Structural stability and evolution of urban systems

EUEVIER

Regional

Science and Urban Economics

27 (1997) 399-442

Structural stability and evolution of urban systems Masahisa

Fujita”‘“,

Tomoya

M~ri”‘~ .

“Institute of Economic Research, Kyoto University, Sakyo-ku, Kyoto, 606, Japan ‘Regional Science Program, University of Pennsylvania, 3718, Locust Walk, Philadelphia, PA 19104-6209, USA Received

31 October

1994; revised

14 February

1996

Abstract This paper proposes an evolutionary approach to urban system theory, which aims to explain the specific spatial configuration of an economy observed today as an outcome of the historical evolutionary process. To illustrate the approach, we consider a simple model of an urban system, and demonstrate that as the population of the economy increases gradually, new cities are created periodically as the result of the catastrophic bifurcation of the existing system, and that as the number of cities increases, the urban system will approach a highly regular central place system. 01997 Elsevier Science B.V. Keywords: JEL

Cities; Urban system; Evolution

classification:

R12;

F12;

014

von Thiinen’s zones, for all they may suggest about Chicago’s later hinterland shed little light on the city’s explosive growth during the 1930s. In just three or four years, a tiny village suddenly increased its population twenty fold, the value of its land grew by a factor of three thousand, and boosters began to speak of it as a future metropolis. . . . To understand these events we have to combine von Thiinen’s abstract geography with booster theories. (Cronon, 1991, p. 52).

*Corresponding

author: Fax +81-7.5-753-7198.

0166~0462/97/$17.00 0 PII SO1 66.0462(96)02

1997 Published 156-4

by Elsevier Science B.V. All rights reserved

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1. Introduction The main concern of this paper is on the methodological approach for the study of economic geography, rather than the proposal of a specific model. Triggered in part by recent events which increased regional integration in several parts of the world, we can observe a renewed interest in the study of economic geography by scholars in various fields of social sciences. Given that in most developed countries and many developing countries, economic activities are largely agglomerated in cities, and that economic linkages among cities (both within each country and beyond national borders) have been increasingly strengthened, the study of economic geography is becoming synonymous with that of urban systems. Although recent developments in urban economics have helped our understanding of economic agglomeration within individuul cities, no systematic approach exists to examine why today’s urban system has taken its present form.’ If there existed no common feature among the various urban systems, then of course, there would be little hope for developing a systematic method to explore such a question. The reality, however, is quite the contrary. In fact, over the past century, economic geographers have been tirelessly advocating their findings regarding the surprising regularity in the nature of urban systems observed throughout the world.’ In particular, their findings have culminated in the “central place theory” pioneered by Christaller (1933) and Losch (1940). Central place theory represents a fundamental insight into the functioning of economic geography. As a basic economic theory of urban systems, however, it leaves room for further development. The basic limitation of the existing central place theory, in our opinion, is not its failure to be expressed in a general equilibrium context of an economy. Rather, its basic limitation in words of Cronon (1991), (p. 282) is that “it (central place theory) is profoundly static and ahistorical.” That is, given the recent development in spatial economics, it would not be too difficult to formalize a general spatial equilibrium model that could yield a regular central place pattern (suggested by Christaller and Losch) as an equilibrium solution. The real difficulty is that such a general equilibrium model would simultaneously yield a continuum of equilibrium solutions having different spatial patterns (thus providing little insight for why such a regular central place pattern can be often observed in reality). Multiple solutions arise because any general equilibrium model capable of generating a systematic pattern of cities on a continuous location space must contain endogenous agglomeration forces as a central element of the model.3 By the very nature of ‘Although many economic models of urban systems have been developed (for a review, see Henderson (1987)). they are mostly aspatial. *For a systematic review of the study on urban systems (both empirical and conceptual theories), see, for example, Marshall (1989). ‘Here, it is implicitly assumed that all agents in the model are atomistic (i.e., of zero measure). If a model contains “large agents”, or it permits coalitions by large groups of individuals, then the degree of multiplicity of equilibria may be reduced.

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endogenous agglomeration forces, however, each city (created in the model) will generate its own “lock-in effect” in the location space; hence its exact location cannot be uniquely specified by the model. Furthermore, when such lock-in effects of many cities interact together, the model would yield a strong indeterminacy in the equilibrium spatial structure. As a way to overcome this difficulty, we propose an evolutionary approach to urban system theory which intends to explain the specific spatial configuration observed today as an outcome of the historical evolutionary process. Mathematically, our approach is essentially an application of bifurcation theory in nonlinear dynamics to the study of economic geography; and hence, it is in line with the recent developments in complexity theory. The focus of this paper, however, is neither in the detail of the mathematical methods of bifurcation analyses nor in the reality of the model. Rather, by using a simple model of an urban system, we aim to illustrate the usefulness of such an evolutionary approach to the study of economic geography. For the study of city formation and the evolution of urban systems, an interesting model must involve two opposing forces, i.e., agglomeration (or centripetal) forces and dispersion (or centrifugal) forces. Our model is built upon the recent work of Fujita and Krugman (1995), in which the agglomeration force is generated through the product variety in (nonland-intensive) manufactured consumption goods (M-goods), while the dispersion force is generated through the transport costs involved in the trade of M-goods and the agricultural good (A-good) between cities and their hinterlands. Fujita and Krugman have shown that von Thiinen’s Isolated State, in which all M-goods are produced in a single city, is in spatial equilibrium when M-goods are sufficiently differentiated from each other and/or when the population of the economy is not too large (and hence the agricultural hinterland of the city is not too large). They have also shown that (ceteris paribus) if the population exceeds a certain critical value (and hence the city’s hinterland becomes too large), then von Thiinen’s monocentric spatial system ceases to be in spatial equilibrium, suggesting the transformation of the single-city spatial system to a multi-city system. In this paper, we extend the monocentric spatial-economy model by Fujita and Krugman to a multi-city model; furthermore, we introduce the adjustment dynamics of the economy’s spatial structure, which enables us to explicitly study the dynamical process of new city formation as well as the resultant new spatial system. In this way, we examine how the spatial system of the economy evolves in the long-run as its population increases gradually. The plan of the paper is as follows. In Section 2, we explain our model informally, and provide an intuitive explanation for why the product variety in M-goods leads to the formation of cities and how new cities will be born periodically when the economy’s population gradually increases. The basic framework of our model is presented in Section 3. In Section 4, we consider the most primitive form of the spatial configuration of an economy, i.e., the monocentric system in which the production of all M-goods takes place. Using the

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concept of potential curves, we show that given parameters in a certain range, the monocentric system is in equilibrium when the population of the economy is sufficiently small, while it ceases to be in equilibrium when the population reaches a certain critical value. Next, in Section 5, we introduce the adjustment dynamics of the spatial system, and examine how the monocentric system will be transformed to a duocentric or tricentric system when the population exceeds the critical value. In particular, we conduct the structural stability analysis of the general three-city economy, and show that the transition of the monocentric spatial system to a duocentric system, or to a tricentric system is achieved through a catastrophic bifurcation of the existing system. In Section 6, we conduct a numerical analysis of the long-run evolutionary process of the urban system. It is shown that as the population increases gradually, new frontier cities are created periodically as a result of the catastrophic bifurcation of the existing system. Furthermore, as the number of cities increases, we show that the urban system will approach a highly regular central place system. Finally, in Section 7, we discuss possible future research directions.

2. Urban agglomeration

and the birth of new cities

Imagine an economy situated on an unbounded homogeneous plain. Suppose that at a given moment of time, the economy has a given population of homogeneous workers (= consumers). Each worker consumes the homogeneous A-good together with a large variety of differentiated M-goods. The A-good production is subject to Leontief technology using labor and land in a fixed proportion, while the production of each M-good exhibits scale economies using labor as its sole input. (In our model, land is used only as an input of A-good production). Due to the scale economies in product-specialization, each variety of M-goods is produced by a single firm that chooses its mill (f.o.b.) price monopolistically (in the sense of Chamberlin). Each worker is endowed with a unit of labor, and is free to choose any location and job (i.e., manufacturing work or agricultural work) in the economy. Likewise, each M-firm is free to choose its location in the economy. Suppose that at present the economy’s population size (i.e., the population size of the workers in the economy), N, is rather small, and hence the economy has a single city in which all M-goods (currently available in the economy) are produced. The location of the city was determined, in the past, rather arbitrarily by historical accidents. The A-good is produced in the agricultural hinterland (Ahinterland) surrounding the city. The reason for the concentration of the entire M-good production at the single city is due to the endogenous agglomeration force generated through the production and the consumption of differentiated M-goods. That is, if a large variety of M-goods is produced in the city, because of the transportation costs, this variety of goods can be purchased at lower prices there in

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comparison with more distant places. Thus, given a nominal wage, because of tastes for variety, the real income of workers rises in the city. This, in turn, induces more workers to migrate to the city. The resulting increase in the number of consumers ( = workers) creates a greater demand for M-goods in the city, which therefore leads more M-firms to locate there. This implies the availability of an even greater variety of M-goods from the city. Thus, the scale economies at the individual firm level are transformed into increasing returns at the city level through the circular causation of spatial agglomeration generated from a forward linkage (where the supply of a greater variety of M-goods in the city increases the real income of workers there) and a backward linkuge (where a greater number of consumers attracts a greater number of specialized firms to the city). When the population size of the economy is small, this positive feedback mechanism will lead to the agglomeration of the entire M-good production in the city. Over time, suppose the population size of the economy increases gradually. Then, the A-hinterland will also expand gradually to support the increasing population of the city, making the location deep inside the expanded hinterland increasingly attractive for M-firms, as there is a relatively uncontested local market (for M-goods consumed by A-workers) in the periphery while the increasing number of M-firms in the city makes the competition there more intense. When the population size N is not sufficiently large, the agglomeration force of the existing city will be stronger than the dispersion force; thus no new city can emerge, and hence the existing city will continue to absorb all newly created M-firms and their workers. However, M-firms and their workers will not continue to agglomerate into the single city indefinitely, for the expanding A-hinterland will become increasingly attractive for M-firms. This suggests that there exists a certain critical population, N, at which location (or locations) deep inside the large A-hinterland becomes equally attractive (for new M-firms) with the existing city. This implies that when the economy’s population reaches #, the existing monocentric spatial system becomes unstable. That is, the relocation of an arbitrarily small number of M-firms from the existing city to the A-hinterland location will trigger the positive feedback mechanism of spatial agglomeration, leading to the explosive growth of a new city there, while the reverse (i.e., negative) feedback mechanism of deagglomeration will work at the existing city. Due to the lock-in effects of the existing city, however, decline of the existing city stops when the two cities become the equal size. That is, at the critical population 15, the monocentric spatial system will be transformed to a (symmetric) duocentric system in the form of a catastrophic bifurcationP Likewise, as the economy’s population continues to increase, new frontier cities will be created periodically (in the expanding agricultural hinterland), while the old cities continue to exist due to their lock-in effects, leading to the outward 4As shown in Section 5, more than one new city may emerge at a time.

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expansion of the urban system. In the long-run, as will be shown later, the urban system of the economy will approach a highly regular central place system a la Christaller and Liisch.’ It is interesting to note that this process of the periodic formation of new frontier cities resembles the development process of the U.S. urban system during its westward expansion in the 19th century (including the explosive growth of new regional capitals such as St. Louis and Chicago in the mid-19th century), which has been eloquently described by Cronon (1991) and many other geographers and historiansP

3. The basic framework In order to formalize the ideas described in the previous section, we now introduce the basic assumption on the economy, and derive their immediate implications. The results in this section are to be used in specific contexts later. Consider a long-narrow spatial economy, the area of which is represented by a one-dimensional unbounded space, X = R. The quality of land is homogeneous and the density of land equals 1 everywhere. At a given moment of time, suppose the economy has a homogeneous workforce of size N. Each worker is endowed with a unit of labor, and is free to choose its location and job. The consumers of the economy consist of the workers and landlords. For simplicity, we assume that all landlords are attached to their land (like weeds), and

‘There are, of course, several major differences between the L&h’s central place system (for the single industry case, depicted in Figures 24-26 in L&h (1940)) and the urban system here. In particular, while in the former each central place is occupied by a single firm (producing a homogeneous product), in the latter each city consists of a large number of firms producing differentiated products. Hence, in the former there exists no trade between central places, while in the latter all cities trade their (differentiated) products to each other. In other words, the urban system here can be considered as the special case of Christaller’s system without hierarchy. Again, however, unlike Christaller’s system, there exists trade among all cities. In this respect, the present urban system is closer to that of Pred (1977) in which trades occur among the cities in the same level as well as different levels. In our model, trade occurs among all cities partly because each M-firm produces a differentiated good and partly because we assume a CES sub-utility function for M-goods (see Eq. (I) below).Since indifference curves of a CES function are always tangent to axes, each consumer turns out to consume a positive amount of every M-good (produced in the economy) regardless of its price level. Hence, in our model, no clear limit exist for the market of M-goods produced in each city. This difference, however, should not be overemphasized, for the market share of the M-goods produced in a city declines rapidly with the distance between each adjacent cities (see Fig. 7 below). ‘For historical studies of the U.S. urban system in the 19th century, in addition to Cronon (1991), see, for example, Abbott (1981), Borchert (1967) and Meyer (1983). Of course, to describe more fully the actual development process of the U.S. urban system, we must introduce many more realistic elements into our model such as the development of transportation systems/technologies and the consideration of many different groups of M-goods, as are discussed further in the conclusion.

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consume the entire revenue from land (i.e., land rent) at their location.’ Each consumer consumes a homogeneous A-good and a continuum of differentiated M-goods of size n where n is determined endogenously. All consumers have the same utility function,

where z, is the amount of the A-good consumed, Z,(W) is the consumption density of M-good indexed by WE [O, n], aA and CY~are positive constants representing respectively the expenditure share of A-good and that of M-goods such that ak + ohl = 1, and p E(0, 1) is a substitution parameter. That is, the preference between the A-good (z,) and the M-goods (zlvl(w)) is of the Cobb-Douglas type, while the CES-function, (J,” z~(u)’ dU)‘lP, represents the sub-utility for differentiated M-goods. Note that a smaller p means that consumers have a stronger preference for variety in M-goods. Given a nominal income Y, and the set of prices, pA and pM(.), a consumer maximizes her utility subject to the budget constraint, pAzA + J‘,” p,(w)z,(o)dw = Y, which yields the following demand functions and the indirect utility function: (2)

P&Y zh4(w)=j$)

for each o E [O, n],

n

I

0

cl =

Yc+Y;~p;Q*

P&J-’

(3)

du

(I:’pp,,(~)-~dw

QM/Y

>

where y=pl( 1 -p). Note from (3) that the demand for each M-good price elasticity, E = I /( 1 - p) = 1 + 7. Hence, E increases as p (or Notice also by (4) that under the same nominal income Y, the utility increases with the size, n, of the M-good varietyB That is, a larger n a higher real income of consumers.

(4) has the same 7) increases. of consumers

contributes

to

‘In the present context of a general equilibrium model with land, the issue of who owns the land (existing at each location) and where they reside must be specified explicitly. One way is to assume the public landownership model in which the total land rent of the economy is equally distributed among all workers in the economy. Another way, which is adopted here for simplicity, is to introduce a (rather fictitious) class of immobile consumers, called landlords, whose sole role is to consume the entire revenue from land at their location. (Like weeds, landlords perish when their land is not used). It is conjectured that either specification of landownership would not affect much the essential results of our study. ‘For example, if pM(o)=pM for all wE[O, n]. then (4) yields that u = YCX~CI~~p;UMnUM’ which r increases with n.

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The A-good production is subject to the Leontief technology requiring aA units of labor and one unit of land to produce one unit of output, which takes place in the hinterland of the cities. The production of each variety of M-goods uses labor as a sole input, and exhibits increasing returns to scale. Namely, the total labor input, L, for producing Q units of any M-good is given by

L =f + aMQ,

(5)

the fixed and marginal labor requirementsP where f and aM are respectively Because of scale economies in production, each M-good is assumed to be produced by a single firm. The transport cost of each good is assumed to take Samuelson’s iceberg form. Namely, if a unit of i-good (i = A or M) is shipped over the distance d, only a fraction e-s,d of the original unit of i-good actually arrives, where 7; is a positive constant. If a firm locates at xEX and produces an M-good, it chooses an f.o.b. price p,,,(n>, so as to maximize its profit at the Chamberlinian equilibrium. By the assumption of iceberg transport technology, the delivered price, p,(ylx), for consumers at location y E X of any M-product produced at location x is given by p,(ylx)

= &_$e’“‘y-X’.

(6)

Hence, given the equilibrium wage rate, W(x), at x, by the equality of the marginal revenue and marginal cost, the optimal f.o.b. price for the firm (at location X) is obtained as lo

which represents the familiar result that each monopolistic firm will charge its f.o.b. price at a markup over the marginal cost a,W(x). Thus, if Q is the output of the firm, its profit equals ti4

=P,WQ

- W4Cf

+ a,Q,

= w-‘W4
- ti~q,A.

(8)

Therefore, given any equilibrium configuration of the economy, if an M-firm operates at X, then by the zero-profit condition, its (equilibrium) output must be Q* = rflaM,

(9)

‘Since Q represents the output per unit offime, f also represents the fixed labor requirement per unit of time. In practice, f may include the labor services provided by backup personnel and core managers. ‘“Given the delivered price function (6), it can be readily verified that the price elasticity of the total demand for any M-good is independent of the spatial distribution of the demand, and equals the price elasticity, E= I/( 1 - p). of the individual demand function (3). Thus, by the equality of the marginal revenue and marginal cost, f,(x)( 1 -E-‘)=a,W(x), the f.o.b. price of the firm at x can be obtained as (7).

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which is a constant independent of location. Hence, by (5) (equilibrium) labor input of each firm is also a constant given by L* =f(l

and

+ y).

4. The monocentric

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(8),

the

(10)

system and the possible

emergence

of new cities

As noted before, in this paper we are interested in examining how the spatial system of the economy evolves as its population increases gradually over time.” In this section, we start with the most primitive spatial-configuration of an economy as introduced in The Isolated State by von Thiinen ( 1826): an economy with a single city in which the production of all M-goods takes place. As represented in Fig. 1, the city is assumed to locate at the origin of the locationcoordinate, and the A-hinterland extends from -1 to 1. For our study of the evolution of the urban system, the scrutiny of this monocentric spatial system is fundamentally important. For if the monocentric system cannot be in equilibrium under any population size N, then the economy can never have a city! Conversely, if the monocentric system is always in equilibrium under any N, then no evolution of the urban system will take place! Hence, it is essential to know the parameter range in which the monocentric system is in equilibrium for small N, while it is not in equilibrium for large N. In the first subsection, assuming that all M-firms locate in the city, we determine all unknowns of the monocentric spatial system. In the second subsection, by using the concept of potential curves, we examine the location equilibrium condition under which indeed no M-firms would desire to deviate from the existing city. In particular, we find the parameter-conditions for the most interesting case in which the monocentric configuration can be in equilibrium when the economy’s population size, N, is sufficiently small, while it

Al. 0

Fig. I. The monocentric

spatial system.

“In this paper, we take the population size of the economy as the parameter of our focus. Although the changes in other parameters, such as the production and transportation technologies and expenditure shares on the A-good and M-goods, can be readily incorporated into our model, the concrete studies of such changes in other parameters are left as future tasks.

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is not in equilibrium for large values of N. Then, focusing on this interesting parameter range, we indicate that when N reaches a critical size, this monocentric configuration becomes unstable; hence, to have a stable equilibrium configuration, the economy needs to create a new city or cities.12 4.1. The monocentric

spatial system

As noted above, suppose that the production of all M-goods takes place in the city located at x = 0 in Fig. 1, and the A-hinterland extends from - 1 to 1. (In the context of Fig. 1, hereafter, we use x and y interchangeably). Let the price of the A-good be normalized such that it is unity in the city. Then, since all excess A-good is to be transported to the city, at each location y EX in the A-hinterland, it must hold that p,(y) = emT+Jyi

(11)

By the zero-profit condition in A-good production, location y in the A-hinterland is given by R(y) = pA( y) - a,W(y)

then, the land rent at each

= e-TA’y’- a,W( y).

(12)

Since R(I)=0 at the agricultural boundary (A-boundary) location, 1, (12) implies that emrA’= a,W(l). Furthermore, by (6) and (7), at each location y, the delivered price of each M-good produced at the city equals p,(y~O)=(a,W(0)lp)e’“‘Y’. Substituting this equation for pM(o) in (3) and (11) for pA, and using the relation, e -‘A’ = a,W(f), the equilibrium wage rate at each y (in the A-hinterland) when the A-boundaly distance equals 1 can be obtained as follows: W(v) = a;’ e-%(r~+rP&I e(aMr~-uAT*)I_V/

(13)

which is uniquely determined by 1. This wage rate W(y) compensates for the price differences (of A-good and M-goods) between location y and the A-fringe location. Next, to determine the mass of firms, n, , in the city, we let NA be the size of A-workers and N, be that of M-workers in the city, then NA =2a,l and N, = N - 2a,l. Hence, using (lo), n, can be obtained as N - 2a,l n 1 =NIL*=f(l+y). I If we know 1, all unknowns will be determined uniquely by (11) through (14). The equilibrium value of 1 can be determined by equating the demand and supply for the A-good in the city. By (2), the excess supply of the A-good, per unit distance, ‘>For a further study of the monocentric section, see Fujita and Krugman (1995).

spatial system (including

the details of calculations

in this

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at each y in the A-hinterland, equals 1- (a,Y( y)lp,(y)) = 1 - cy~= LYE (where Y(y) =a,W( y) +R(y) =p,(y) by (12)). Thus, considering the consumption of the A-good in transportation, the total net supply of the A-good from the A-hinterland to the city is given by $!, oM eeTAlr’dy = 2oM7i’( 1 - em’“‘). While, by (2) and (13), the demand for the A-good in the city equals aAW(O = oAa;’ e-aM(rA+rM)’ (N - 2a,l). Hence, equality of the demand and supply requires that

aAa,‘(N-

2aAl)e-nMcrA+rM)t = 2a,r,‘(l

- e-‘“‘).

(15)

We can readily see that this equation uniquely determines the equilibrium boundary distance, 1” =E*(N), as a continuously increasing function of the population size, N, such that I*(O) = 0 and 1*(m) = ~0. Therefore, in the following discussion, we can treat the two parameters, 1 and N, rather interchangeably. Using 1*(N), all other unknowns can be determined uniquely as continuous functions of the population N. In particular, using (4) (7) and (13), the equilibrium utility level, uz,,,, (N), for the present monocentric system can be obtained as follows: uz,,,<,(N) = A;‘@&$M’Ya;O1~[N

- 2aAl*(N)]OIM’Ye~aAaM(7AtlH)I*(N), (16)

where A,=~~aA~~aM[f(l

wLo(N) &V

=A2

+ y)]UM’Y(aMlp)aM, which together with (15) yields cu, -P ___ l-p

___ ?4

+rA+rM

e

-T*/*(N)

l-e-Q*(N)

(17)

where A, is another positive constant. Hence, if (YeLp, then duz,,,,(N)ldN>O for all N. That is, if M-goods are highly differentiated so that CX~2 p, then the scale economies of population in the M-good consumption are so strong that they overwhelm the scale diseconomies of population in the A-good supply. Thus, u,$,,, always increases with N. Conversely, if q,,
~=~ 1-p

aM

rA

e-x’*(N)

7A + rM 1 _ e-?d*‘N)’

(18)

It can be readily shown that N increases as p becomes smaller; that is, the optimal population size is greater when M-goods are more differentiated. If we define i=I*(fi), then / also increases as p becomes smaller. Fig. 2 demonstrates the effect of N and p on the equilibrium utility level, uz,,, while other parameters are fixedat r1~=a,=0.5, r-=0.8, rM=1,aA=0.5,uM=1, andf=l.Wecanconfirm

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Fig. 2. The effect of N and p on IA:,,,,,,.

by the figure that if p > a,,,, = 0.5, the optimal population indeed decreases with p, while if ~ILX~ =OS, then u:,,,,(N) increases for all N. (Notice that $ is “optimal” for the workers of the economy (not necessarily optimal for landlords)). 4.2. The location equilibrium

and the possible

emergence

of new cities

So far, we have assumed a priori that all M-firms locate in the city. To claim, however, that this monocentric configuration is really in equilibrium, we must make sure that no firm would desire to deviate from the city. That is, it must be confirmed that all the existing M-firms earn zero-projt, and that no M-jirms could increase its projit by moving away from the city. To express this location equilibrium condition precisely, suppose an M-firm locates at any xEX, and let L&,(x) represent the total demand for the product of the firm from the entire economy (under the optimal f.o.b. price (7)). Then, setting Q=&,,(x) in (8), and using (9), the profit of the firm is given by ~(x)=a~f ’W(x)(L&,(x)-Q*), which implies that ~&x)SO as D,(x)SQ*. For convenience, let us define L+) =&,(x)/Q* Then, it follows that r(x)50 that

(19) as n(x)5

1. Hence, the location equilibrium

requires

n(x) 5 1 for all x E X, and

(20)

O(O) = 1,

(21)

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which together assure that no firm can earn positive profit at any location, while the existing firms in the city earn zero profit. We call a(x) the (market) potential function of the M-industry, which represents the relative profitability of each location for M-firms. To examine whether this location equilibrium condition is satisfied or not, for convenience we express the potential R (defined by (19)) generally as a function of two parameters, x and 1. That is, 0(x; I) represents the market potential at each xEX when the agricultural fringe distance is 1.13(Recall that all unknowns of the monocentric system can be expressed uniquely as functions of the fringe distance 1 (while the equilibrium 1* is uniquely determined as a function of N)). Through the calculations in Appendix D, the precise form of the potential function (associated with the present monocentric system) can be obtained as follows: I4

for x E [0, I],

(22)

where 7,1=( 1 + y)(a,~~ - LYST-)+ y~~.To examine the basic characteristics of this function, first, notice by (22) that ar each given x>O, the potential 0(.x; 1) is continuously increasing in 1. Next, by differentiating (22) with respect to x, we have n’(x;

I) = - 7$&x; I) + g(x; 1), where

(23)

I aArAyTM &;I)=

e(2yr,-?)x

1 _ e-TA'

Since L?(O; I)= 1 by definition, follows from (23) that n’(o+;r)=

-‘I+cyAyTM=

e Ix

-Td'd

(24)

Y.

if we define n’(O+;

I)=lim,&‘(x;

-{(l+p)(Yh.17M-LyA7A}l(1-p)’

1), then it

-6,

(25)

which represents the slope of the potential curve at the right edge of the city. Notice that this slope is independent of 1 (and hence IV). Furthermore, if we define the (upper) limit curve by

(26) then the limit curve is finite at each x, while its slope at the right edge of the city is given by

“Since I and N are in one-to-one correspondence through the relation, l= I*(N), in this paper we use the two expressions, LQ; I) and 0(x; N), interchangeably. “?ince the potential curve is symmetric with respect to x=0, and since the curve beyond the fringe distance 1 has no importance for our purposes, hereafter we focus on the behavior of the curve over LO, 11.

412

M. Fujita, T. Mori I Regional Science and Urban Economics l=Ir(O+)=

-

27 (1997)

6,

399-442

(27)

where 8 is the same constant defined in (25). Using (23) through readily conclude the following:

Lemma 4.1. In the context of the monocentric

(27), we can

spatial system;

1. Ifcx*r,>(l +p)(~~r~ (i.e., r]<(~~y~~), then n’(O,; I)>0 for any l>O; hence, the monocentric system can never be in equilibrium for any N>O. 2. Zf cy~r~s(l +p)oMrM (i.e., ~]?cI~uA~/T~)and aM2p (i.e., 2yr,~7--+77), then 0(x; l)< 1 for all xE(0, 11; hence, the monocentric system is always in equilibrium under and N>O.

Proof. (i) If ak~A>(l+p)a,r,, then by (25) we have that L?‘(O+; l)>O for all 1>O. Therefore, given J&O; l)= 1 by definition, for any given 1>O, the potential curve 0(x; 1) exceeds 1 in a vicinity of the city. As 1 and N are in one-to-one correspondence through the relation, 1=1*(N), it follows that the monocentric configuration can never be in equilibrium for any N >O. (ii) If 7~ a*y~‘yr~and 2~7,s rA + 11. then by differentiating (26) with respect to x, we can readily show that n’(x) < 0 for all x >O. Then, given n(O) = 1 and ax) > 0(x; 1) for all x >O (since function (22) is strictly increasing in l), we can conclude as (ii) above. Q.E.D. Notice that both (i) and (ii) are not interesting cases for our study of the evolution of an urban system. In case (i), the fact that the monocentric configuration can never be in equilibrium even for very small N, suggests that the economy could never develop cities. This can happen if the transport cost for the A-good (weighted by the expenditure share a*) is very high in comparison with that of M-goods. Conversely, in case (ii), because of the relatively low transport cost of the A-good (i.e., ‘~~7~I( 1 +p)q,,rM) and the relatively low price-elasticity of M-goods (i.e., psq,,), the centripetal force created by the agglomeration of M-good production in the city is so strong that no other city can emerge (however large N becomes). (Recall that if p(q,, then uz,,, continues to increase for all N). Therefore, for the purposes range defined by (a) myrrh < (1 + p)q,,~~

of this study, we always

focus on the parameter

(i.e., YJ> aAyrM), and

(b) arvl < P (i.e., 2~7, > rA + v),

>

(28)

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which is mutually exclusive from both (i) and (ii) in Lemma 4.1.15 Notice that (28) means that the A-good transport cost is not too high while the price elasticity of M-goods is not too low (i.e., M-goods are not too highly differentiated), Recall also from the discussion following (17) that condition (b) assures that the equilibrium utility level, uz,,,, (N), achieves the maximum at the finite optimal population, t?. For an illustration of the behavior of potential curves under assumption (28), Fig. 3 depicts the potential curves for the monocentric configuration under various N (i.e., various /=1*(N)), while all other parameters are fixed at cYA= (Ylvl= 0.5, T* = 0.8, rr,,, = 1, p = 0.75, (i.e., y = 3), aA = 0.5, uM=l,f=l,

(29)

which satisfy both conditions (a) and (b) in (28). We can see from the figure that when N (i.e., I) is small, the potential curve is below 1 for all x#O. As N (i.e., I) continues to increase, the potential curve gradually shifts upward everywhere except x=0; and, it eventually starts exceeding 1 in the periphery. Specifically, I) such that the there exists a critical population, fi (i.e., a critical fringe-distance, associated potential curve just hits 1 at a distance, 2, far from the city. We call 2 the characteristic distance of the M-good industry.

‘(t = 1.398)

I

Fig. 3. Potential

curves for the monocentric

‘?n (28), if we generalize that ~T~S( cannot be assured to be strictly positive.

system under various N.

I+ p)q.,~~and (Ye
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Generalizing the results of Fig. 3, it is not difficult characteristics of potential curves under (28).

to obtain following

Lemma 4.2. Given (28), the potential curves associated configuration have the following characteristics.

basic

with the monocentric

1. At each distance x>O, potential 0(.x; 1) increases continuously with 1, while its upper limit, 0(x), is finite. 2. Each potential curve is first convex, then concave. 3. At (the right edge of} the city, all potential curves have the same negative slope, - 8, given by (25). 4. There exists an urban-shadow distance x>O such that for any 1>O, 0(x; 1) is less than I for all O
I< f*

0(x; 1) < 1 for all x # 0, and

1 < f + 0(x; 1) < 1 in a neighborhood 6. Therefore, the monocentric 7. l?>A and i>f.

(31)

of x = 1.

system is in equilibrium

(32) ijf N&

In terms of Fig. 3, we can consider that the slope, 6, represents the strength of the lock-in effect of the existing city (located at x=0). The greater 8 is, the more disadvantageous it is for any M-firm (in the city) to relocate into the nearby hinterland of the city. In particular, the area, (0, x), represents the urban shadow of the existing city, an area into which, regardless of the economy’s population, M-firms would never choose to relocate. That is, no new city would be viable in this urban shadow.16 Notice, however, that the lock-in effect of the existing city above is a local phenomenon. That is, if an M-firm moves far away from the city (and captures the local demand by A-workers in a periphery), then the firm may be able to earn a higher profit. This becomes the reality when the population reaches the critical j6We can readily see by (25) that this lock-in effect of the existing city is stronger, as p, Q, or 7M is greater while as cr, or +-nis smaller.

not surprisingly,

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415

value fi (or, when it exceeds 13). Indeed, as suggested before, when N reaches 15, the monocentric system becomes unstable, so that the relocation of an arbitrarily small number of M-firms from the existing city to the location, i or -2, would trigger the positive feedback mechanism of spatial agglomeration there, resulting in a catastrophic transition of the urban system.” In order to make this idea precise, however, we must introduce the adjustment dynamics of the spatial system (in the general context with multiple cities), and define the stability of spatial equilibria precisely. This is to be achieved in the next section.

5. The initial bifurcation: system

from the monocentric

system to a multicentric

In the first subsection, we introduce the adjustment dynamics which specifies the behavior of the spatial system when it is not in equilibrium, and define the stability of the spatial equilibrium at each moment of time. In the second subsection, we introduce the transition rule which indicates that when the spatial system becomes unstable, it should move to an adjacent stable equilibrium. Finally, in the third subsection we use the dynamical framework discussed above to examine in detail how the monocentric spatial system (studied in the previous section) will transit to a duocentric or tricentric system when the economy’s population reaches the critical value 13. 5.1. The stability of spatial equilibria As noted before, in our model, new cities emerge when the existing spatial system of the economy becomes unstable. In defining the stability of spatial system at each point of time, we imagine that the economy’s population, N(t), increases rather slowly over the historica time t E iF8+ . At any given historical time t, the spatial population-distribution of the economy may experience a small random fluctuation from equilibrium. If the population distribution moves back to the original equilibrium, we call it (locally) stable; otherwise, it will move away towards another stable equilibrium, and the original equilibrium is called unstable. We assume that in comparison with the historical population process N(s), this adjustment process takes place so fast that it can be considered to take place over jictitious time denoted by vER+, while historical time t stops proceeding momentarily. Since unstable equilibria would rarely be observed in the real world, the stability of the spatial system at each moment of historical time is a natural requirement for an observable evolutionary path. To define the stability of the spatial system precisely, let us consider a general “By Lemma 4.2 (vii), this transition from the monocentric system to a multicentric only after the economy’s population N exceeds the optimal size fi (not surprisingly).

system occurs

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situation in which the equilibrium spatial system at a historical time t contains K(t) cities. (See Appendix A for the definition of spatial equilibria in the general situation with K cities). Let (1, 2, . . . , k, . . . , K(t)}=%(t) be the corresponding set of existing cities, (n,, x2, . . . , xk, . . . , xKcrj)=x be their location, and (N:(t), be the associated population-distribution of iv?(t), . . . ) N;(t), . . . , N&,)=N*(t) the economy.‘* Suppose, the economy experiences a small perturbation in population distribution such that the population of each existing city kEX(t) changes from N:(t) to N:(t) +ANk, while AK new cities, k= K(t) t 1, K(t)+ 2, . * . ) K(t)+AK, are created respectively at .x~(~)+,, x~(,)+*, . . . , x~(~,+~~; each new city kE{K(t)+ 1, K(t)+2, . . . , K(t) + AK}= AX has a population ANk >O. Here, AK is an arbitrary nonnegative integer. Let X’={X(t), AX}={l, 2, . . . , K(t), K(t)+l, . . . . K(t) + AK} be the new set of cities and $(x(t), Ax) = (x,, x1, . . . , ) be their location. Given this small perturbation of XK(r)+ I’ ’ ’ ’ 7 XK(r)+AK ‘K(r)* population distribution at historical time t, the economy is assumed to conduct the (fast) adjustment process over the fictitious time V, following the dynamics defined by I9 n’,(v) = cN,( v){u,( V) - U(v)}N(t) for k E X’

(33)

where, c is a positive constant, uk(v) is the temporary-equilibrium utility level of workers in each city kEX’, U*(V) is that of A-workers, and fi(v)=(N,(v)u,(v) + ~,JVk(v)uk(v))lN(t) is the average utility level, at each time Y (Here iVA(v) =N(t) EkNk(v) by definition). These temporary-equilibrium utility levels at each v are to be determined by using the temporary equilibrium conditions in which the population of each city k and that of A-workers are temporarily fixed at N,(V) and N,(V) respectively. (See Appendix C for the precise definition of temporary equilibria). We say that the spatial equilibrium (having the associated population distribution X*(t)E[WT”‘) at historical time t is (locally asymptotically) stable if the following condition is met: given any finite AK and &, there exists a small neighborhood of X’* =(N*(t), 0, 0, . . . , O)ER:(‘)fAK such that from any point in the neighborhood, the adjustment mechanism (33) will eventually move N’(V)= N’*(t) (i.e., (NI(y)* N,(V)? . . . 9 &(,)+AK (Y)) back to the original equilibrium lim _,N,(v)=N,(t) for kEX(t), and limV,_J~(v)=O for kEAX). Notice by (33) that instead of all migrants choosing the single city with the highest utility level, all cities having above-average utility levels receive positive net-inflows of migrants such that the larger a city’s population, the greater the magnitude of the migration towards it.‘” This reflects an implicit assumption that

‘“For simplicity,

we omit

N:(t) in the vector N*(t). Given X*(r), we have by definition that

N:(r) =N(t) - Z,qyt). ‘“Notice that Eq. (33) can be equivalently rewritten as follows: Ili,(v)=c X,IEx.UII) ~V~(v)(u~(v)y(v))N,(v) for kEYC’. “Since Rix(v) = -Z&(v) =cN,(v){u,(v) - ti(v)}N(t), it also follows that the A-worker population increases iff uA(v)>J(v).

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417

during the fast adjustment process, workers (and also firms) behave under “bounded rationality” (i.e., under imperfect information and foresight); the dynamical system (33) reflects the aggregate result of such behaviors. However, notice also by (33) that the result of the adjustment process (which takes place under the fixed historical time t) is not inconsistent with our previous assumption of costless mobility of workers. That is, when the adjustment process ends, all workers (in cities and in the agricultural area) enjoy the same utility level (which is the average utility level, U of the economy). In determining the stability of spatial equilibria and the possible emergence of new cities, the following result is useful. Lemma 5.1. Suppose that a spatial equilibrium (at a given moment of historical time) is stable when the formation of new city is not considered. Let O(e) be the associated potential curve. Then, it holds that: (i) no new city (initiated from a population close to zero) is viable at any location x such that O(x)< 1, (ii) hence, if the potential curve is less than I everywhere except the locations of existing cities, the spatial system is stable even if we consider the possible formation of new cities at any finite set of locations. An intuitive explanation of this lemma is as follows2 Notice, by the definition of the potential function in (19) or (A.7) that &X)-C 1=7r(x)
path

Suppose that we have specified exogenously the history of the population size (of the total labor force) by a continuous function, N(t), of time t E R + . Since our model does not contain any truly endogenous elements of dynamics (such as capital accumulation or durable urban infrastructure), it might be sufficient to replace N by N(t) in order to determine the equilibrium spatial system at each time t. As illustrated in Appendix F, however, there may possibly exist a continuum of spatial equilibria under the same N(t). Therefore, if we require only that the spatial system be in equilibrium at each moment of time, there exists quite a large number of possible paths for the spatial system of the economy. Hence, in order to single out a unique evolutionary path, we introduce three additional requirements: Rule 1 (stability): Except at possibly zero-measure points of time, the spatial system must be stable. “For

a formal proof of Lemma 5.1, see Appendix

2 of Fujita et al. (1995).

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Rule 2 (continuity): As long as possible, at each moment, the evolutionary path should follow its continuous extension under the same set of cities at the same locations. Rule 3 (transition rule): When the continuity of the path becomes impossible, it should move to an adjacent stable equilibrium by following a well-defined rule. The motivation of Rule 1 was already explained. Concerning Rule 2, because Rule 1 already requires that the spatial equilibrium at each time on the evolutionary path be stable, no new city is viable from an arbitrarily small perturbation of the present population distribution. Therefore, given the basic premise of our model that any new city must start from an arbitrarily small one (since no agent is assumed to be able to coordinate the behavior of a large group of firms and workers), Rule 2, which requires the continuity of the evolutionary path under the same set of cities at the same locations, is a natural outcome of Rule 1. To elaborate upon Rule 3, in general, let 0(x; t) be the potential curve associated with the equilibrium spatial system at time t which has the city set 3%(t). Define the critical location set at time t by Ax(r) = {x E X(0(x; t) = 1, x # xk for any k E X(t)},

(34)

which represents the set of locations (other than those of existing cities) at which the potential function 0(.x; t) takes the value of 1. Recall by the location equilibrium condition (20) or (A.8) that 0(,x; t) exceeds 1 nowhere. Therefore, considering Lemma 5.1(i) we can conclude that a new city calz possibly emerge only in the critical location set AX(t). In fact, as will be shown later, this set AX(t) contains at most two elements. That is, consider a critical time t at which Ax(t) is not empty. Then, if the equilibrium spatial system at time t is symmetric (with respect to some point in X), then Ax(t) will contain two elements; otherwise, it contains only one element. Therefore, without loss of generality, we can consider the following alternative ways of the creation of new cities at each critical time t: Transition Rule A: From each existing city, E firms (together with EL* workers) move to each point in Ax(t), where E is a given, arbitrarily small positive number. Transition Rule B: From each existing city, E firms (together with EL* workers) move to the (farthest) right point in Ax(t). In either way, once a new city or cities are initiated (as an e-deviation from the equilibrium population distribution N*(t)), then the adjustment process given by (33) sets off with the new set of cities (which proceed over the fictitious time vE[W+ while the historical time is fixed at t). When this adjustment process ends (possibly at v==), a new stable spatial system will emerge. 5.3. The structural

stability analysis of the tricentric system

Returning to the monocentric spatial system in Section 4, recall that when the population N reaches a critical value @, the associated potential curve hits 1 at +-I

M. Fujita, T. Mori I Regional Science and Urban Economics 27 (1997) 399-442

That is, if we define the first critical time, ?, , by the relation, terms of (34) we have that

Wi, ) = { - f, f}

419

N( f, ) =I’?, then in

(35)

It was suggested at the end of Section 4 that at this critical moment, the relocation of an arbitrarily small number of M-firms from the existing city to the location, 2 and/or -2, would trigger the positive feedback mechanism of spatial agglomeration there resulting in a catastrophic transition of the spatial system from the monocentric to a duocentric or tricentric one. Using the adjustment dynamics introduced in the previous subsections, here we can examine this transition process precisely. Notice that a monocentric system (resp., duocentric system) can be considered mathematically, as a special case of the general tricentric system in which two cities (resp., one city) have zero population. Therefore, in the rest of this subsection, we examine in detail how the stability of the general tricentric system changes as the population, N, gradually increases. This structural stability unalysis of the tricentric system will provide us with a clear understanding of how the initial monocentric system bifurcates to a duocentric or tricentric one. The study is conducted by combining theoretical analyses with numerical ones. In all numerical examples below, the values of all parameters (except N) are fixed at those in (29). Under this parameter set, we have that A = 4.36,

.40,x”= 1.1 0, x = 0.645, e = 1.90, and (36)

fi = 1.03, i = 0.455. Therefore, the locations fixed respectively at x,=O,x,=l.lO,andx_,=

of the possible three cities, city 1, city 2 and city - 2, are -X=

-1.10.

(37)

Notice in (36) that the critical population fl is more than three times that of the optimal population fi. In this context of the tricentric system, for each given value of the total population N, we set N(t) =N and 3C’= { 1, 2, - 2) in (33) and solve the dynamical system (33) under every possible initial population-distribution, N,(O)rO, N,(O)20 and N_,(O)sO, such that N,(O)+N,(O)+N_,(O)
‘*See Appendix 3 of Fujita and Mori (1994) for the numerical system (33) in the present context.

procedure

of running the dynamical

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Fig. 4. The stable subset of the tricentric

system.

directed.23 It follows that to examine the stability of the tricentric system, we can focus on the dynamics in this subset. The results are summarized in Fig. 5 (see Appendix E for the detail of the stability analyses of the tricentric system). In each phase diagram in Fig. 5, all stable- and unstable manifolds are depicted by solid lines, while broken lines depict representative trajectories; furthermore, each “e” represents a stable equilibrium, while each “0” an unstable equilibrium. In each diagram, the monocentric equilibrium with city i (i = 1, 2, and - 2) is denoted by Mi, which corresponds to the point M, in Fig. 4. The spatial configuration is duocentric along each side of M,M,M_,, and it is tricentric in the interior of M,M,M_,. We also note that each phase diagram is symmetric with respect to M, D,, since city 2 and city - 2 are located symmetrically with respect to city 1. Diagram (a) depicts the case of N < 1.8, where only monocentric configurations can be in stable equilibrium. Although there are symmetric duocentric equilibria at D,, D: and D,, and a tricentric equilibrium T,, all of them are unstable. Thus, starting from any point in M,M,M_, (except those points exactly on the duocentric and tricentric equilibria), the economy will eventually reach one of the monocentric equilibria. At N= 1.8, new duocentric equilibria D, and D, bifurcate from D,, making D, stable, while D, and D, inherit the instability of D, . Diagram (b) shows the case of 1.8
“If rA = rM =O, then the equation of this subset, M,M,M_,, is given by N, +Nz +N_, =a,N, which represents the clearance condition of the A-good market under the assumption of the costless mobility of all goods. In the present context of positive transport costs (rA, rM>O), however, this subset is convex to the origin, located farther from the origin than the N, + NL + N_, = Lu,N-plane.

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D4

MZ (a)

M-2

M2

D5

Ds

Do

oj

Dj

Fig. 5. Phase diagrams

M-2

M2

Ds

D4

D4

Dj

M-2

(f) 3.1
M

“N=4.36
Ds

M-2

(e) 2.97
M

(g)

D4

M2

(c) 2
(b) 1.8
(d) N=2.97

M2

M-2

D4

M2

N<1.8

421

q

(h) 5.1
dynamics

of the tricentric

system.

the other city with low transport cost. On the other hand, when the cities are close, given the iceberg transport technology, an increase in population of a city raises A-price significantly enough to dominate the marginal utility of the additional M-variety available in the city. Since the variety size and the population size are proportional, it can be concluded that when the cities are close, the net effect of a duocentric population increase tends to be negative. Thus, an asymmetric equilibrium tends to emerge at a smaller N when the cities are closer to each other. Returning to (c), a further increase in N does not change the phase diagram until N=2.97 at which a new tricentric equilibrium T, appears between M, and T,, which further bifurcates into T2 and T, as N increases. This process is shown in (d) and (e). A comparison of diagrams (e) and (f) shows that at N=3.1, T4 and Ti bifurcate from T,, and T, and Ti from T,. In particular, the emergence of T4 and T: makes T3 stable as shown in (f). Now N is large enough to have a stable tricentric equilibrium, TX. The relative city sizes, NZlN, , at tricentric equilibria, T, , T2 and T3 are respectively 4.99, 0.02 and 0.66. The dynamics is explained by diagram (f) until N=4.36. When N reaches the critical size fi=4.36, major changes in the phase diagram

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occur. The comparison of diagrams (f) and (g) shows that at this critical population size, the two unstable duocentric equilibria, D, and Di, and another unstable tricentric equilibrium, T,, merge into M,, unstable D, merges into M,, while unstable 0: into M_,, which make all three (previously stable) monocentric equilibria, M, ,M, and M_,, unstable. If the economy has been previously at M, in diagram (f), for example, then at this critical moment, the potential curve 0(x) just reaches 1 at x= ?f, which will trigger the formation of new cities at X and/or -2, as explained previously. Under transition rule A (resp., B), the spatial system will be transformed from M, to the stable tricentric equilibrium T3 (resp. duocentric equilibrium D,),in diagram (g). At N = 5.1, tricentric equilibria, T,, T, and T5,merge into duocentric equilibria, D,, D, and Di, respectively. For 5.1 CNC7.2, (h) describes the phase diagram. Finally, a comparison of diagrams (h) and (i) shows that at N=7.2, the manifold M,D, (resp., M,D,) merges into the manifold M,M2 (resp., M, D_2),with only T, remaining as the stable equilibrium. This indicates, for example, that if the spatial system previously transformed from M, to the duocentric equilibrium D, at N= fi= 4.36, it must be transformed again to a tricentric equilibrium T, at N=7.2.24 Notice that although diagram (i) seems to indicate that the tricentric configuration T, would remain in equilibrium for any N >7.2, this is true only if the creation of no additional new city were considered. Actually, as will be shown in the next section, when N reaches 7.47, the potential curve 0(x) hits 1 again at x = 22.11, suggesting the emergence of new cities there.

6. The long-run evolutionary

process: an example

Based on the preliminary analyses above, in this section we study the long-run evolutionary process of the urban system. To do so, we need to choose either one of the two transition rules, A and B. By the structural stability analyses above, however, we can readily infer that the two rules would bring about little difference in the equilibrium spatial configuration of the economy in the long-run. Therefore, in this section, we adopt the transition rule A, which keeps the equilibrium urban system always symmetric with respect to the first city located at x, =O. (In the subsequent discussion, each new city in the RHS (resp., LHS) of X, =0 is indexed by 2, 3, 4, . . . (resp., -2, -3, -4,. . . )). Furthermore, although a different set of parameter values may generate minor differences in the result, it turns out that as long as the assumption (28) is met, the long-run evolutionary process of the urban system is qualitatively the same under any parameter set. Therefore, in this section

24Notice that our discussion here is limited to the three-city D, to be transformed to a quadcentric configuration.

system. In general, it is also possible for

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also, we use the parameter set specified in (29) which is accompanied with those characteristic values listed in (36). In this context, Fig. 6 describes how the urban system evolves over time as N increases gradually. First, the diagram (a- 1) depicts the potential curve (right half) associated with the monocentric equilibrium under the initial population size, N = 3, while diagram (a-2) shows the associated land rent curve of the economy. Since L!(X)< 1 for all X#X, ~0, this monocentric equilibrium is stable.” However, as shown in diagram (b-l), when N reaches the critical value, 4.36~15, the potential curve hits 1 at distance 1.10=X (and at - l.lO=f); hence, the monocentric system becomes structurally unstable. Therefore, following transition rule A, we transfer an arbitrarily small size of M-workers from the existing city (at X, = 0) to each location, x2 = 1.10 and x_ z = - 1.10, and then set off the adjustment dynamics described by (33). Diagrams (c-l) and (c-2) describe the new (stable) spatial system which has emerged at the end of this adjustment dynamics. A comparison of the two land-rent curves in diagrams (b-2) and (c-2) indicates that a, large transformation of spatial system has occurred at this bifurcation point. In particular, since the land rent at the location of each city is roughly proportional to the city’s population,z6 diagram (c-2) indicates that the new frontier city 2 (and city -2) has a slightly larger population than the original city 1. (In fact, as will be seen later in Fig. 9(a), just after the bifurcation, we have that N, =0.74 and N,=N_,=0.97). Diagrams (d-l) and (d-2) describe the tricentric urban system at N = 6, which is halfway between the first bifurcation and the next bifurcation. Diagram (d-2) indicates that the two frontier cities have become much larger than city 1 at the center. Next, diagram. (e-l) shows that the second bifurcation is ready at N= 7.47. In fact, at this moment, the frontier potential curve has just hit 1 at x2 = 2.11 (and at x_~= - 2.11) and the urban system becomes structurally unstable again. As indicated by diagram (e-2), just before the second bifurcation each frontier city has a much larger population than the city at the center. (In fact, as can been seen from Fig. 9(a), at this moment, N?lN, =2.10/0.63 =3.08). Diagrams (f-l) and (f-2) describe the five-city urban system which has emerged just after the catastrophic bifurcation. In a similar manner, as the population, N(t), increases further, a pair of new frontier cities will emerge periodically as the result of catastrophic bifurcations of the existing spatial system. Diagrams (g-l) to (h-2) in Fig. 6 describe another

“Using

the fact that the RHS of (15) (i.e., the supply curve of the A-good) is steeper than the LHS

(the demand curve) at the equilibrium, always stable when the formation

it can be readily shown that the monocentric equilibrium

of no new city is considered.

monocentric equilibrium depicted in Fig. 6.

I

Hence,

by Lemma

5.1 (ii),

is the

(a- I ) is stable even if we consider the possible formation

of new cities. IhThis is true in the same land-rent diagram. Since the A-good price in the city at the farthest right is always normalized at I, the direct comparison of different land-rent diagrams is not meaningful.

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27 (1997) 399-442

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Fig. 6. (continued)

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example of such bifurcation, in which a 7-city system has been transformed into a 9-city system when the frontier potential curve on each side hits 1 at x5 =4.11 and x-j = - 4.11 respectively. These diagrams suggest that as the number of cities increases (in association with increasing N), the spatial system will approach a highly regular central place system in which sizes of all cities are roughly the same. (More precisely, as illustrated in diagram (d-2), the two frontier cities are always the largest (since no competing city exists in their forelocation), and the two cities next to them are the smallest (reflecting the strongest competition from the frontier cities), while the middle cities have almost identical sizes). In particular, we can see from diagram (h-l) that x2 -

x, =.C=

1.10,x,

-x2

= 1.01,x,

-xj

=x5 -x4

=xg -x5

= 1.00, (38)

which shows that the distance between each pair of adjacent cities is almost the same. Furthermore, if we measure the strength of the lock-in effect of each city k by n’(x,-)a’(~~+), then these diagrams indicate that it always remains at roughly the same value, i.e., fi’(x,~)n’(x,+)-24= 3.80 for each k, which represents a very sharp kink in the potential curve at the location of each city. Therefore, once a city is created at a location, it remains there forever, while no new cities emerge in its close vicinity. Fig. 7 depicts the market shure curve of M-goods produced by each city k (k= 1, 2, 3, 4, 5) in the 9-city equilibrium at N= 13.62, which corresponds to the spatial system represented by Fig. 6 (h-l) and (h-2). In the figure, at each consumption location x, the market share, MS(k, x), of M-goods produced in city k

Fig. 7. Market share curves of the ‘$-city equilibrium

at A’= 13.62.

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(in terms of delivered prices at x) can be obtained (by using (3), (6), (7), (A.l) and (A.3)) as follows:

where X=(-5, -4, - 3, - 2, 1, 2, 3, 4, 5) in the present context. The figure indicates that with the exception of city 4 (and city -4), each city imports less than 10% of its M-good consumption from other cities (mostly from the directly adjacent cities). Because the frontier city 5 (and -5) is the largest, having no competing city in its foreland, it imports the least percentage of M-goods from other cities. Conversely, because city 4 (and -4) is the smallest (due to being in the shadow of the frontier city 5), it imports the largest percentage of M-goods from other cities. Notice by these market share curves that the trade pattern of M-goods realized in our model is different from that of the classical central place theory a la Christaller and L&h. In the former, market share curves are bell-shaped, having no clear limit to the distance for trade; while, in the latter each city (or central place) has a clearly defined market area for its goods. This difference arises from the fact that, in our model, each city produces a group of M-goods which are differentiated from other cities’ M-goods, while in the classical central place theory the same order of cities produce the same group of 27 (nondifferentiated) goods. Next, Fig. 8 presents the wage curve in the 9-city equilibrium at N= 13.62. We can see that the wage rate achieves a local minimum at the location of each city, reflecting the real-wage effect of M-goods supplied from that city. Furthermore, the wage rate also achieves a (deep) local minimum at the boundary of each adjacent pair of regions. This is due to the fact that the boundary location has a good accessibility to the M-goods supplied from both cities while the A-good price curve is the lowest there; hence, the compensating wage, W(x), can be the lowest there. Furthermore, as the number of cities increases, all cities (except the two frontier ones) tend to have about the same wage rate. Fig. 9 provides a summary of the evolutionary process. Diagram (a) depicts the changes in the city size distribution along the evolutionary path. We can see that a new frontier city (more precisely, a pair of new frontier cities) is created periodically as a result of catastrophic bifurcation of the existing spatial system, and that the new frontier city is always the largest and grows fastest; but it

2’As is well known, the market-area structure assumed in the classical central place theory, called Economic Law of Market Areas (LMA), was outlined by Launhardt (1885) and rediscovered by Fetter (1924). In contrast, the market-area structure represented by Eq. (39) resembles the Law of Retail Gmvitation (LRG) which was proposed by Reilly (1931) as an empirical regularity. Hence, we can consider that our model (based on monopolistic competition with differentiated goods) provides a theoretical justification to the LRG.

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Fig. 8. The wage curve in the 9-city equilibrium

becomes the example, we frontier city, the next new

at N= 13.62.

smallest when the next new frontier city appears in its foreland. For can see by Diagram (a) that at N =7.47, city 3 emerges as the new and it remains the largest and grows fastest until N= 10.52 at which frontier city, city 4, appears as a result of a catastrophic bifurcation.

Fig. 9. Summary

measures

of the evolutionary

process.

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The diagram also indicates that in the long-run, as the number of cities increases, most cities (except the frontier city and its adjacent city) have approximately the same size. Next, Diagram (b) depicts the associated cyclical change in the equilibrium utility level of workers, in which the broken curve represents the equilibrium utility level of workers that would be achieved if the economy was forced to remain monocentric. It shows that the equilibrium utility level changes cyclically over time, where the periodic creation of new frontier cities enables the economy to overcome the diseconomies of population-increase, maintaining a constant equilibrium utility level in the long run. Diagram (c) depicts the cyclical change in the total real land rent (TRLR) defined by I’ TRLR = (40) ROW dy, I -/* which is the sum of land rents normalized by the equilibrium wage rate at each location. Since each income of W(y) at location y yields an equilibrium utility of u* for a worker there, R(y)/ W(y) represents the welfare measure of landlords at y in terms of u*-units. Hence, TRLR represents the aggregate welfare measure of all landlords in terms of u*-units. Not surprisingly, each TRLR-curve in diagram (c) (the solid line for the actual spatial system and the broken one for the base case of the monocentric economy) exhibits the opposite trend to the corresponding curve in diagram (b). Finally, if we define the overall social welfare of the economy by SW= N* + TRLRu*

(41)

then the solid line in diagram (d) shows that it is increasing almost proportionally with the population N. If we compare this curve with the broken line (for the monocentric case) in the same diagram, we can understand that “constant returns” in the overall performance of the economy in the long run is sustained by the periodic creation of new frontier cities.

7. Conclusion In this paper, we have proposed an evolutionary approach to urban system theory which aimed to explain the specific spatial configuration of an economy observed today as an outcome of the historical evolutionary process. Specifically, in the context of a simple model with a single (consumption good) manufacturing industry and the agricultural sector, we have examined how the spatial configuration of the economy evolves as its population increases gradually over time. That is, suppose at any given historic time there exists a set of cities at specific locations in the homogeneous location space. Agglomeration economies of these existing cities will exert strong lock-in effects in the location space, preventing the formation of new cities, while absorbing all newly created manufacturing firms

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and their workers. As the existing cities grow further, however, the agricultural hinterland will also expand outwards. This makes the transportation of the agricultural good to the cities more costly on one hand, while on the other hand, it makes the expanded large hinterland (with no city) more attractive as a possible location for new firms (because of the large uncontested local demand for consumption goods). When the latter dispersion force starts overwhelming the agglomeration forces of existing cities, new cities will be created at some locations deep inside the hinterland, causing a catastrophic transformation of the existing spatial system. In this way, new frontier cities will be created periodically, expanding the existing spatial system outwards. Furthermore, new frontier cities are the largest and fastest growing until the next new frontier cities emerge. In the long-run, as predicted by the study, the spatial system of the economy will approach a highly regular central place system a la Christaller and Liisch, with the past history of the evolutionary process clearly imprinted on it. It is interesting to note that, although the main concern of this paper has not been on the reality of the model, the evolutionary process of the spatial system predicted above has some resemblance to that of the U.S. urban system during its westward expansion in the 19th century. In particular, our definition of the stability of the spatial system and the associated mechanism of the emergence of new cities above are not inconsistent with the “booster theories” of urban and regional growth, advocated by the historians of the U.S. economic geography. William Cronon (1991, p. 34), for example, writes as follows: “In the speculators’ dreams lay the urban promise - and the urban imperative - of frontier settlement and investment. The search for the great western cities of the future drove nearly all nineteenth-century townsite speculation, and the accompanying rhetoric always inclined toward enthusiastic exaggeration and self-interested promotion”. This statement is consistent with our assumption that at each moment of time, spatial population-distribution of the economy is subjected to small random fluctuations everywhere, and hence every townsite can be the potential site of a new city. Although every townsite may be in the speculators’ dreams, the history of the frontier development in the U.S. during the 19th century indicates that only a relatively small number of townsites actually succeeded in growing as large cities. This is also consistent with the mechanism of the formation of new cities in our model, which suggests that only those towns (e.g., Chicago and St. Louis) that happened to be situated near the location where the potential curve just hit 1 actually succeeded in becoming real cities.28 It also may be noted that this paper’s model of the spatial economy is consistent with the conceptual foundation of Cronon (1991) which emphasizes the symbiotic relationship between cities and their surrounding countrysides. In reference to the work by Scott (1859), Cronon 280f course, for the growth of Chicago and St. Louis, in the mid 19th century, the consideration of the water-based transport network is also indispensable. For the introduction of transport networks into a monopolistic competition model of urban systems, see Fujita and Mori (1996).

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(1991, p. 40) notes that “ . . . , cities grow in tandem with the increasing size and density of regional population. Geography was secondary to population increase, channelling rather than creating the underlying demographic pressures that led cities to expand.” Our study in this paper, however, is rather preliminary, and a great deal of work is left for the future. First, although this paper focused on the evolutionary process of a spatial system associated with a gradual increase of the population size of the economy, it would be interesting to apply the same approach to the study of the impact of gradual changes in other parameters such as transport costs and expenditure shares of each group of consumption goods (in particular, a gradual decrease of the expenditure share on agricultural goods). For this purpose, the mathematical method proposed by Papageorgiou and Smith (1983) will become useful. Next, in order to make the preliminary model here more realistic, a variety of extensions is needed. In particular, the consideration of only one group of M-goods (which share the same technological/taste parameters) limits significantly the explanatory power of the model in this paper. That is, whenever a new city emerges, it produces the M-goods in the same group with existing cities. Hence, all cities essentially belong to the same type, and tend to have similar sizes. In order to generate hierarchical urban system a la Christaller, we need to introduce multiple groups of manufactured goods having different price-elasticities and different transport costs.29 Furthermore, although the agglomeration forces in the present model were created through product variety in consumption goods, variety in intermediate inputs are equally important in explaining the formation of specialized cities in reality. Finally, we must eventually introduce various elements of the first nature (such as natural harbors, rivers and mineral deposits) as well as the consideration of endogenous formation of various infrastructures (such as transport networks and housing). It is our hope to be able to report some of our progress along these lines in the near future.

8. Notational glossary X: N: n.

P: LY;: U: 7,:

aA: “For

the location space (=R); total population size of the economy; size of the available M-good variety in the economy; substitution parameter ( E (0, 1)); expenditure share of good i (i = A, M), q, + q, = 1; utility level of workers; transport rate of good i (i = A, M); labor input per output of the A-good; a preliminary

study on hierarchical

urban systems,

see Fujita et al. (1995).

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Y(y): WY>: R(Y): r(Y): ni: 1: P,(Y)1 Ph4(Y>I PM(Yb

D(x): f&x): fi: A (resp., f): -. X. x_: 8: &):

marginal labor input for the M-good production; fixed labor input for the M-good production; nominal income at location y; wage rate at location y; land rent at location y; potential profit of an M-firm at location y; size of the available M-good variety in city i; A-boundary distance of a monocentric system; A-good price at location y; f.o.b. price of an M-good at location y; the delivered price of an M-good at location y produced

at location

x; potential demand for the product of an M-industry at location X; the market potential of the M-industry at location x(= D,(x)lQ* where Q* is the zero profit output of an M-firm); the optimal population size of the monocentric system; critical population size (resp., critical A-boundary distance) of the monocentric system; characteristic distance of the M-industry; urban-shadow distance of the monocentric system; the slope of the potential curve, 0(x; 1), at the right of the city of a monocentric system; upper limit of the potential curves for all 1>O.

Acknowledgments An earlier version of this paper was presented at the Special Conference in honor of David Pines at Tel-Aviv University, May 29-30, 1994. The authors are grateful to Richard Arnott, Toni Horst, Paul Krugman, Eytan Sheshinski, Jacques Thisse and the three anonymous referees for valuable comments.

Appendix A

Spatial equilibria under a given total population size Given a fixed size, N, of the total work force of the economy, we say that a system is in equilibrium if (i) all markets are cleared for all goods, (ii) all workers in the economy attain the same utility level, and (iii) all existing firms earn zero profit at their present locations, and no jrm has an incentive to change its location. spatial

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433

In obtaining an equilibrium spatial system under given N, the first step is to specify the set of cities, { 1, 2, . . . , k, . . . , K}=X, which are supposed to exist at the equilibrium, and their location (x,, x2, . . . , xk, . . . , xK)=x. Given the specification of {X, x}, the next step is to assume a regional division of the economy, {ti,, 9e2, . . . , d,, . . . , dR}, in terms of A-good trade. Here, each regional area, 93,. = [l’, 1, ‘1 Cx, is supposed to represent a minimum interval of M in which A-good trade is balanced. For convenience, region 1 is assumed to locate at the furthest left, region 2 at the next furthest left, and so on, which implies that 1!<1:~12_<1:5 . . . %“
=P~(.Q(,)) e-TA’v-XI-(r)’for y E &r

(A.1)

where PA = (+,,) re p resents the A-good price at the central city k(r), which is an unknown to be determined later. That is, the A-good price curve is single-peaked in each region. Furthermore, when two regions are adjacent to each other, the A-price curve must be continuous at the boundary: p,(l:)=p,(l?‘) if 1: =lL” forr=l,2, . . . . R - 1. We normalize the A-good price at the central city of region R at unity: pA(xLcR))= 1. Next, let W, be the wage rate at city k. Then by (6), the f.o.b. price, pMk=pM(xk), of each M-good produced in city k is given by pMk = a,W,lp

fork E X.

Fig. 10. The spatial configuration

(A.2)

of a region

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Furthermore, if we let Nk denote the size of M-workers in each city k, and nk be the number of firms (i.e., the size of the product variety) in the manufacturing industry (M-industry) in city k, then, since each M-firm employs a constant labor force given by (lo), it must hold that Nk=rr,J(l + y) for kEX, and that Y,, the total income of each city k, is Yk= W,N, = W,n,f( 1 + y) for kEX. Suppose U* represents the equilibrium utility level which is to be enjoyed by all workers in the economy. Then, the supply wage, W(y), at each location y EX which is necessary for workers to achieve the equilibrium utility, u*, at y can be obtained by using (4), (6) and (A.2) as follows: W(y) = u*(YAaAaM”“(aMlp)aMp*(y)aA

2

n,W,

Y e -Y%IYpx~l

-aM/Y

k=l =

W,(y: u*,x,n,LJ,

p,(y))

f0rkE.K

where _W-(W, , W,, . . . , W,) and n= (rz,, n2, . . , nK). By definition, equilibrium it must hold in each city k that W, = W&x,: u*, 5, n, w, p*(x,)) fork E X

(A.3) then, in

(A.4)

Furthermore, at each location y, since the entire surplus generated from A-good production goes to the landlords there, the land rent R(y) can be obtained as R(Y) = m&p,(y)

- ~,W(Y), 01 for Y E X

64.5)

which implies that R(Z!)=O and R(lT)=O; R(l’)=O if l’>l:-’ for r=2, 3, . . . , R(I:)=Oif1:I’f’forr=l,2, . . . . R - 1. That is, if a border location is adjacent to a vacant land, the land rent there must be zero. It can also be readily verified that in equilibrium, no vacant land remains inside any region. Next, in each region r, the equality of demand and supply of the A-good must hold. To express this condition, notice from (2) that the excess supply of A-good per unit distance at each non-city location y equals 1 - (cx,,Y(y)lp,(y)) = 1 - cu, = CX~(because Y(y) =a,W(y) + R(y) =p,(y) by (AS)). Let X(r) be the set of cities in region r. Then, by (A.l) the demand for the A-good by workers in each city kEX(r) equals cu,YkIRA(xk)= a*N,W,/p,(x,). Since all excess supply of A-good is to be transported towards the city at xkCr) (recall Fig. lo), considering the consumption of A-good in transportation, the equality of the demand and supply of A-good at city k(r) can be expressed as

In addition, for the A-good flows depicted in Fig. 10 to be feasible in each region r, it must be confirmed that at any city in region r, the total supply of A-good to that city must not be less than the demand of A-good there.

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Since no vacant land remains inside each region and the labor input per unit agricultural land equals aA, the total labor force in agriculture is given by XIEs Hence, the economy-wide labor constraint is given by E’kt3YNk+aA a,(11 -1:). X:lE!z(l: -l’_)=N. Finally, we must make sure that all the existing M-firms earn zero-profit at their present locations, and that no existing M-firm could increase its profit by moving away from its present location. To express this location equilibrium condition precisely, suppose an M-firm locates at X, and let D(x) represent the total demand for the product of the firm from the entire economy. Then, setting Q = D(x) in (8) and using (9), the profit of the firm is given by r(x) = a, y -‘W(x)@(x) - Q*), which implies that r(x)50 as D(x)sQ*. For convenience, let us define L?(X)= D(x)/Q*

(A.7)

Then, it follows that r+)Z?O as a(x)% 1. Therefore, if n, is the equilibrium number of M-firms in each city k, the location equilibrium requires that L?(X)5 1 for all x,

(A.8)

and for each kEX, L!(x) = 1 if nk > 0,

(A.9)

which together assure that no firm can earn positive profit at any location, and existing firms earn just zero profit. Following Krugman (1993), we call L&c) the (market) potentialfunction of M-industry, which represents the relative profitability of each location for M-firms. For the precise forms of functions D(x) and a(x), see respectively (B.1) and (B.2) in Appendix B below. In sum, given the total work force, N, suppose we specify a priori the spatial structure of the economy by the number of cities, K, and their locations, x = (x,, x21 . . . XI,> . . . 7 x,), together with the number of regions, R, and their central-city locations, x’ = (xkC,), xkc2), . . . , x,+), . . , xkcRj,Then, the equilibrium spatial configuration of the economy is characterized by the set of unknowns, (u*, {W,>, {N,), {n,>, {Y,), 0’1, KI,

{Pi,

(A. 10)

which is to be determined by the equilibrium conditions noted above. As illustrated in Appendix F, in general, the equilibrium spatial configuration of the economy tends to be highly nonunique (due to the lock-in effect of cities). That is, under a fixed population size N, a different specification of city-location _wmay yield a different equilibrium. Furthermore, even under the same set of N and x, a different specification of central-city location, x’, may yield a different equilibrium. (Here, the notion of multiple equilibria excludes the obvious nonuniqueness due to the assumption of boundless homogeneous land. That is, all equilibria obtained by a parallel transfer or by a symmetric rotation of an equilibrium are considered the same).

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Appendix

B

Equations

of D(X) and O(X)

Given the location of an M-firm at any XCZX, using (3) (6) and (7), the total demand (including transport consumption) for the firm’s product (of the firm at x) can be obtained as follows:

D(x) =

c

W(x)-’ e-Y’M’Xk-X’

pct,W,N,

c

a,W(x)

kE.75

nh,y~MIvw

hE%

+C rea

I

&gr

w(x)-Y e-Yq"lY-~l

P%lPA(Y) 2

a,W(x)

n,,qY7e-yrMJy--xhJ

dy.

03.1)

In the RHS of (B.l) above, the first term represents the sum of the demand for the product from each city k, while the second term is the sum of the demand from the agricultural area in each region r. Substituting (9) and (B.l) into (A.7), the potential function for the M-industry can be obtained as follows: n(x> = (1 - p)/f

c

W(x>Y+ ’

a,W,N, 2

kE%

nhW,Y

e-y’M’xk-x’ ,-Yq.&-~J

hEl aMp*(y)

+C &St

2

e--yvd~-~l

nhw,Y e-Y'd-*hl

dy1b

hEX

03.2)

1

Each term in (B.2) can be interpreted as follows: W(x)‘+’ represents the labor-cost disadvantage of location x. Each a;, WkNk = aM Y, and crMpA ( y) = cu, Y(y) represents the market size (for M-goods) at city and at agricultural location y, respectively. Each e -y$,lxk-xt and e-,‘T~t?-xt represents the accessibility of production location x to the market at xk and at respectively. Finally, each Z,,, nhWLY e-Y’M’“r-*’ and 1 h~Si nhW,Y e-Y?IY-% Y*represent the intensity of competition in the M-good market at xk and at y, respectively.

Appendix

C

The temporary

equilibrium

at each adjustment

time v

In the temporary equilibrium at each adjustment time V, it is assumed that M-workers in each city cannot move to other cities or to the agricultural area, while A-workers can move freely within the agricultural area but cannot become

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M-workers in cities. Therefore, the utility level may differ among cities and/or the agricultural area. We say that a spatial configuration is in temporary equilibrium tf given the j?xed locations of the cities and the fixed population distribution of the economy, all M-workers in the same city achieve a common utility level, all A-workers attain a common utility level irrespective of their locations, each active firm earns zero profit, and the market clears for all goods. In the context of the adjustment process given by (33) in order to determine the temporal equilibrium at each v, we need the following changes in the equilibrium conditions in Appendix A (in addition to obvious notational changes): Condition (A.4) is to be changed as W, = Ws(x,, uk, SC,n, W, pA(xk)) for kEX’. Define W,(y)= Ws(y, uA, z, n, W, p,(y)>, and replace W(y) by W,(y) in (AS). Finally, drop condition (A.8), and when we apply condition (A.9), define R(x,)=D(x,)lQ* for each kEX’. Here, instead of (B.l), each D(x,) should be defined as follows:

DW=~

PffMYY

jEW'

a

w M

k

-YT&-xkl

Wi'e

2

,-mlr,--;,,I

nhw,Y

hEZ'

where a. time v.

represents

the set of regions

corresponding

to the regional

division

at

Appendix D

Derivation of O(x) for the monocentric In the context of the monocentric

system

system, Eq. (B.2) becomes

as

dy Since n,f( 1 + -y)=N, =N -2a,l, rewritten as

using (13) and (15), the above equation

(D.1) can be

where 7 is defined in (22). To rewrite (D.2) in a more convenient form, first we differentiate it with respect to x, and obtain n’(x) = -770(x) + g(x) for x>O, where g(x) is defined by (24). The solution of this differential equation can be expressed

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as n(x)=e-““{fl(O)+.f,,” which leads to (22).

Appendix

e”g(y)

dy}=e-‘X{l+_f,”

eqy g(y) dy} since @O)=l,

E

Local stability

analysis

of tricentric

equilibria

In the present context, the linearization A&=cN:(A~~-A.~)N

fork=

of the dynamical l,+-2,

system (33) is given by (E.1)

where c is a positive constant, AN,=N, -N:, Au, =u,(X)u* (k= 1, +2), and Afi=XJN~AuilN (j= 1, 22, and A). In particular, Au, (k= 1, +2, and A) can be approximated by a linear function of AX=(AA_,, AN,, AN>) when (IAJV/(is arbitrarily small. Rewriting (E.l) in a matrix form, we obtain Ati = @AN,

(E.2)

where AFV=(ANZ, AN,, AA_,)T, and @ represents the coefficient matrix. Since it can be shown that the differential Eq. (33) is continuously differentiable in some neighborhood of an equilibrium, it follows from the Grobman-Hartman Theorem that if @ has no eigenvalue with zero real part, the phase diagram of the dynamical system (33) is qualitatively the same as that of the linearized system (E.2) near the equilibrium. (For the Grobman-Hartman Theorem, see, for example, Perk0 (1991). Now, it is well known that if all the real parts of the eigenvalues of @ are negative, the equilibrium of the linearized system (E.2) is locally (asymptotically) stable. Thus, an equilibrium of the dynamical system (33) is locally (asymptotically) stable if all the real parts of @ are negative. At a symmetric tricentric equilibrium, @ has the following form.

a=($

where 5, follows:

K,

i

,u,

6

A,=[-/,&

{)

(E.3)

and 0 are real numbers.

&=f{t+/L+e+

(t+rf-/X)*+8K&

A,=f{~+p+f3-l/((B-~-p)2+8~fi.

The corresponding

eigenvectors

The eigenvalues

are

of @ are obtained

as

M. Fujita, T. Mori / Regional Science and Urban Economics 27 (1997) 399-442

439

Notice that v, is orthogonal to the N2 =N_,-plane which contains both u2 and uj. Thus, we can see that A, determines the stability between city 2 and city -2. Although the signs of the elements in @ are not obvious, the numerical examples show that all of them are negative. It follows that the sign of A, can be either positive or negative while As is always negative. In fact, we can see that the sign of h, determines the stability between city I and the two peripheral cities city 2 and city -2 while that of A, determines the stability between the cities and the agricultural area.

Appendix

F

An illustration

of the existence

of a continuum

of stable equilibria

Here, we demonstrate that under a fixed population size N, different spatial configurations can be in stable equilibrium, and that such equilibrium configurations may exist as a continuum. For simplicity, we show only the co-existence of symmetric stable configurations of monocentric, duocentric and tricentric equilibria when N=3.5. (For the cases of other population sizes, see Fujita and Mori ( 1995), (Appendix 5)). In Fig. 11, diagrams (a), (b) and (c) show respectively (the right half of) the potential curve of the monocentric equilibrium, those of duocentric equilibria, and those of tricentric equilibria at N= 3.5 under the different specifications of city-location 3. The dots on the horizontal axis represent the city locations in the corresponding equilibria. In diagram (a), we can see that at N = 3.5, the monocentric equilibrium is stable (i.e., the potential value is less than 1 everywhere except at the city). With this population size of the economy, if all the firms locate in a single city, the lock-in effect of the firms’ agglomeration in the city makes it impossible for a small number of firms to move to the A-hinterland and make a nonnegative profit. (Since the location space is homogeneous, any point on this space can be the location of the monocenter, and hence the location of the city is purely subject to historical accidents). If there exist already two cities of similar sizes, then each city may generate its own lock-in effect that is strong enough to sustain its existence. Diagram (b) shows that even though the monocentric equilibrium is stable (under N=3.5), it is also possible to have a continuum of stable duocentric equilibria in which the distance between two cities is not less than 0.68 (= 0.34 X 2). For example, when two cities locate respectively at x = 0.8 and - 0.8, as demonstrated in the diagram,

440

M. Fuji&, T. Mori I Regional Science and Urban Economics 27 (1997) 399-442

(a)

Potentlol

curve

of

the

monocentrlc

equilibrium

,. Mono

(b)

(c)

Potential

Potential

curves

curves

of

ot

duocentric

trlcentric

equllibrlc

equilibria

,

1

Fig.

11.Existence of a continuum of stable equilibria.

the associated potential curve is below 1 everywhere except at the cities’ locations, and hence the duocentric system is in a stable equilibrium. The same is true, for another example, when two cities locate respectively at x = 1.2 and - 1.2. (When

M. Fujita, T. Mori I Regional Science and Urban Economics 27 (1997) 399-442

441

two cities become too close, however, the location between the two cities becomes rather indifferent for firms and consumers in terms of the accessibility to the M-markets, while the A-good price becomes lower towards the mid point of the two cities; hence, the wage rate also becomes lower towards the mid point of the two cities. It follows that the agricultural area between the two cities offers a cost advantage to firms. In fact, our preliminary analysis has shown that no matter how many (discrete) cities exist, if the distance between the farthest two cities is smaller than 0.68, it is always profitable for firms to locate between cities, which suggests the formation of an industrial belt between the two cities). Finally, as demonstrated in diagram (c), a continuum of stable symmetric equilibria also exists under the same population size N = 3.5 when the distance between the nearest two cities is greater than 0.82. For example, when the central city locates at x =0 while two fringe cities locate respectively at x= 1.2 and - 1.2, then the potential curve is below 1 everywhere except at the cities’ locations, and hence the tricentric system is in stable equilibrium. (As in the case of the duocentric system explained above, when the distance between the nearest two cities is smaller than 0.34, the agricultural area between them becomes profitable for firms to locate in, which leads to the formation of an industrial belt. When the distance between the nearest two cities is between 0.34 and 0.76, three cities cannot exist together, or, the spatial system changes to either a monocentric or duocentric configuration. This happens because the cities are neither close enough to develop an industrial belt between them, nor far from each other so that each city is out of the urban shadow of another city).

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