City with Green Energy, and Smart Grid Technology

Note:
DB : Dispersion Battery, SP : Solar Power, WP : Wind Power, SB : Storage Battery, FB : Fuel Battery, HS : Home
School, EV : Electric Vehicle, SG : Smart Grid, MT : Monitor, PS : Power Supply, SCD : System Cooperation Device,
OHS : Office, Hospital, Shop, DH : Demand foe Heat, PC : Power Conditioner, RSP : Rooftop Sunlight Panel,
SECS : System of Electricity in Commercial Sector, PH : Power House, PS : Power Supply.
References
Van Wee, Bert (2007), Environmental Effects of urban Traffic. In: Threats from car traffic to the quality of urban life
problems, causes, and solutions. Gärling & Steg ed. 11-32. Elsevier.
Gärling & Loukopolos (2007), Effectiveness, Public Acceptance, and political Feasibility of Coercive Measures for
Reducing car Traffic. In: Threats from car traffic to the quality of urban life problems, causes, and solutions. Gärling &Steg ed., 313-324. Elsevier.
Morten Rask, Poul Houman Andersen, Mai Skjøtt Linneberg & Poul Rind Christensen Local Design & Global
Dreams – Emerging Business Models creating the Emergent Electric Vehicle Industry, Aarhus School of Business,
Aarhus University, Denmark
Toyohashi City HP : http://www.city.toyohashi.aichi.jp/-6-