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A Dynamic Analysis of Multiple-Center Formation in a City
JOURNAL OF URBAN ECONOMICS ARTICLE NO.
40, 257]278 Ž1996.
0032
A Dynamic Analysis of Multiple-Center Formation in a City KOMEI SASAKI
AND
SE-IL MUN
Graduate School of Information Sciences, Tohoku Uni¨ ersity, Sendai, Japan Received November 24, 1994; revised May 18, 1995 An optimizing model of urban land development is presented where formation and growth processes of subcenters are described. The model determines the location of new firms and households so as to maximize the net rental revenue from land over the planning horizon. The conditions on parameter values for which subcenters can be formed are investigated by the simulation method. Furthermore, how the optimal subcenter location varies in response to changes in relevant parameters is examined. Q 1996 Academic Press, Inc.
1. INTRODUCTION A dynamic urban process would be described in terms of growth in the number of firms and households located in a city; at any given period, firms are newly located in a specific region, and this is followed by new household entrants in response to an increased demand for labor. At an early stage of urban growth, where the number of firms is fewer, firms tend to locate around the center of a region in order to benefit from agglomeration economies. Therefore the area outside a business district is used for household residences, i.e., the formation of a monocentric city. In a monocentric city, however, as the number of firms increases, the industrial district area is expanded and the average commuting distance increases. At a later stage, other business districts are naturally formed around subcenters and the average transport cost for commuting might be lower. White w9x examined the effects of establishing a subcenter in a closed city setting. Within the framework of a non-monocentric approach, Ogawa and Fujita w5x and Fujita and Ogawa w3x classified patterns of equilibrium urban configurations according to the relative size of parameters. They showed numerically as well as theoretically that multiple centers are formed when commuting cost is relatively high. Sullivan w7x analyzed numerically the effects of land-use policy on the equilibrium urban configuration in the setting of a multiple-center city. It is hypothesized that office-sector firms are located in the CBD, the central business district, and are involved in the production and sale of office output subject to 257 0094-1190r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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external agglomeration economies, and manufacturing firms are located in the SBD, the suburban business district, the area surrounding a circumferential highway. A city is open such that the profits of firms and utility levels of residents are exogeneously given. The general equilibrium of a city is defined, and a computation of an equilibrium of a city was performed under given parameters. Based on this model, some simulation analyses were conducted to examine the effects of a land-use policy on the equilibrium configuration of a multiple-center city. Sasaki w6x also undertook the numerical analysis within the framework of a multicentric approach. The difference from Sullivan w7x is that a city is closed and hence the utility level of residents is endogenously determined. The profit and net rental revenue are distributed to residents. In Sasaki w6x, the performance of the endogenous variables such as the utility level of a resident, firm profit, personal income, city size, and the area of land for each activity, both with and without a subcenter, are compared to evaluate the effects of subcenter establishment. However, most of the research so far has not been dynamic analysis but comparative static analysis of the effects of changes in the parameters on an equilibrium urban configuration. Recently, Helsley and Sullivan w4x intended to describe a dynamic process of subcenter formation in a city. The city planner’s problem there is to allocate a given population growth to two contiguous locations, a central city and a subcenter, and thereby to maximize the aggregate social value of output in each period. Their analysis is very clear, but some problems might be pointed out. First, the spatial aspect is neglected in their model. Stated differently, the land used for business and residence is not treated explicitly, and therefore the land rent structure in a city is not analyzed. The formation of subcenters affects the land use pattern, land rent structure, and commuting costs, and thus the spatial aspect of a city should be incorporated in the analysis. Second, the behavior of residents is suppressed. The utility level of households should be treated in a model since the effect of subcenter formation on the welfare of residents is of great concern to a city planner. Thirdly, a planner in their model behaves myopically and optimizes at each period, but optimization over time might be more realistic. Intending to overcome the drawbacks above in Helsley and Sullivan w4x, we will model the formation of city subcenters, describing the growth process of subcenters, and investigate the conditions associated with a particular process. It is stressed that it is only on the basis of such analysis that an appropriate urban policy for decentralizing industrial activities and developing subcenters can be made. In Section 2, the model is presented, and the formation process of subcenters is analyzed in Section 3. This theoretical analysis is complemented by the numerical simulation analyses in Section 4. Section 5
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develops an alternative model and makes a comparison between two models. 2. THE MODEL 2.1. The Setting Throughout the paper, a linear city is assumed, i.e., a city growing on a long-narrow homogeneous agricultural land whose width is unity. A particular location in this city is represented by its coordinate; < x < represents the distance from the center of the strip whose coordinate is 0, x ) 0 Ž x - 0., indicating that the location lies to the right Žleft. of the center. It is presumed that the city government rents all land used for industry and residence at each time from absentee landlords, and rents it to firms and households in a city. The goal of the city government is to bring about an urban configuration which will permit net revenue from the rentals, i.e., the difference between the rental revenue and the agricultural land rent, to be maximized given that households and firms in a city attain the predetermined utility and profit levels, respectively. We assume that each firm employs a certain amount of labor, n, and uses land of area 1 as inputs for production. The number of workers employed by a firm, n, is assumed to be constant throughout time. Furthermore, the area of residential land of a household, q, is also assumed to be constant and the same regardless of location. Let us suppose at the beginning of the planning period Ž t s 0. that no firms or households exist in this city. Also assume the location of the center of the central business district ŽCBD. is prespecified as 0, and the fringes of other business districts Žsecond business district, SBD. closer to the city center are specified as C and yC; i.e., firms entering the city are located in one of the three business districts. The CBD expands symmetrically around 0 while an SBD develops outward from the fringe C Žand yC .. Residential districts are formed in the areas between the CBD and the SBDs, and outside the SBDs. It is assumed that in this city, the business district areas never shrink, and a residential area can be converted to business use as the city grows.1 We assume the land-use pattern 1
This assumption implies that office buildings are durable while residential buildings are malleable and can be redeveloped without cost. We should recognize that this malleable structure assumption might limit the model’s applicability to reality, although some of the existing dynamic land-use models assume costless redevelopment Že.g., Fisch w2x, and Brueckner w1x.. On the other hand, it is widely observed in Japan that housing in the central area of large cities has been easily converted to office buildings. This is because the demolition cost of housing is very low in Japan since most residences are small and made of wood. Thus the development process derived under malleable housing assumptions might not be very far from reality in Japan.
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is symmetrical with respect to the city center: thus the land-use pattern on the left side is the same as that on the right side of the city. 2.2. Land Use Pattern and De¨ elopment Process We start by defining the symbols in the following way: 2 = l Ž t .: the number of firms newly located in a city at time t. aŽ t .: the right-side fringe of the CBD at t Žthe left-side fringe is yaŽ t . from the assumption of symmetry.. C: the left-side fringe of the SBD, which is predetermined. dŽ t .: the right-side fringe of the SBD Žfrom the assumption of symmetric land use, the right and left-side fringe of another SBD is yC and ydŽ t ... LŽ t .: the total number of firms operating in a city at t, i.e., LŽ t . s 2H0t l Ž tX . dtX . ¨ Ž t .: the ratio of firms newly located in the CBD at t to the total new firms l Ž t .. w aŽ t ., bŽ t .x: the residential area for households commuting to the CBD from the right. w eŽ t ., C x: the residential area for the households commuting to the SBD from the left. w dŽ t ., f Ž t .x: the residential area for households commuting to the SBD from the right Žfrom the assumption wybŽ t ., yaŽ t .x is the residential area for households commuting to the CBD from the left; wyf Ž t ., ydŽ t .x and wyC, yeŽ t .x are residential areas for households commuting to another SBD from the left and right, respectively.. Three different patterns of land use in a city at time t are shown in Fig. 1. The land use pattern will change with time in the order of stage 1, stage 2, and stage 3, but all three patterns do not necessarily emerge within a certain planning time period. We will explain each of the three stages. Stage 1. In the early stages of urban development, when the size of a city is not very huge, the residential districts are formed symmetrically around the CBD and the SBDs, and there is vacant land between the two residential districts, i.e., w bŽ t ., eŽ t .x Žand wyeŽ t ., ybŽ t .x.. The land use pattern of stage 1 continues until the vacant land between bŽ t . and eŽ t . is completely developed. While the land is used as in stage 1, it holds that aŽ t . s
t
X
X
H0 ¨ Ž t . l Ž t . dt
dŽ t . s C q
t
X
X
X
H0 Ž 1 y ¨ Ž t . . l Ž t . dt
b Ž t . s Ž 1 q nq . a Ž t .
X
MULTIPLE-CENTER FORMATION
261
FIG. 1. Land-use patterns in each stage of development.
eŽ t . s C y
1 2
Ž d Ž t . y C . nq
f Ž t . s dŽ t . q
1 2
Ž d Ž t . y C . nq
Ž 1.
and
Ž 1 q nq . aŽ t . F C y
1 2
Ž d Ž t . y C . nq
since bŽ t . F eŽ t .. Stage 2. When a city continues to grow even after all vacant land between the CBD and a SBD is used up, the residences of workers who will be employed by new firms in the SBD cannot be increased any more on the left side of the SBD. Thus, the residential district for such workers has to be expanded on the right side of the SBD. In this situation, if new firms are located in the CBD, then the location bŽ t .Žs eŽ t .. will move toward the right, and the residences of some employees in the SBD will be
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moved to the area on the right side of the SBD, whereby the city boundary f Ž t . will move outward. As far as stage 2 is concerned, the following holds: b Ž t . s Ž 1 q nq . a Ž t . eŽ t . s bŽ t . f Ž t . s Ž d Ž t . y C q a Ž t . . Ž 1 q nq . ,
Ž 2.
and since stage 2 continues to emerge until bŽ t . reaches C, Cy
1 2
Ž d Ž t . y C . nq F Ž 1 q nq . aŽ t . F C.
Stage 3. If the CBD continues to expand even after the right-side fringe of the residential district for the CBD workers, bŽ t ., coincides with the left-side fringe of the SBD, C, some CBD workers will reside on the right side of the SBD. Residences of SBD workers will be moved to the outside of the CBD workers’ residences. Each boundary location at stage 3 is represented as b Ž t . s Ž 1 q nq . a Ž t . q Ž d Ž t . y C . eŽ t . s C f Ž t . s Ž a Ž t . y C q d Ž t . . Ž 1 q nq . ,
Ž 3.
and the condition of emergence of stage 3 is C F Ž 1 q nq . a Ž t . . 2.3. Beha¨ ior of a Firm Firms in the city are assumed to be homogeneous with respect to their size and technology. Each firm produces homogeneous products using the land of area 1 and employing n workers. A firm’s productivity increases as the size of a city increases. That is, agglomeration economies work as the number of firms in a city increases because of better access to information on business, speedy diffusion of new technology, coordinated purchase of materials, and so on. The production function of each firm is assumed to be represented as b
Q Ž t . s L Ž t . F Ž n, 1 .
Ž 4.
Let the wage rates applied in the CBD and the SBD, respectively, be denoted by w C Ž t . and w S Ž t ..
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Besides labor and land input costs, each firm incurs communication costs in order to enjoy the agglomeration economies described above. That is, each firm must carry on communication with other firms for collecting and exchanging information on future prospects in the industry, prices of products and materials, wage rates, land rents, and development of new technology. This communication is conducted in face-to-face meetings, and thus its cost depends on the distance traveled for communication. Without communication with other firms, a firm cannot gain useful business information and consequently will sustain substantial losses. Letting r denote the cost of communication between unit distance per time period, a firm at location x incurs the cost r < x y xX < in order to communicate with a firm at xX . Each firm is assumed to make a certain number of trips to other firms for communication Žas is hypothesized in Ogawa and Fujita w3x as well.. Therefore the total communication cost incurred by firm at x in the CBD Ž x g w0, aŽ t .x. at time t is
t C Ž t, x. s r
yC
X
0
X
X
Hxa t Ž x y x . dx q HCd t Ž x y x . dx
X
q 2
X
x
HydŽ t . Ž x y x . dx q HyaŽ t . Ž x y x . dx q H0 Ž x y x . dx Ž .
X
X
X
Ž .
X
X
2
s r x 2 q aŽ t . q d Ž t . y C 2 .
Ž 5.
Likewise, the total communication cost for a firm at x in the SBD Ž x g w C, dŽ t .x. is 2
t S Ž t , x . s r x 2 y 2 Ž C y aŽ t . . x q d Ž t . .
Ž 6.
Under the assumption of a symmetric land use pattern, we can treat only the section to the right of the city center without losing generality. The profit attained at equilibrium is the same among firms regardless of location because of the homogeneity of firms. Where the profit level at time t is p Ž t ., the maximum amount a firm at location x can pay for land input, i.e., the bid-rent, is represented in the form b
2
2
r fC Ž t , x . s L Ž t . F Ž n, 1 . y w C Ž t . n y r x 2 q a Ž t . q d Ž t . y C 2 yp Ž t .
for x g 0, a Ž t .
b
r fS Ž t , x . s L Ž t . F Ž n, 1 . y w S Ž t . n y r x 2 y 2 Ž C y a Ž t . . x q d Ž t . yp Ž t .
for x g C, d Ž t . ,
Ž 7. 2
Ž 8.
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in which the product price is fixed at unity. Hereafter, r f Ž t, x . is referred to as the rent. 2.4. Beha¨ ior of a Household Suppose just one household member is employed by a firm in either the CBD or the SBD. It is also assumed that the number of households migrating into the city at time t is exactly equal to the number of workers employed by firms newly located in the city at t.2 Households are homogeneous and their utility function is represented in the form u s u Ž z, q . ,
Ž 9.
where z is the amount of composite goods, and q is the residence lot size. Throughout the analysis in the present paper, the lot size q is assumed to be fixed regardless of a household’s location. As households are homogeneous; the utility level attained in the equilibrium is the same regardless of where they live and work. Under the assumption of the same lot size q, this implies the same amount of composite good z regardless of a household’s location. Given the utility level u, the highest rent a household can pay for the unit size of lot at location x, i.e., the bid-rent, is defined in Ž10.. r hj Ž t , x . s
y j Ž t , x . y z Ž u, q . q
,
j s C, S,
Ž 10 .
in which y j Ž t, x . is the net income of a household residing at x and commuting to the j business district, defined for a household employed in the CBD as y C Ž t , x . s w C Ž t . y k Ž x y aŽ t . .
Ž 11 .
and for a household employed in the SBD as y S Ž t , x . s w S Ž t . y k min < C y x < , d Ž t . y x 4 .
Ž 12 .
In Ž11. and Ž12., k denotes the transport cost for commuting unit distance. The formulation in Ž11. and Ž12. assumes that the commuting cost of a household is equal to the transport cost from its residential location to the 2 One interpretation of this assumption is that firms newly located in this city have operated somewhere else, and all of their employees move to the city with firms. An alternative interpretation is that full employment equilibrium is ensured in a nation where workers can migrate freely between cities so that every household can attain the same utility level as prespecified in this city: thus, in this city, the work force grows in step with the number of firms.
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boundary of the business district where that household is working and does not depend on the location of that household’s workplace in the business district. This assumption is not as unrealistic as it may first appear. In fact, the fare system of public transport in many metropolitan areas is such that the fare from a particular location in the suburb into any part of the CBD is the same. 3. OPTIMAL LAND USE AND THE PROCESS OF SBD FORMATION 3.1. Problem This study will deal with a centralized optimizing model, but it will help to clarify the conditions for Pareto-efficient land use in a decentralized society. The city government allocates new firms to the CBD and SBDs, namely, determines ¨ Ž t ., given the utility level of a household and the profit of a firm and other parameters, so as to maximize the present value of the net rental revenue stream from land in a city. For simplicity, a firm’s profit is assumed to be zero over the planning horizon, T. The objective function is written as Vs
t1
H0
t2
Ž R1 Ž t . y S Ž t . . eyr t dt q H Ž R 2 Ž t . y S Ž t . . eyr t dt t1
q
T
Ht
Ž R 3 Ž t . y S Ž t . . eyr t dt,
Ž 13 .
2
where r is the discount rate, t 1 and t 2 are referred to as the switching times at which the land-use pattern shifts from stage 1 to stage 2, and from stage 2 to stage 3, respectively. R i Ž t . Ž i s 1, 2, 3. denotes the total rental revenue of land in a city at time t when the stage of urban development is i, and SŽ t . is the total amount the city government has to pay to absentee landlords. R i Ž t ., SŽ t . are represented as follows: Ri Ž t . s
H0a t r
Ž . C f
Ž t , x . dx q H
HCd t r
q
bŽ t . C
aŽ t .
Ž . S f
Ž t , x . dx q H
r h Ž t , x . dx q
f Žt. S
dŽ t .
r h Ž t , x . dx
S Ž t . s s Ž a Ž t . q d Ž t . y C . Ž 1 q nq . ,
C
HeŽ t .r
S h
Ž t , x . dx Ž 14 . Ž 15 .
where aŽ t ., bŽ t ., dŽ t ., eŽ t ., and f Ž t . are determined by systems Ž1., Ž2., and Ž3. corresponding to stage i. s denotes a constant, agricultural land rent. We note that the numbers of firms and households in a city at time t are
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given exogenously, and thus the value of SŽ t . does not depend on a particular development plan Žstage.. At each switching time, t 1 and t 2 , the following relations hold from Ž1. and Ž2.:
Ž 1 q nq . aŽ t1 . s C y 12 Ž d Ž t1 . y C . nq Ž 1 q nq . aŽ t 2 . s C.
Ž 16 .
In order to examine the meaning of the objective function, the rents of firms and households, Ž7., Ž8., and Ž10. are substituted into Ž14. and manipulated properly. As a result, we obtain, for example, for stage 1, R1 Ž t . y S Ž t . s Y 1 Ž t . y Z 1 Ž t . y GC1 Ž t . y GS1 Ž t . y IC1 Ž t . y IS11Ž t . y IS12Ž t . y S Ž t . ,
Ž 17 .
where Y 1Ž t. s
H0a t L Ž t . Ž .
b
F Ž n, 1 . dx q
HCd t L Ž t . Ž .
b
F Ž n, 1 . dx
b
s Ž a Ž t . q d Ž t . y C . L Ž t . F Ž n, 1 . Z1 Ž t . s
HabŽ t .t
Ž .
zŽ t. q
dx q
C
HeŽ t .
zŽ t. q
dx q
HaŽf t .t
Ž .
zŽ t. q
dx
s Ž a Ž t . q d Ž t . y C . nz Ž t . GC1 Ž t . s
H0a t t
Ž t , x . dx
GS1 Ž t . s
HCd t t
Ž t , x . dx
Ž . C
Ž . S
IC1 Ž t . s
HabŽ t .t
IS11Ž t . s
HeŽ t .
IS12Ž t . s
HdfŽ t .t
Ž .
C
k Ž x y aŽ t . . q
kŽC y x.
Ž .
q
dx
dx
k Ž x y dŽ t . . q
Ž 18 .
dx.
The meaning of each term in Ž17. is obvious: Y 1 Ž t . is the total industrial output, Z 1 Ž t . is the total composite good consumption, GC1 Ž t . and GS1 Ž t . are the total communication costs of firms operating in the CBD and the
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SBD, respectively, IC1 Ž t . is the total commuting cost of households working in the CBD, and IS11Ž t . and IS12Ž t . are the total commuting costs of households commuting to the SBD from the left and right, respectively. Therefore, the term Ž R i Ž t . y SŽ t .. expresses the social surplus in a city. Considered another way, the problem of the government is to maximize the present value of the stream of urban social surplus. The city government determines ¨ Ž t . and the switching time periods t 1 and t 2 so as to maximize Ž13. subject to the following constraints: a ˙Ž t . s ¨ Ž t . l Ž t .
Ž 19 .
d˙Ž t . s Ž 1 y ¨ Ž t . . l Ž t .
Ž 20 .
aŽ 0. s 0
Ž 21 .
d Ž 0. s C
Ž 22 .
¨ Ž t. G 0
Ž 23 .
¨ Ž t . F 1.
Ž 24 .
3.2. Optimal Conditions For solving this problem, the optimal control theory is applied. The Hamiltonian is defined as H i Ž t . s R i Ž t . y S Ž t . 4 eyr t q l1i Ž t . ¨ Ž t . l Ž t . q li2 Ž t . Ž 1 y ¨ Ž t . . l Ž t .
Ž i s 1, 2, 3 . , Ž 25. in which l1i Ž t . and li2 Ž t . are adjoint variables satisfying the following adjoint equations:
˙i1 Ž t . s y l ˙i2 Ž t . s y l
J iŽ t. aŽ t . J iŽ t. dt
Ž i s 1, 2, 3 .
Ž 26 .
Ž i s 1, 2, 3 . .
Ž 27 .
J i Ž t . in Ž26., Ž27. is the Lagrangian J i Ž t . s H i Ž t . q u 1Ž t . ¨ Ž t . q u 2 Ž t . Ž 1 y ¨ Ž t . . ,
Ž 28 .
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where u 1Ž t . and u 2 Ž t . are, respectively, the Lagrange multipliers corresponding to the constraint in Ž23. and Ž24.. The optimum condition with respect to ¨ Ž t . is
l1i Ž t . y li2 Ž t . q u 1 Ž t . y u 2 Ž t . s 0
Ž 29 .
u 1 Ž t . G 0,
u 1Ž t . ¨ Ž t . s 0
Ž 30 .
u2Ž t. G 0
u 2 Ž t . Ž 1 y ¨ Ž t . . s 0,
Ž 31 .
l1i Ž t . s li2 Ž t .
when 0 - ¨ Ž t . - 1
Ž 32 .
l1i
when ¨ Ž t . s 1
Ž 33 .
when ¨ Ž t . s 0.
Ž 34 .
which implies that
Ž t. G
li2
Ž t.
l1i Ž t . F li2 Ž t . The transversality conditions are
l1i T Ž T . s 0
Ž 35 .
li2T Ž T . s 0,
Ž 36 .
in which i T indicates the stage emerging at the terminal point T. Furthermore, according to Tomiyama w8x, it holds at each switching time period that
l1i Ž t i . s l1iq1 Ž t i .
Ž i s 1, 2 .
Ž 37 .
li2 Ž t i . s liq1 Ž ti . 2
Ž i s 1, 2 .
Ž 38 .
J i Ž t i . s J iq1 Ž t i .
Ž i s 1, 2 . .
Ž 39 .
It is noted that a switch in the land-use pattern in the optimization might not happen during the planning period. For instance, when t 1 s T, only the stage 1 pattern appears through the planning period, and it holds that J 1 ŽT . ) J 2 ŽT .. Likewise, where t 2 s T, the land-use pattern switches from stage 1 to stage 2, but stage 3 never appears during the planning period, and it holds that J 2 ŽT . ) J 3 ŽT .. The adjoint equation is developed. For example, for i s 1 Žstage 1., we obtain
J1Ž t. aŽ t .
b
b
s L Ž t . F Ž n, 1 . q b L Ž t . F Ž n, 1 . y nz Ž t . 2
2
y2 r Ž 2 a Ž t . q d Ž t . y C 2 . ykn2 qaŽ t . y s Ž 1 q nq . eyr t
Ž 40 .
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MULTIPLE-CENTER FORMATION
J1Ž t. dŽ t .
b
b
s y L Ž t . F Ž n, 1 . q b L Ž t . F Ž n, 1 . y nz Ž t . y4 r d Ž t . Ž a Ž t . q d Ž t . y C . 1 y kn2 q Ž d Ž t . y C . y s Ž 1 q nq . eyr t . 2
Ž 41 .
The first term in Ž40. and Ž41. denotes the value of product per firm; the second, the external agglomeration effect due to an increase in the number of firms; the third, the consumption of n households; the fourth, the increase in the communication cost of all the firms due to the location of a new firm at aŽ t . Žor dŽ t .., which includes the external effects to existing firms; the fifth, the increase in the commuting cost of households due to the location of a new firm in the CBD Žor SBD.; the sixth, the increase in the opportunity cost of land due to a new firm’s location in a city. Integrating Ž26. and Ž27. from t to T and taking the transversality condition in Ž35. and Ž36. we obtain
l11 Ž t . s l12 Ž t . s
t1
Ht
t1
Ht
J 1 Ž tX . aŽ tX . J 1 Ž tX . d Ž tX .
dtX q
t2
Ht
J 2 Ž tX . aŽ tX .
1
dtX q
t2
Ht
J 2 Ž tX .
1
d Ž tX .
dtX q
T
Ht
2
dtX q
T
Ht
2
J 3 Ž tX . aŽ tX . J 3 Ž tX . d Ž tX .
dtX
Ž 42 .
dtX .
Ž 43 .
Considering Ž40. and Ž41., l11Ž t . and l12 Ž t . are the present value of the stream of the increase in the social surplus in a city when an additional firm locates in the CBD and SBD, respectively, at time t when the stage 1 land-use pattern is prevailing. Now we are ready to give an economic interpretation to the optimum conditions Ž32. through Ž34.. When the CBD and the SBD are developed simultaneously at a particular time period, ¨ Ž t . Ž0 - ¨ Ž t . - 1. is determined such that the present value of marginal social surplus is equal between the two centers. When only the CBD ŽSBD. is developed, i.e., ¨ Ž t . s 1 Ž ¨ Ž t . s 0., the marginal social surplus is larger at the CBD ŽSBD. than at the SBD ŽCBD.. We can analyze the optimal conditions for stages 2 and 3 by a similar procedure. 3.3. Optimal Location of a Subcenter An important parameter in the model is the location of a subcenter, i.e., C, the left-side fringe of the SBD. Its location affects the value of the optimized social surplus V U and, hence, the determination of where to establish a subcenter is an important urban policy. By the envelope theorem, it holds at an optimal location C that V U r C s 0. Using Ž13.
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the condition that V Ur C s 0 is expressed as where U
Ri Ž t . C
sy
GCi Ž t . C
y
GSi Ž t . C
y
ICi Ž t . C
y
ISi 1Ž t . C
y
s yDG i Ž t . y DI i Ž t . .
ISi 2Ž t . C
Ž 45 .
In Ž45., DG i Ž t .Žs Ž GCi Ž t .r C . q Ž GSi Ž t .r C .. represents a change in the total communication cost of firms when the location is moved outward by one unit distance, which is found to be positive for all i: DI i Ž t . Žs Ž ICi Ž t .r C . q Ž ISi Ž t .r C . q Ž ISi Ž t .r C .., a change in the total 1 2 commuting cost of households when C moves outward by one unit distance, which is found to be nonpositive for all i. Thus, the optimal location of a subcenter is such that the increase in the communication cost is just offset by the decrease in the commuting cost. 4. SIMULATION ANALYSIS The theoretical model in the previous section is too complicated to trace the properties of optimal solution. Thus, in this section, simulation analysis is carried out to analyze solution properties. Figure 2a indicates the process of urban development under the following parameter values: C s 30,
k s 3.0,
r s 0.05,
q s 2.0,
n s 1.0,
T s 10.
Throughout the planning period, both the CBD and the SBD are growing, i.e., 0 - ¨ Ž t . - 1, although the growth rate varies with time: in initial development, most additional new firms are allocated to the CBD, but the ratio of allocation to the SBD increases with time. It holds that bŽT . eŽT .. That is, only the stage 1 type of land use is realized during the planning period, whereby some vacant land is left between the CBD and the SBD. Figure 2b shows the different processes under the following alternative parameter values: C s 25, k s 3.0, r s 0.1, q s 3.0, n s 1.0, T s 10. In the beginning, ¨ Ž t . s 1 so that all of the new firms are allocated to the CBD and thereby a monocentric land-use pattern is realized. In the middle of the planning period, the SBD is also growing together with the CBD, and in the last half periods, almost all new firms are allocated to the SBD. The land-use pattern shifts from stage 1 to stage 2 in the last half period, and after that the residences of workers in the SBD are built in the right side of the SBD. Simulations were performed for some other parameter values, but we could not find any case where the stage 3 land-use pattern emerges.3 3 As shown in Section 5, a stage 3 land-use pattern emerged under the alternative specification of firm’s behavior.
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FIG. 2. Ža. Optimal solution and spatial development process Ž1.. Žb. Optimal solution and spatial development process Ž2..
Next we investigate the effect of parameter change on the possibility of subcenter formation; i.e., it is examined whether a subcenter is formed or not during the planning period. Figures 3a and 3b show the simulation results. In both parts of the figure, the location of C is measured along the horizontal axis, and simulations were performed for the various combina-
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tions of values of C and other parameters so as to specify the conditions for SBD formation. The curve in each figure separates the area into two parts Žhereafter referred to as ‘‘separation curve’’.: one is where the SBD is formed, whether its size is large or small, during the planning period, and the other is where no firm is allocated to the SBD during the planning period so that a city grows around a monocenter. We can abstract some observations from Figures 3a and 3b. 1. Other things being equal, the closer C is located to the city center, the more likely an SBD is to be formed. Since each firm carries on communication with all other firms, the average communication cost becomes higher with the distance between the CBD and the SBD. Thus, if C is rather far from the CBD, all firms will be located in the CBD to save the communication cost. 2. The higher the commuting cost k, the more likely it is that an SBD will be formed. This is because the average commuting distance is shortened by forming the SBD, hence the average commuting cost is lowered. 3. The lower the communication cost r , the more likely it is that an SBD will be formed since the average communication cost is hardly increased by SBD formation. Although the figures are not shown, we examined the effects of parameter q and n on the possibility of subcenter formation. The results are summarized as follows. 4. A larger residential lot size q is likely to promote the formation of the SBD since it has the same effect as a higher commuting cost.
FIG. 3. Ža. Communication cost Ž r . and subcenter formation. Žb. Commuting cost Ž k . and subcenter formation.
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5. When the size of a firm, n, is increased, it is likely that an SBD will be developed since the residential district is enlarged, whereby it has the same effect as a larger residential lot size. We finally examine how optimal SBD location changes in response to changes in parameters. In Fig. 4a the optimal value of the objective function is plotted against the location C, where the other parameters are held constant at the values for the case in Fig. 2b. The curve has a single peak around C s 13, the optimal location of an SBD. The curve becomes flat beyond C s 40, which implies that it is not optimal to form an SBD if
FIG. 4. Optimal location of the subcenter.
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C is located so far from the city center. It is also verified from Fig. 4b that, at optimal C, change in communication cost Ž DG . is just equal to the change in commuting cost Ž DI . by moving C to the right. Figure 4 indicates that if the location of a subcenter is appropriately determined, then multicentric land use brings about larger social benefit than a monocentric one. The effects of changes in parameter values on optimal subcenter location are summarized as follows. The optimal location of C moves further from a city center as k, n, or q increase. An interpretation is that when commuting cost is heightened due to an increase in k, n, or q, the average commuting cost is decreased by moving C to the right and thereby increasing the number of commuters from the left of an SBD. When r is small, moving C to the right is optimal because it increases the communication cost by a relatively small amount but significantly decrease the commuting cost by increasing the number of commuters from the left of an SBD. 5. ALTERNATIVE MODEL The previous model has assumed that every firm in a city benefits equally, regardless of location, from agglomeration economies. However, it might be more natural and realistic that firms within the same industrial district strongly affect each other, but the spillover of agglomeration economies between different districts is weak. In particular, it might be the case that the benefit from some types of social overhead capital Žroads, transportation systems, water supply., the lower price of materials due to the coordinated purchase, the lower repair costs due to better access to related industries, and so on, can be realized only within a certain area. In the alternative models, therefore, the extent of agglomeration economies is assumed to depend on the number of firms located in a particular industrial district rather than the total number of firms in a city; i.e., a production function for a firm is specified as b
Q Ž t . s Ž 2 a Ž t . . F Ž n, 1 .
for a CBD firm
and b
Q Ž t . s Ž d Ž t . y C . F Ž n, 1 .
for an SBD firm.
Another departure from the previous model is concerned with the communication behavior of a firm. In the previous model, every firm makes a certain number of trips to all the other firms while in the alternative model each firm makes a given number of trips to the main and local trading floors, respectively, where the main trading floor is at the city
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center and local trading floors are at the city center and at C.4 That is, communication costs of a firm are specified as
t C Ž t , x . s Ž r˜ q g . x
for x g 0, a Ž t .
t S Ž t , x . s r˜ x q g Ž x y C .
for x g C, d Ž t . ,
where r˜ and g are the communication costs per unit distance per unit time period at the main trading floor and a local trading floor, respectively.5 Thus, in the alternative model the bid-rent of a firm is expressed in the form b
r fC Ž t , x . s Ž 2 a Ž t . . F Ž n, 1 . y w C Ž t . n y Ž r˜ q g . x b
r fS Ž t , x . s Ž d Ž t . y C . F Ž n, 1 . y w S Ž t . n y r˜ x y g Ž x y C . . Under this circumstance, the same problem as Ž13. was solved, similar optimum conditions were derived, and similar analyses were done. Figures 5a]5c summarize the dynamic land-use process under the alternative model for three different parameter values. In Fig. 5a, all the new firms are allocated in the CBD up to the seventh period, and after that, SBDs are exclusively developed: even at the terminal period, a stage 1 pattern of land use is realized. In Fig. 5b, during the middle time period, both districts simultaneously grow, but at later periods only the SBD does. At the eighth period, the land-use pattern shifts to stage 2 and continues until the terminal point. The case in Fig. 5c generates a stage 3 type of land use which could not be observed in the previous model.6 In this case, the CBD grows exclusively first, the SBD being exclusively developed next, and both districts grow simultaneously in the latter periods. We conduct a simulation similar to that in the previous section to clarify the condition for SBD formation and summarize the results in similar figures Žalthough the figures are not shown.. Some important observations and, in particular, differences from the previous model are summarized as follows. 1. Other things being equal, the further C is located from the city center, the more likely it is that an SBD will be formed in the alternative 4
So, the main trading floor coincides with the local trading floor of the CBD. This is the same specification as the one in Sasaki w6x. 6 Although it is not easy to explain rigorously why the stage 3 land-use pattern emerged not in the first model but in the alternative model, the following conjecture might be helpful: In the first model, the possibility of subcenter formation is solely determined by the trade-off between communication cost Žcentralizing factor. and commuting cost Ždecentralizing factor.. On the other hand, in the alternative model, localized agglomeration economies work as an additional centralizing factor, strengthening the tendency of firms’ concentration in the CBD so that the stage 3 land-use pattern emerges in a particular environment. 5
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FIG. 5. Ža. Optimal solution and spatial development process Ž1.. Žb. Optimal solution and spatial development process Ž2..
model. This result is contrary to that of the previous model. An interpretation of this is that when C is rather close to the city center, the savings of commuting and communication costs due to SBD formation are not large enough to offset the increased agglomeration economies due to the concentration of firms in the CBD.
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FIG. 5ŽC.. Optimal solution and spatial development process Ž3..
2. As b increases, a monocentric land-use pattern is likely to emerge in order to fully enjoy agglomeration economies. And where b exceeds a certain level, a subcenter is never formed, regardless of the location of C. 3. The separation curve for g and C is monotonic, but that of r and C has a parabolic shape. This implies that when the subcenter location is very far from a city center, the average communication cost of SBD firms at the main trading floor becomes high enough to offset the savings of communication cost at a local trading floor and the lowered commuting cost due to SBD formation, whereby a monocentric land use is realized. 4. The separation curves for k, q, and n have a flat part which was not observed in the previous model; i.e., there are threshold levels of n, q, and k to induce a subcenter formation. The simulated results concerning the optimal location of C are as follows. First, the optimal location of C moves further from the center with the increasing degree of agglomeration economies so as to fully enjoy the externality by more concentration in one district. Second, the impacts on the optimal C of n and q are the same as in the previous model: the optimal location of C moves further out with their increased values. However, the effect of increasing k is different between the two models. In the alternative model, the optimal location of C moves closer to a city center as commuting costs increase. We can interpret this result as follows. The share of CBD firms becomes smaller as k increases, and thus the
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required land space for CBD activities becomes smaller, and consequently the optimal location of C moves closer to a city center. Third, an increased communication cost at the main trading floor induces firms to locate in the CBD, and thus the optimal location of C moves further out to retain sufficient land space for CBD activities. On the other hand, increased communication cost at a local trading floor is favorable to SBD formation, and thereby the optimal location of C moves closer to a city center. The investigation of the alternative model so far suggests that variations in the specification of agglomeration economies and communication costs result in different dynamic patterns of land use and different optimal locations of a subcenter. This implies that the urban policy of establishing a subcenter is rather sensitive to firm behavior. Therefore, in order to make an appropriate urban policy, policymakers need to know the economic circumstances around the relevant agents in a city.
ACKNOWLEDGMENTS An earlier version of this paper was presented at the 6th Annual Conference of the Pacific Rim Council on Urban Development, Taipei, November 1]3, 1994, and at the Applied Regional Science Conference, Kobe, December 3]4, 1994. We thank Drs. Tatsuaki Kuroda, Yukio Itagaki, and the participants of both conferences for their useful comments. We are also grateful to the editor and two anonymous referees of this journal for their constructive comments on the first version. This research was supported by grant from the Ministry of Education, which is gratefully acknowledged.
REFERENCES 1. J. K. Brueckner, A vintage model of urban growth, Journal of Urban Economics, 8, 389]402 Ž1980.. 2. O. Fisch, Dynamics of the housing market, Journal of Urban Economics, 4, 428]447 Ž1977.. 3. M. Fujita and H. Ogawa, Multiple equilibria and structural transition of non-monocentric urban configurations, Regional Science and Urban Economics, 12, 161]196 Ž1982.. 4. R. W. Helsley and A. M. Sullivan, Urban subcenter formation, Regional Science and Urban Economics, 21, 255]275 Ž1991.. 5. H. Ogawa and M. Fujita, Equilibrium land use patterns in a non-monocentric city, Journal of Regional Science, 20, 455]475 Ž1980.. 6. K. Sasaki, The establishment of subcenter and urban spatial structure, En¨ ironment and Planning A, 22, 369]383 Ž1990.. 7. A. M. Sullivan, A general equilibrium model with agglomerative economies and decentralized employment, Journal of Urban Economies, 20, 55]74 Ž1986.. 8. K. Tomiyama, Two-stage optimal control problems and optimality conditions, Journal of Economic Dynamics and Control, 9, 317]337 Ž1985.. 9. M. J. White, Firm suburbanization and urban subcenters, Journal of Urban Economics, 3, 323]343 Ž1976..
40, 257]278 Ž1996.
0032
A Dynamic Analysis of Multiple-Center Formation in a City KOMEI SASAKI
AND
SE-IL MUN
Graduate School of Information Sciences, Tohoku Uni¨ ersity, Sendai, Japan Received November 24, 1994; revised May 18, 1995 An optimizing model of urban land development is presented where formation and growth processes of subcenters are described. The model determines the location of new firms and households so as to maximize the net rental revenue from land over the planning horizon. The conditions on parameter values for which subcenters can be formed are investigated by the simulation method. Furthermore, how the optimal subcenter location varies in response to changes in relevant parameters is examined. Q 1996 Academic Press, Inc.
1. INTRODUCTION A dynamic urban process would be described in terms of growth in the number of firms and households located in a city; at any given period, firms are newly located in a specific region, and this is followed by new household entrants in response to an increased demand for labor. At an early stage of urban growth, where the number of firms is fewer, firms tend to locate around the center of a region in order to benefit from agglomeration economies. Therefore the area outside a business district is used for household residences, i.e., the formation of a monocentric city. In a monocentric city, however, as the number of firms increases, the industrial district area is expanded and the average commuting distance increases. At a later stage, other business districts are naturally formed around subcenters and the average transport cost for commuting might be lower. White w9x examined the effects of establishing a subcenter in a closed city setting. Within the framework of a non-monocentric approach, Ogawa and Fujita w5x and Fujita and Ogawa w3x classified patterns of equilibrium urban configurations according to the relative size of parameters. They showed numerically as well as theoretically that multiple centers are formed when commuting cost is relatively high. Sullivan w7x analyzed numerically the effects of land-use policy on the equilibrium urban configuration in the setting of a multiple-center city. It is hypothesized that office-sector firms are located in the CBD, the central business district, and are involved in the production and sale of office output subject to 257 0094-1190r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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external agglomeration economies, and manufacturing firms are located in the SBD, the suburban business district, the area surrounding a circumferential highway. A city is open such that the profits of firms and utility levels of residents are exogeneously given. The general equilibrium of a city is defined, and a computation of an equilibrium of a city was performed under given parameters. Based on this model, some simulation analyses were conducted to examine the effects of a land-use policy on the equilibrium configuration of a multiple-center city. Sasaki w6x also undertook the numerical analysis within the framework of a multicentric approach. The difference from Sullivan w7x is that a city is closed and hence the utility level of residents is endogenously determined. The profit and net rental revenue are distributed to residents. In Sasaki w6x, the performance of the endogenous variables such as the utility level of a resident, firm profit, personal income, city size, and the area of land for each activity, both with and without a subcenter, are compared to evaluate the effects of subcenter establishment. However, most of the research so far has not been dynamic analysis but comparative static analysis of the effects of changes in the parameters on an equilibrium urban configuration. Recently, Helsley and Sullivan w4x intended to describe a dynamic process of subcenter formation in a city. The city planner’s problem there is to allocate a given population growth to two contiguous locations, a central city and a subcenter, and thereby to maximize the aggregate social value of output in each period. Their analysis is very clear, but some problems might be pointed out. First, the spatial aspect is neglected in their model. Stated differently, the land used for business and residence is not treated explicitly, and therefore the land rent structure in a city is not analyzed. The formation of subcenters affects the land use pattern, land rent structure, and commuting costs, and thus the spatial aspect of a city should be incorporated in the analysis. Second, the behavior of residents is suppressed. The utility level of households should be treated in a model since the effect of subcenter formation on the welfare of residents is of great concern to a city planner. Thirdly, a planner in their model behaves myopically and optimizes at each period, but optimization over time might be more realistic. Intending to overcome the drawbacks above in Helsley and Sullivan w4x, we will model the formation of city subcenters, describing the growth process of subcenters, and investigate the conditions associated with a particular process. It is stressed that it is only on the basis of such analysis that an appropriate urban policy for decentralizing industrial activities and developing subcenters can be made. In Section 2, the model is presented, and the formation process of subcenters is analyzed in Section 3. This theoretical analysis is complemented by the numerical simulation analyses in Section 4. Section 5
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develops an alternative model and makes a comparison between two models. 2. THE MODEL 2.1. The Setting Throughout the paper, a linear city is assumed, i.e., a city growing on a long-narrow homogeneous agricultural land whose width is unity. A particular location in this city is represented by its coordinate; < x < represents the distance from the center of the strip whose coordinate is 0, x ) 0 Ž x - 0., indicating that the location lies to the right Žleft. of the center. It is presumed that the city government rents all land used for industry and residence at each time from absentee landlords, and rents it to firms and households in a city. The goal of the city government is to bring about an urban configuration which will permit net revenue from the rentals, i.e., the difference between the rental revenue and the agricultural land rent, to be maximized given that households and firms in a city attain the predetermined utility and profit levels, respectively. We assume that each firm employs a certain amount of labor, n, and uses land of area 1 as inputs for production. The number of workers employed by a firm, n, is assumed to be constant throughout time. Furthermore, the area of residential land of a household, q, is also assumed to be constant and the same regardless of location. Let us suppose at the beginning of the planning period Ž t s 0. that no firms or households exist in this city. Also assume the location of the center of the central business district ŽCBD. is prespecified as 0, and the fringes of other business districts Žsecond business district, SBD. closer to the city center are specified as C and yC; i.e., firms entering the city are located in one of the three business districts. The CBD expands symmetrically around 0 while an SBD develops outward from the fringe C Žand yC .. Residential districts are formed in the areas between the CBD and the SBDs, and outside the SBDs. It is assumed that in this city, the business district areas never shrink, and a residential area can be converted to business use as the city grows.1 We assume the land-use pattern 1
This assumption implies that office buildings are durable while residential buildings are malleable and can be redeveloped without cost. We should recognize that this malleable structure assumption might limit the model’s applicability to reality, although some of the existing dynamic land-use models assume costless redevelopment Že.g., Fisch w2x, and Brueckner w1x.. On the other hand, it is widely observed in Japan that housing in the central area of large cities has been easily converted to office buildings. This is because the demolition cost of housing is very low in Japan since most residences are small and made of wood. Thus the development process derived under malleable housing assumptions might not be very far from reality in Japan.
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is symmetrical with respect to the city center: thus the land-use pattern on the left side is the same as that on the right side of the city. 2.2. Land Use Pattern and De¨ elopment Process We start by defining the symbols in the following way: 2 = l Ž t .: the number of firms newly located in a city at time t. aŽ t .: the right-side fringe of the CBD at t Žthe left-side fringe is yaŽ t . from the assumption of symmetry.. C: the left-side fringe of the SBD, which is predetermined. dŽ t .: the right-side fringe of the SBD Žfrom the assumption of symmetric land use, the right and left-side fringe of another SBD is yC and ydŽ t ... LŽ t .: the total number of firms operating in a city at t, i.e., LŽ t . s 2H0t l Ž tX . dtX . ¨ Ž t .: the ratio of firms newly located in the CBD at t to the total new firms l Ž t .. w aŽ t ., bŽ t .x: the residential area for households commuting to the CBD from the right. w eŽ t ., C x: the residential area for the households commuting to the SBD from the left. w dŽ t ., f Ž t .x: the residential area for households commuting to the SBD from the right Žfrom the assumption wybŽ t ., yaŽ t .x is the residential area for households commuting to the CBD from the left; wyf Ž t ., ydŽ t .x and wyC, yeŽ t .x are residential areas for households commuting to another SBD from the left and right, respectively.. Three different patterns of land use in a city at time t are shown in Fig. 1. The land use pattern will change with time in the order of stage 1, stage 2, and stage 3, but all three patterns do not necessarily emerge within a certain planning time period. We will explain each of the three stages. Stage 1. In the early stages of urban development, when the size of a city is not very huge, the residential districts are formed symmetrically around the CBD and the SBDs, and there is vacant land between the two residential districts, i.e., w bŽ t ., eŽ t .x Žand wyeŽ t ., ybŽ t .x.. The land use pattern of stage 1 continues until the vacant land between bŽ t . and eŽ t . is completely developed. While the land is used as in stage 1, it holds that aŽ t . s
t
X
X
H0 ¨ Ž t . l Ž t . dt
dŽ t . s C q
t
X
X
X
H0 Ž 1 y ¨ Ž t . . l Ž t . dt
b Ž t . s Ž 1 q nq . a Ž t .
X
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FIG. 1. Land-use patterns in each stage of development.
eŽ t . s C y
1 2
Ž d Ž t . y C . nq
f Ž t . s dŽ t . q
1 2
Ž d Ž t . y C . nq
Ž 1.
and
Ž 1 q nq . aŽ t . F C y
1 2
Ž d Ž t . y C . nq
since bŽ t . F eŽ t .. Stage 2. When a city continues to grow even after all vacant land between the CBD and a SBD is used up, the residences of workers who will be employed by new firms in the SBD cannot be increased any more on the left side of the SBD. Thus, the residential district for such workers has to be expanded on the right side of the SBD. In this situation, if new firms are located in the CBD, then the location bŽ t .Žs eŽ t .. will move toward the right, and the residences of some employees in the SBD will be
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moved to the area on the right side of the SBD, whereby the city boundary f Ž t . will move outward. As far as stage 2 is concerned, the following holds: b Ž t . s Ž 1 q nq . a Ž t . eŽ t . s bŽ t . f Ž t . s Ž d Ž t . y C q a Ž t . . Ž 1 q nq . ,
Ž 2.
and since stage 2 continues to emerge until bŽ t . reaches C, Cy
1 2
Ž d Ž t . y C . nq F Ž 1 q nq . aŽ t . F C.
Stage 3. If the CBD continues to expand even after the right-side fringe of the residential district for the CBD workers, bŽ t ., coincides with the left-side fringe of the SBD, C, some CBD workers will reside on the right side of the SBD. Residences of SBD workers will be moved to the outside of the CBD workers’ residences. Each boundary location at stage 3 is represented as b Ž t . s Ž 1 q nq . a Ž t . q Ž d Ž t . y C . eŽ t . s C f Ž t . s Ž a Ž t . y C q d Ž t . . Ž 1 q nq . ,
Ž 3.
and the condition of emergence of stage 3 is C F Ž 1 q nq . a Ž t . . 2.3. Beha¨ ior of a Firm Firms in the city are assumed to be homogeneous with respect to their size and technology. Each firm produces homogeneous products using the land of area 1 and employing n workers. A firm’s productivity increases as the size of a city increases. That is, agglomeration economies work as the number of firms in a city increases because of better access to information on business, speedy diffusion of new technology, coordinated purchase of materials, and so on. The production function of each firm is assumed to be represented as b
Q Ž t . s L Ž t . F Ž n, 1 .
Ž 4.
Let the wage rates applied in the CBD and the SBD, respectively, be denoted by w C Ž t . and w S Ž t ..
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Besides labor and land input costs, each firm incurs communication costs in order to enjoy the agglomeration economies described above. That is, each firm must carry on communication with other firms for collecting and exchanging information on future prospects in the industry, prices of products and materials, wage rates, land rents, and development of new technology. This communication is conducted in face-to-face meetings, and thus its cost depends on the distance traveled for communication. Without communication with other firms, a firm cannot gain useful business information and consequently will sustain substantial losses. Letting r denote the cost of communication between unit distance per time period, a firm at location x incurs the cost r < x y xX < in order to communicate with a firm at xX . Each firm is assumed to make a certain number of trips to other firms for communication Žas is hypothesized in Ogawa and Fujita w3x as well.. Therefore the total communication cost incurred by firm at x in the CBD Ž x g w0, aŽ t .x. at time t is
t C Ž t, x. s r
yC
X
0
X
X
Hxa t Ž x y x . dx q HCd t Ž x y x . dx
X
q 2
X
x
HydŽ t . Ž x y x . dx q HyaŽ t . Ž x y x . dx q H0 Ž x y x . dx Ž .
X
X
X
Ž .
X
X
2
s r x 2 q aŽ t . q d Ž t . y C 2 .
Ž 5.
Likewise, the total communication cost for a firm at x in the SBD Ž x g w C, dŽ t .x. is 2
t S Ž t , x . s r x 2 y 2 Ž C y aŽ t . . x q d Ž t . .
Ž 6.
Under the assumption of a symmetric land use pattern, we can treat only the section to the right of the city center without losing generality. The profit attained at equilibrium is the same among firms regardless of location because of the homogeneity of firms. Where the profit level at time t is p Ž t ., the maximum amount a firm at location x can pay for land input, i.e., the bid-rent, is represented in the form b
2
2
r fC Ž t , x . s L Ž t . F Ž n, 1 . y w C Ž t . n y r x 2 q a Ž t . q d Ž t . y C 2 yp Ž t .
for x g 0, a Ž t .
b
r fS Ž t , x . s L Ž t . F Ž n, 1 . y w S Ž t . n y r x 2 y 2 Ž C y a Ž t . . x q d Ž t . yp Ž t .
for x g C, d Ž t . ,
Ž 7. 2
Ž 8.
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in which the product price is fixed at unity. Hereafter, r f Ž t, x . is referred to as the rent. 2.4. Beha¨ ior of a Household Suppose just one household member is employed by a firm in either the CBD or the SBD. It is also assumed that the number of households migrating into the city at time t is exactly equal to the number of workers employed by firms newly located in the city at t.2 Households are homogeneous and their utility function is represented in the form u s u Ž z, q . ,
Ž 9.
where z is the amount of composite goods, and q is the residence lot size. Throughout the analysis in the present paper, the lot size q is assumed to be fixed regardless of a household’s location. As households are homogeneous; the utility level attained in the equilibrium is the same regardless of where they live and work. Under the assumption of the same lot size q, this implies the same amount of composite good z regardless of a household’s location. Given the utility level u, the highest rent a household can pay for the unit size of lot at location x, i.e., the bid-rent, is defined in Ž10.. r hj Ž t , x . s
y j Ž t , x . y z Ž u, q . q
,
j s C, S,
Ž 10 .
in which y j Ž t, x . is the net income of a household residing at x and commuting to the j business district, defined for a household employed in the CBD as y C Ž t , x . s w C Ž t . y k Ž x y aŽ t . .
Ž 11 .
and for a household employed in the SBD as y S Ž t , x . s w S Ž t . y k min < C y x < , d Ž t . y x 4 .
Ž 12 .
In Ž11. and Ž12., k denotes the transport cost for commuting unit distance. The formulation in Ž11. and Ž12. assumes that the commuting cost of a household is equal to the transport cost from its residential location to the 2 One interpretation of this assumption is that firms newly located in this city have operated somewhere else, and all of their employees move to the city with firms. An alternative interpretation is that full employment equilibrium is ensured in a nation where workers can migrate freely between cities so that every household can attain the same utility level as prespecified in this city: thus, in this city, the work force grows in step with the number of firms.
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boundary of the business district where that household is working and does not depend on the location of that household’s workplace in the business district. This assumption is not as unrealistic as it may first appear. In fact, the fare system of public transport in many metropolitan areas is such that the fare from a particular location in the suburb into any part of the CBD is the same. 3. OPTIMAL LAND USE AND THE PROCESS OF SBD FORMATION 3.1. Problem This study will deal with a centralized optimizing model, but it will help to clarify the conditions for Pareto-efficient land use in a decentralized society. The city government allocates new firms to the CBD and SBDs, namely, determines ¨ Ž t ., given the utility level of a household and the profit of a firm and other parameters, so as to maximize the present value of the net rental revenue stream from land in a city. For simplicity, a firm’s profit is assumed to be zero over the planning horizon, T. The objective function is written as Vs
t1
H0
t2
Ž R1 Ž t . y S Ž t . . eyr t dt q H Ž R 2 Ž t . y S Ž t . . eyr t dt t1
q
T
Ht
Ž R 3 Ž t . y S Ž t . . eyr t dt,
Ž 13 .
2
where r is the discount rate, t 1 and t 2 are referred to as the switching times at which the land-use pattern shifts from stage 1 to stage 2, and from stage 2 to stage 3, respectively. R i Ž t . Ž i s 1, 2, 3. denotes the total rental revenue of land in a city at time t when the stage of urban development is i, and SŽ t . is the total amount the city government has to pay to absentee landlords. R i Ž t ., SŽ t . are represented as follows: Ri Ž t . s
H0a t r
Ž . C f
Ž t , x . dx q H
HCd t r
q
bŽ t . C
aŽ t .
Ž . S f
Ž t , x . dx q H
r h Ž t , x . dx q
f Žt. S
dŽ t .
r h Ž t , x . dx
S Ž t . s s Ž a Ž t . q d Ž t . y C . Ž 1 q nq . ,
C
HeŽ t .r
S h
Ž t , x . dx Ž 14 . Ž 15 .
where aŽ t ., bŽ t ., dŽ t ., eŽ t ., and f Ž t . are determined by systems Ž1., Ž2., and Ž3. corresponding to stage i. s denotes a constant, agricultural land rent. We note that the numbers of firms and households in a city at time t are
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given exogenously, and thus the value of SŽ t . does not depend on a particular development plan Žstage.. At each switching time, t 1 and t 2 , the following relations hold from Ž1. and Ž2.:
Ž 1 q nq . aŽ t1 . s C y 12 Ž d Ž t1 . y C . nq Ž 1 q nq . aŽ t 2 . s C.
Ž 16 .
In order to examine the meaning of the objective function, the rents of firms and households, Ž7., Ž8., and Ž10. are substituted into Ž14. and manipulated properly. As a result, we obtain, for example, for stage 1, R1 Ž t . y S Ž t . s Y 1 Ž t . y Z 1 Ž t . y GC1 Ž t . y GS1 Ž t . y IC1 Ž t . y IS11Ž t . y IS12Ž t . y S Ž t . ,
Ž 17 .
where Y 1Ž t. s
H0a t L Ž t . Ž .
b
F Ž n, 1 . dx q
HCd t L Ž t . Ž .
b
F Ž n, 1 . dx
b
s Ž a Ž t . q d Ž t . y C . L Ž t . F Ž n, 1 . Z1 Ž t . s
HabŽ t .t
Ž .
zŽ t. q
dx q
C
HeŽ t .
zŽ t. q
dx q
HaŽf t .t
Ž .
zŽ t. q
dx
s Ž a Ž t . q d Ž t . y C . nz Ž t . GC1 Ž t . s
H0a t t
Ž t , x . dx
GS1 Ž t . s
HCd t t
Ž t , x . dx
Ž . C
Ž . S
IC1 Ž t . s
HabŽ t .t
IS11Ž t . s
HeŽ t .
IS12Ž t . s
HdfŽ t .t
Ž .
C
k Ž x y aŽ t . . q
kŽC y x.
Ž .
q
dx
dx
k Ž x y dŽ t . . q
Ž 18 .
dx.
The meaning of each term in Ž17. is obvious: Y 1 Ž t . is the total industrial output, Z 1 Ž t . is the total composite good consumption, GC1 Ž t . and GS1 Ž t . are the total communication costs of firms operating in the CBD and the
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SBD, respectively, IC1 Ž t . is the total commuting cost of households working in the CBD, and IS11Ž t . and IS12Ž t . are the total commuting costs of households commuting to the SBD from the left and right, respectively. Therefore, the term Ž R i Ž t . y SŽ t .. expresses the social surplus in a city. Considered another way, the problem of the government is to maximize the present value of the stream of urban social surplus. The city government determines ¨ Ž t . and the switching time periods t 1 and t 2 so as to maximize Ž13. subject to the following constraints: a ˙Ž t . s ¨ Ž t . l Ž t .
Ž 19 .
d˙Ž t . s Ž 1 y ¨ Ž t . . l Ž t .
Ž 20 .
aŽ 0. s 0
Ž 21 .
d Ž 0. s C
Ž 22 .
¨ Ž t. G 0
Ž 23 .
¨ Ž t . F 1.
Ž 24 .
3.2. Optimal Conditions For solving this problem, the optimal control theory is applied. The Hamiltonian is defined as H i Ž t . s R i Ž t . y S Ž t . 4 eyr t q l1i Ž t . ¨ Ž t . l Ž t . q li2 Ž t . Ž 1 y ¨ Ž t . . l Ž t .
Ž i s 1, 2, 3 . , Ž 25. in which l1i Ž t . and li2 Ž t . are adjoint variables satisfying the following adjoint equations:
˙i1 Ž t . s y l ˙i2 Ž t . s y l
J iŽ t. aŽ t . J iŽ t. dt
Ž i s 1, 2, 3 .
Ž 26 .
Ž i s 1, 2, 3 . .
Ž 27 .
J i Ž t . in Ž26., Ž27. is the Lagrangian J i Ž t . s H i Ž t . q u 1Ž t . ¨ Ž t . q u 2 Ž t . Ž 1 y ¨ Ž t . . ,
Ž 28 .
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where u 1Ž t . and u 2 Ž t . are, respectively, the Lagrange multipliers corresponding to the constraint in Ž23. and Ž24.. The optimum condition with respect to ¨ Ž t . is
l1i Ž t . y li2 Ž t . q u 1 Ž t . y u 2 Ž t . s 0
Ž 29 .
u 1 Ž t . G 0,
u 1Ž t . ¨ Ž t . s 0
Ž 30 .
u2Ž t. G 0
u 2 Ž t . Ž 1 y ¨ Ž t . . s 0,
Ž 31 .
l1i Ž t . s li2 Ž t .
when 0 - ¨ Ž t . - 1
Ž 32 .
l1i
when ¨ Ž t . s 1
Ž 33 .
when ¨ Ž t . s 0.
Ž 34 .
which implies that
Ž t. G
li2
Ž t.
l1i Ž t . F li2 Ž t . The transversality conditions are
l1i T Ž T . s 0
Ž 35 .
li2T Ž T . s 0,
Ž 36 .
in which i T indicates the stage emerging at the terminal point T. Furthermore, according to Tomiyama w8x, it holds at each switching time period that
l1i Ž t i . s l1iq1 Ž t i .
Ž i s 1, 2 .
Ž 37 .
li2 Ž t i . s liq1 Ž ti . 2
Ž i s 1, 2 .
Ž 38 .
J i Ž t i . s J iq1 Ž t i .
Ž i s 1, 2 . .
Ž 39 .
It is noted that a switch in the land-use pattern in the optimization might not happen during the planning period. For instance, when t 1 s T, only the stage 1 pattern appears through the planning period, and it holds that J 1 ŽT . ) J 2 ŽT .. Likewise, where t 2 s T, the land-use pattern switches from stage 1 to stage 2, but stage 3 never appears during the planning period, and it holds that J 2 ŽT . ) J 3 ŽT .. The adjoint equation is developed. For example, for i s 1 Žstage 1., we obtain
J1Ž t. aŽ t .
b
b
s L Ž t . F Ž n, 1 . q b L Ž t . F Ž n, 1 . y nz Ž t . 2
2
y2 r Ž 2 a Ž t . q d Ž t . y C 2 . ykn2 qaŽ t . y s Ž 1 q nq . eyr t
Ž 40 .
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J1Ž t. dŽ t .
b
b
s y L Ž t . F Ž n, 1 . q b L Ž t . F Ž n, 1 . y nz Ž t . y4 r d Ž t . Ž a Ž t . q d Ž t . y C . 1 y kn2 q Ž d Ž t . y C . y s Ž 1 q nq . eyr t . 2
Ž 41 .
The first term in Ž40. and Ž41. denotes the value of product per firm; the second, the external agglomeration effect due to an increase in the number of firms; the third, the consumption of n households; the fourth, the increase in the communication cost of all the firms due to the location of a new firm at aŽ t . Žor dŽ t .., which includes the external effects to existing firms; the fifth, the increase in the commuting cost of households due to the location of a new firm in the CBD Žor SBD.; the sixth, the increase in the opportunity cost of land due to a new firm’s location in a city. Integrating Ž26. and Ž27. from t to T and taking the transversality condition in Ž35. and Ž36. we obtain
l11 Ž t . s l12 Ž t . s
t1
Ht
t1
Ht
J 1 Ž tX . aŽ tX . J 1 Ž tX . d Ž tX .
dtX q
t2
Ht
J 2 Ž tX . aŽ tX .
1
dtX q
t2
Ht
J 2 Ž tX .
1
d Ž tX .
dtX q
T
Ht
2
dtX q
T
Ht
2
J 3 Ž tX . aŽ tX . J 3 Ž tX . d Ž tX .
dtX
Ž 42 .
dtX .
Ž 43 .
Considering Ž40. and Ž41., l11Ž t . and l12 Ž t . are the present value of the stream of the increase in the social surplus in a city when an additional firm locates in the CBD and SBD, respectively, at time t when the stage 1 land-use pattern is prevailing. Now we are ready to give an economic interpretation to the optimum conditions Ž32. through Ž34.. When the CBD and the SBD are developed simultaneously at a particular time period, ¨ Ž t . Ž0 - ¨ Ž t . - 1. is determined such that the present value of marginal social surplus is equal between the two centers. When only the CBD ŽSBD. is developed, i.e., ¨ Ž t . s 1 Ž ¨ Ž t . s 0., the marginal social surplus is larger at the CBD ŽSBD. than at the SBD ŽCBD.. We can analyze the optimal conditions for stages 2 and 3 by a similar procedure. 3.3. Optimal Location of a Subcenter An important parameter in the model is the location of a subcenter, i.e., C, the left-side fringe of the SBD. Its location affects the value of the optimized social surplus V U and, hence, the determination of where to establish a subcenter is an important urban policy. By the envelope theorem, it holds at an optimal location C that V U r C s 0. Using Ž13.
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the condition that V Ur C s 0 is expressed as where U
Ri Ž t . C
sy
GCi Ž t . C
y
GSi Ž t . C
y
ICi Ž t . C
y
ISi 1Ž t . C
y
s yDG i Ž t . y DI i Ž t . .
ISi 2Ž t . C
Ž 45 .
In Ž45., DG i Ž t .Žs Ž GCi Ž t .r C . q Ž GSi Ž t .r C .. represents a change in the total communication cost of firms when the location is moved outward by one unit distance, which is found to be positive for all i: DI i Ž t . Žs Ž ICi Ž t .r C . q Ž ISi Ž t .r C . q Ž ISi Ž t .r C .., a change in the total 1 2 commuting cost of households when C moves outward by one unit distance, which is found to be nonpositive for all i. Thus, the optimal location of a subcenter is such that the increase in the communication cost is just offset by the decrease in the commuting cost. 4. SIMULATION ANALYSIS The theoretical model in the previous section is too complicated to trace the properties of optimal solution. Thus, in this section, simulation analysis is carried out to analyze solution properties. Figure 2a indicates the process of urban development under the following parameter values: C s 30,
k s 3.0,
r s 0.05,
q s 2.0,
n s 1.0,
T s 10.
Throughout the planning period, both the CBD and the SBD are growing, i.e., 0 - ¨ Ž t . - 1, although the growth rate varies with time: in initial development, most additional new firms are allocated to the CBD, but the ratio of allocation to the SBD increases with time. It holds that bŽT . eŽT .. That is, only the stage 1 type of land use is realized during the planning period, whereby some vacant land is left between the CBD and the SBD. Figure 2b shows the different processes under the following alternative parameter values: C s 25, k s 3.0, r s 0.1, q s 3.0, n s 1.0, T s 10. In the beginning, ¨ Ž t . s 1 so that all of the new firms are allocated to the CBD and thereby a monocentric land-use pattern is realized. In the middle of the planning period, the SBD is also growing together with the CBD, and in the last half periods, almost all new firms are allocated to the SBD. The land-use pattern shifts from stage 1 to stage 2 in the last half period, and after that the residences of workers in the SBD are built in the right side of the SBD. Simulations were performed for some other parameter values, but we could not find any case where the stage 3 land-use pattern emerges.3 3 As shown in Section 5, a stage 3 land-use pattern emerged under the alternative specification of firm’s behavior.
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FIG. 2. Ža. Optimal solution and spatial development process Ž1.. Žb. Optimal solution and spatial development process Ž2..
Next we investigate the effect of parameter change on the possibility of subcenter formation; i.e., it is examined whether a subcenter is formed or not during the planning period. Figures 3a and 3b show the simulation results. In both parts of the figure, the location of C is measured along the horizontal axis, and simulations were performed for the various combina-
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tions of values of C and other parameters so as to specify the conditions for SBD formation. The curve in each figure separates the area into two parts Žhereafter referred to as ‘‘separation curve’’.: one is where the SBD is formed, whether its size is large or small, during the planning period, and the other is where no firm is allocated to the SBD during the planning period so that a city grows around a monocenter. We can abstract some observations from Figures 3a and 3b. 1. Other things being equal, the closer C is located to the city center, the more likely an SBD is to be formed. Since each firm carries on communication with all other firms, the average communication cost becomes higher with the distance between the CBD and the SBD. Thus, if C is rather far from the CBD, all firms will be located in the CBD to save the communication cost. 2. The higher the commuting cost k, the more likely it is that an SBD will be formed. This is because the average commuting distance is shortened by forming the SBD, hence the average commuting cost is lowered. 3. The lower the communication cost r , the more likely it is that an SBD will be formed since the average communication cost is hardly increased by SBD formation. Although the figures are not shown, we examined the effects of parameter q and n on the possibility of subcenter formation. The results are summarized as follows. 4. A larger residential lot size q is likely to promote the formation of the SBD since it has the same effect as a higher commuting cost.
FIG. 3. Ža. Communication cost Ž r . and subcenter formation. Žb. Commuting cost Ž k . and subcenter formation.
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273
5. When the size of a firm, n, is increased, it is likely that an SBD will be developed since the residential district is enlarged, whereby it has the same effect as a larger residential lot size. We finally examine how optimal SBD location changes in response to changes in parameters. In Fig. 4a the optimal value of the objective function is plotted against the location C, where the other parameters are held constant at the values for the case in Fig. 2b. The curve has a single peak around C s 13, the optimal location of an SBD. The curve becomes flat beyond C s 40, which implies that it is not optimal to form an SBD if
FIG. 4. Optimal location of the subcenter.
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C is located so far from the city center. It is also verified from Fig. 4b that, at optimal C, change in communication cost Ž DG . is just equal to the change in commuting cost Ž DI . by moving C to the right. Figure 4 indicates that if the location of a subcenter is appropriately determined, then multicentric land use brings about larger social benefit than a monocentric one. The effects of changes in parameter values on optimal subcenter location are summarized as follows. The optimal location of C moves further from a city center as k, n, or q increase. An interpretation is that when commuting cost is heightened due to an increase in k, n, or q, the average commuting cost is decreased by moving C to the right and thereby increasing the number of commuters from the left of an SBD. When r is small, moving C to the right is optimal because it increases the communication cost by a relatively small amount but significantly decrease the commuting cost by increasing the number of commuters from the left of an SBD. 5. ALTERNATIVE MODEL The previous model has assumed that every firm in a city benefits equally, regardless of location, from agglomeration economies. However, it might be more natural and realistic that firms within the same industrial district strongly affect each other, but the spillover of agglomeration economies between different districts is weak. In particular, it might be the case that the benefit from some types of social overhead capital Žroads, transportation systems, water supply., the lower price of materials due to the coordinated purchase, the lower repair costs due to better access to related industries, and so on, can be realized only within a certain area. In the alternative models, therefore, the extent of agglomeration economies is assumed to depend on the number of firms located in a particular industrial district rather than the total number of firms in a city; i.e., a production function for a firm is specified as b
Q Ž t . s Ž 2 a Ž t . . F Ž n, 1 .
for a CBD firm
and b
Q Ž t . s Ž d Ž t . y C . F Ž n, 1 .
for an SBD firm.
Another departure from the previous model is concerned with the communication behavior of a firm. In the previous model, every firm makes a certain number of trips to all the other firms while in the alternative model each firm makes a given number of trips to the main and local trading floors, respectively, where the main trading floor is at the city
MULTIPLE-CENTER FORMATION
275
center and local trading floors are at the city center and at C.4 That is, communication costs of a firm are specified as
t C Ž t , x . s Ž r˜ q g . x
for x g 0, a Ž t .
t S Ž t , x . s r˜ x q g Ž x y C .
for x g C, d Ž t . ,
where r˜ and g are the communication costs per unit distance per unit time period at the main trading floor and a local trading floor, respectively.5 Thus, in the alternative model the bid-rent of a firm is expressed in the form b
r fC Ž t , x . s Ž 2 a Ž t . . F Ž n, 1 . y w C Ž t . n y Ž r˜ q g . x b
r fS Ž t , x . s Ž d Ž t . y C . F Ž n, 1 . y w S Ž t . n y r˜ x y g Ž x y C . . Under this circumstance, the same problem as Ž13. was solved, similar optimum conditions were derived, and similar analyses were done. Figures 5a]5c summarize the dynamic land-use process under the alternative model for three different parameter values. In Fig. 5a, all the new firms are allocated in the CBD up to the seventh period, and after that, SBDs are exclusively developed: even at the terminal period, a stage 1 pattern of land use is realized. In Fig. 5b, during the middle time period, both districts simultaneously grow, but at later periods only the SBD does. At the eighth period, the land-use pattern shifts to stage 2 and continues until the terminal point. The case in Fig. 5c generates a stage 3 type of land use which could not be observed in the previous model.6 In this case, the CBD grows exclusively first, the SBD being exclusively developed next, and both districts grow simultaneously in the latter periods. We conduct a simulation similar to that in the previous section to clarify the condition for SBD formation and summarize the results in similar figures Žalthough the figures are not shown.. Some important observations and, in particular, differences from the previous model are summarized as follows. 1. Other things being equal, the further C is located from the city center, the more likely it is that an SBD will be formed in the alternative 4
So, the main trading floor coincides with the local trading floor of the CBD. This is the same specification as the one in Sasaki w6x. 6 Although it is not easy to explain rigorously why the stage 3 land-use pattern emerged not in the first model but in the alternative model, the following conjecture might be helpful: In the first model, the possibility of subcenter formation is solely determined by the trade-off between communication cost Žcentralizing factor. and commuting cost Ždecentralizing factor.. On the other hand, in the alternative model, localized agglomeration economies work as an additional centralizing factor, strengthening the tendency of firms’ concentration in the CBD so that the stage 3 land-use pattern emerges in a particular environment. 5
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FIG. 5. Ža. Optimal solution and spatial development process Ž1.. Žb. Optimal solution and spatial development process Ž2..
model. This result is contrary to that of the previous model. An interpretation of this is that when C is rather close to the city center, the savings of commuting and communication costs due to SBD formation are not large enough to offset the increased agglomeration economies due to the concentration of firms in the CBD.
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FIG. 5ŽC.. Optimal solution and spatial development process Ž3..
2. As b increases, a monocentric land-use pattern is likely to emerge in order to fully enjoy agglomeration economies. And where b exceeds a certain level, a subcenter is never formed, regardless of the location of C. 3. The separation curve for g and C is monotonic, but that of r and C has a parabolic shape. This implies that when the subcenter location is very far from a city center, the average communication cost of SBD firms at the main trading floor becomes high enough to offset the savings of communication cost at a local trading floor and the lowered commuting cost due to SBD formation, whereby a monocentric land use is realized. 4. The separation curves for k, q, and n have a flat part which was not observed in the previous model; i.e., there are threshold levels of n, q, and k to induce a subcenter formation. The simulated results concerning the optimal location of C are as follows. First, the optimal location of C moves further from the center with the increasing degree of agglomeration economies so as to fully enjoy the externality by more concentration in one district. Second, the impacts on the optimal C of n and q are the same as in the previous model: the optimal location of C moves further out with their increased values. However, the effect of increasing k is different between the two models. In the alternative model, the optimal location of C moves closer to a city center as commuting costs increase. We can interpret this result as follows. The share of CBD firms becomes smaller as k increases, and thus the
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required land space for CBD activities becomes smaller, and consequently the optimal location of C moves closer to a city center. Third, an increased communication cost at the main trading floor induces firms to locate in the CBD, and thus the optimal location of C moves further out to retain sufficient land space for CBD activities. On the other hand, increased communication cost at a local trading floor is favorable to SBD formation, and thereby the optimal location of C moves closer to a city center. The investigation of the alternative model so far suggests that variations in the specification of agglomeration economies and communication costs result in different dynamic patterns of land use and different optimal locations of a subcenter. This implies that the urban policy of establishing a subcenter is rather sensitive to firm behavior. Therefore, in order to make an appropriate urban policy, policymakers need to know the economic circumstances around the relevant agents in a city.
ACKNOWLEDGMENTS An earlier version of this paper was presented at the 6th Annual Conference of the Pacific Rim Council on Urban Development, Taipei, November 1]3, 1994, and at the Applied Regional Science Conference, Kobe, December 3]4, 1994. We thank Drs. Tatsuaki Kuroda, Yukio Itagaki, and the participants of both conferences for their useful comments. We are also grateful to the editor and two anonymous referees of this journal for their constructive comments on the first version. This research was supported by grant from the Ministry of Education, which is gratefully acknowledged.
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