Interregional Computable General Equilibrium Model ofPrefectural Cross-Border Areas in Japan
Hossain Nahid*
Ph.D. Student at Graduate School of Environmental & Life Engineering, Toyohashi University of Technology
Ha Thi Thu Trang
Ms. Student at Graduate School of Knowledge Based Information Engineering, Toyohashi University of Technology
Yuzuru Miyata
Graduate School of Architecture and Civil Engineering, Toyohashi University of Technology
1. Introduction
The global financial crisis and a collapse in domestic demand shrunk Japanese economy 1.2% in 2008 and5.0% in 2009. Furthermore, the negative impact of the rapid strengthening of the yen against foreign
currencies is likely to be felt until the end of 2010, and export growth is highly expected to slow (The Japan
Research Institute, Dec, 2009). In addition, Japan is experiencing a decreasing trend in population growth
and aging problem. The situations putting Japan’s national budget into a situation where, in one side, thedemand for investment in social welfare sectors like medical expenses, pension, nursing, child care areincreasing and, on the other side, sectors like transportation, construction, real estate, labor asserting
expansionary demand for economic growth. As a result, proper economic forecast of transport network, realestate location choice, labor demand both at firm and at household level is drawing attention of the planners.
The present presentation/article therefore, aims to draw an outline of new integrated prefectural crossborder economic model of transport, land choice and economy at firm and household level. Although thismodel is based on Ito (Osaka Institute of Technology) and Muto (Yamanashi University), some extensions
and alternation were made and some shortcomings are being addressed. For example, firms’ profit
maximization and location choice behavior based on its transport demand (both at input and output level)
and households’ utility maximization behavior and corresponding residential choice behavior are illustrated
in the model. The implication of the model is tested with the published data of Aichi, part of Nagano and
Shizuoka prefectures (known as ‘San-En-Nanshin’ region) and considered as study region.
2. Research Objective:
The general research idea is to develop a model and to test empirically the impact of distance/commuting
time upon firm’s profit maximization and households’ utility maximization behavior. The research also
intends to draw the location choice behavior and corresponding market equilibrium condition based ontransportation demand.
The research is opted to focus on the following issues:
1. Transport demand forecasting;
2. Firms’ and households’ behavior to choose commodity, labor and capital based on respective
transport arrangement;
3. Land use and location choice based on economic preferences and;4. Corresponding equilibrium conditions in various market.

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3. Structure of the Model
Based on the assumption that firms behave to maximize profit and households maximize utility, the model
aims to determine firms’ and households’ respective behaviors depending on their respective transportation
choice. The model also assumes that firms and households choice their location to produce and to reside
based on their respective transportation cost function. Some parameters are being used to solve the model.
Each behavior of the firm and household is solved in stages where associated primary input andintermediate input costs represent the cost functions. The model solves the firm and household behaviors forfollowing two situations:
1. profit (firm) and for utility (household) maximization and,2. location choice.
In the later part, the model draws the general equilibrium condition of the markets (commodity, capital, landand labor).
4. The Model
4.1 Firm’s Behavior
A representative firm in zone s is defined for one worker, and aims at profit maximization.

max π s ≡ py s − tc s

(1)

subject to⎡ pc s z s ⎤y ≡ min ⎢, ⎥⎣⎢ a0 a1 ⎦⎥stcs≡ pcs pc s +nmi =1r =1∑∑(2)σF⎤⎡⎧⎛ t rs ⎞ ⎫⎪⎪⎢⎨ p + pTF µTF ⎜ ⎟ ⎬ PTF rs z s ⎥⎜ t ss ⎟⎥⎢⎪⎝ ⎠ ⎪⎭⎦⎣⎩m⎡ s s ⎛⎞ s ⎤spcs ≡ ⎢ pva
va + ⎜⎜ pTP + w
t sr PTPB srδ b ⎟⎟ zTP⎥ / pc
⎢⎣
r =1
⎝⎠ ⎥⎦

∑

[]spva≡ w ⋅ ld s + r ⋅ kd s + hFs ⋅ ad s / va s

αs 1− αTP(6)( ) (kd ) (ad )va s ≡ η3 ld srsβls βk[s βars
ξTF− (t rs / t ss )σ F + 100

PTF ≡ m∑ξ [(4)(5)( ) (z )pc s ≡ η 2 va s

(3)(7)]

σs(8)]σsrs
rs
ss σ F+ 100
TF − (t / t )r =1

where πs : firm’s profit in zone s, p : price of commodity, ys : firm’s output in zone s, tcs : firm’s production
cost in zone s, pcs : firm’s production capacity in zone s, a0 : firm’s production capacity rate, zs :intermediate input of commodity in a firm in zone s, a1 : intermediate input coefficient, pcs : price ofproduction capacity, pTF : price of commodity transport service, µTF : input of unit commodity transport
service associate with input of commodity, t rs : time distance between zone r and zone s, PTF rs :probability of inputting zone r’s commodity in zone s, pvas : price of value added in a firm in zone s, vas :value added in a firm in zone s, pTP : passenger transport price, w : wage rate, PTPBsr : choice probability of-2-

passenger transport destination, δb : number of business trips per unit passenger transport service, zTPs : inputof passenger transport service in a firm in zone s, η2, η3, ξTF rs : parameters, ld s : labor input a firm in zone s ,kd s : capital input in a firm in zone s , ad s : land input in a firm in zone s , βl, βk, βa : elasticity parameter( βl + βk + βa = 1 ), σS and σF, : elasticity parameters
In order to solve the above-mentioned profit maximization problem, we hierarchically solve the costfunction, the conditional factor demand function under fixing output ys.
(1) The First Stage
Since the production function is a Leontief type, the production capacity and the intermediate inputs are
obtained as,

pc s = a0 y sz s = a1 y s

(9)

(10)

(2) The Second Stage
Given the production capacity pcs, we solve the following cost minimization problem:( )( )m⎡⎤ sss αs 1−αzTP
min c s ≡ pvas va s + ⎢ pTP + w∑ t sr PTPB srδ b ⎥ zTP(11) subject to pc ≡ η2 va
(12)
r =1
⎣
⎦

Following this, the value added input and passenger transport input associated with the production
capacity pcs can be obtained as well.
α1−α1 α ⎡ pvas ⎤ ⎡ θ s ⎤va =η 2 pvas ⎢⎣ α ⎥⎦ ⎢⎣1 − α ⎥⎦sαpcs1−α

1 1 − α ⎡ pvas ⎤ ⎡ θ s ⎤z =
η 2 θ s ⎢⎣ α ⎥⎦ ⎢⎣1 − α ⎥⎦sTP(13)pc s(14)mθ s ≡ pTP + w∑ t sr PTPB sr δ b(15)r =1

And the price index pcs of production capacity is also obtained as follows:
α1−α

ps α ⎡ ps ⎤ ⎡ θ s ⎤p = va s ⎢ va ⎥ ⎢η2 pva ⎣ α ⎦ ⎣1 − α ⎥⎦scα1− αθ s 1 − α ⎡ pvas ⎤ ⎡ θ s ⎤+η 2 θ s ⎢⎣ α ⎥⎦ ⎢⎣1 − α ⎥⎦α1− α1 ⎡ ps ⎤ ⎡ θ s ⎤= ⎢ va ⎥ ⎢η 2 ⎣ α ⎦ ⎣1 − α ⎥⎦(16)

Although the choice probability of passenger transport destination PTPB sr is not defined above, it ispossible to formulate it by the gravity model or the logit model. This is left as a future task.
(3) The Third Stage
Given the value added vajs, the following cost minimization problem is solvedmin cvas ≡ w ⋅ ld s + r ⋅ kd s + hFs ⋅ ad s

(17)( ) (kd ) (ad )subject to va = η3 ldss βls βks βa(18)From this, the conditional inputs of labor, capital, and land associated with the value added vajs can beobtained as,βl

βkβa

1 β l ⎡ w ⎤ ⎡ r ⎤ ⎡ hFs ⎤sld =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ va
η3 w ⎣ β l ⎦ ⎣ β k ⎦ ⎣ β a ⎦sβlβkβa

1 β k ⎡ w ⎤ ⎡ r ⎤ ⎡ hFs ⎤skd =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ va
η3 r ⎣ β l ⎦ ⎣ β k ⎦ ⎣ β a ⎦s(19)-3-(20)βlβkβa

1 β a ⎡ w ⎤ ⎡ r ⎤ ⎡ hFs ⎤sad =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ va
η3 hFs r ⎣ βl ⎦ ⎣ β k ⎦ ⎣ β a ⎦s(21)Price index of value added pvajs is obtained as follows:βlβk1 ⎡ w ⎤ ⎡ r ⎤ ⎡ hs ⎤p = ⎢ ⎥ ⎢ ⎥ ⎢ F⎥
η3 ⎣ β l ⎦ ⎣ β k ⎦ ⎣ β a ⎦βasva(22)

From the above, we can obtain the minimized cost tcjs associated with the firm’s output ys determining its
profit.
4.2 Firms’ Location Choice Behavior
Each firm determines its location choice probability to maximize the following expected weighted profit
based on the expected profit in each zone.
nmax SF ≡ ∑ PF rξ Fr π r −r =11n

∑ PF rln PF r

θ F r =1n(23)subject to∑ PF = 1r

(24)

r =1

where SF : firm’s expected profit with weight obtained by location choice, PF r : probability of locating inzone r in a firm, ξFjr : weight for firm’s profit in zone r, πjr : firm’s profit obtained in zone r, θF : Logitparameter
Solving this optimization problem, firm’s location choice probability for zone r is obtained as,

PF = nrexp θF ξ Fr π r∑ exp θ ξ πF

s =1sF(25)sThe maximum expected profit with weight in a firm is calculated as follows:

SF =1θFnln ∑ exp θ F ξ Fr π r

(26)

r =1

4.3 Households Behavior
Each household behaves to maximize the utility function. Households’ utility maximization behavior canbe hierarchically obtained as well.
Households are assumed to be homogeneous in the study area, and we consider a representative household
who resides in zone r and work in zone s. Each household is assumed to behave to maximize its utility
function of the current consumption, future consumption, and land under the budget condition. The current
consumption good consists of production commodity, passenger transport service, and leisure, while thefuture consumption is derived from the savings. The utility maximization then obtained as,max u (G rs , H rs , a Hrs ) ≡ G rsα G H rsα G a Hrsα a(27)subject to p Gr G rs + p Hr H rs + hHr a Hrs = w(T − n rs t rs ) + r ⋅ ks rs(28)where u : household utility function, G rs : current consumption in zone r, H rs : future consumption in zone r,aHrs : land consumption in zone r, αG，αH，αa : elasticity parameter ( αG + αH + αa = 1 ), pGr : price of
current consumption, pHr : price of future consumption, hHr : residential land rent in zone r, w : wage rate,
T : total available time endowed by a household, n rs : number of commuting trips from zone r to zone s,-4-t rs : commuting time between zone r and zone s, r : capital return rate, ks rs : capital stock endowed by ahousehold in zone r
Solving this utility maximization problem, we obtain the following demand functions:(29)[w(T − n t ) + r ⋅ ks ](30)

[w(T − n t ) + r ⋅ ks ]

(31)

rs rs

pGr

H rs =a Hrs =[w(T − n t ) + r ⋅ ks ]αGG rs =αHp Hr

αars

rs rs

rs

rs rshHrrs

By solving these three equations (29), (30) and (31) hierarchically, we can obtain the household utility
maximization behavior.
4.4 Households Residential Location Choice Behavior
The choice probability of household’s residential location in zone r and working in zone s can be derived
from the following maximization problem for the expected utility function with weight.

max SH s ≡n∑

PH rs ξ Hrs v rs −r =11n∑

θ Hs r =1

PH rs ln PH rsn(32)subject to∑ PH = 1rs(33)r =1where SH s : expected utility from choosing a residential place by a household working in zone s, PH rs :probability of residing in zone r of household working in zone s, ξHjrs : weight corresponding to indirect
utility value in a household working in zone s and residing in zone r, v rs : indirect utility value in ahousehold working in zone s and residing in zone r, θHs : Logit parameter
Solving this optimization problem, household probability of residing in zone r and working in zone s isobtained as,PH rs =exp θHs ξ Hrs v rsn

∑

(34)

ξexp θHs Hks v ksk =1

The household maximum expected utility value with weight for working in zone s and residing in zone r iscalculated as follows:SH s =1lns

θHn∑ exp θ ξ vsHrs rsH(35)r =1

The number of firms in zone s, that is, the number of employees is expressed by E s =m∑ E . Thus N ≡rssjj =1

PH rsE s expresses not only the number of households working zone s and residing in zone r, but also thenumber of commuting trips. The number of households in zone r is expressed as N r =n

∑N .
s =1-5-rs

4.5 Behavior of Real Estate Industry
The real estate industry in zone s provides differentially the land for business use (include land used byitself) ASFs and residential use ASHs. The amount of real estate industry’s services is assumed to be expressed
as hFs·ASFs + hHs·ASHs by using the initial land rents by zone. Corresponding to this service amount, theintermediate inputs, passenger transport input, labor input, capital input, and land input are determined so asto minimize the production cost. Because of the similarity with firms’ behavior, we omit its explanation here.
5. Market Equilibrium ConditionsMarket equilibrium conditions are denoted as follows:
Commodity market

Production in Individual Firm = Demand for Individual Firm

(36)

Labor marketnnn⎡⎛ nr =1

⎣ s =1n∑ ∑ T ⋅ N − ∑ ⎢⎜⎝ ∑ ∑ t PTPFr =1rss =1rk

k =1nnnms =1

r =1

r =1

j =1

rsk

⎞ ⎛ n rs rs ⎞⎤ n n rs rs⎟δ f ⎜ ∑ xTP N ⎟⎥ − ∑ ∑ xTP N⎠ ⎝ s =1⎠⎦ r =1 s =1(37)

− ∑ ∑ δ C t rs N rs = ∑ ∑ n r ld r
Capital marketnnnmr=s =1r=j =1∑ ∑ ksrs N rs = ∑ ∑ nr kd r

(38)

Land market for business use

AS Fr =m∑ ad Erjrjj =1(r = 1, L , n)(39)Land market for residential use
AS Hr =n∑a Ns =1

rsHrs

(r = 1, L , n)(40)

According to the model, the volume of traffic distribution by business objective, private objective, andcommuting objective can be determined when (generalized) transport cost is given. Moreover adding a
traffic assignment model to the current one, we can obtain simultaneous equilibrium of traffic, economy,
and location.

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