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On the evolution of hierarchical urban systems1
European Economic Review 43 (1999) 209—251
On the evolution of hierarchical urban systems Masahisa Fujita *, Paul Krugman, Tomoya Mori Institute of Economic Research, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan Department of Economics, Massachusetts Institute of Technology, Cambridge, MA 02139-4309, USA Accepted 1 June 1998
Abstract The rapid urbanization trend of the world economy implies an increasing importance of cities as basic units of national and international trade. Given that the cities within an economy constitute some form of hierarchical structure, we model the endogenous formation of a hierarchical urban system. To overcome the multiplicity of equilibria, we propose an evolutionary approach which combines a general equilibrium model with an adjustment dynamics. It is demonstrated that as the economy’s population size increases gradually, the urban system self-organizes into a highly regular hierarchical system a la Christaller. 1999 Elsevier Science B.V. All rights reserved. JEL classification: R11; F12; D5 Keywords: Urban system; Self-organization; Evolution
1. Introduction Given that national boundaries no longer provide the most natural unit of economic analysis, what could replace them? The next citation from * Corresponding author. Tel.: #81 75 753 7122; fax: #81 75 753 7198; e-mail: fujita@kier. kyoto-u.ac.jp. The first version of the paper was presented at the 41st North American Meetings of Regional Science International, Niagara Falls, Ontario, Canada, 17—20 November, 1994. 0014-2921/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 4 - 2 9 2 1 ( 9 8 ) 0 0 0 6 6 - X
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¹he Economist (1995, 29 July, survey p. 18) provides a clue: The liberalization of world trade and the influence of regional trading groups such as NAFTA and the EU will not only reduce the powers of national governments, but also increase those of cities. This is because an open trading system will have the effect of making national economies converge, thus evening out the competitive advantages of countries, while leaving those of cities largely untouched. So in future the arenas in which companies will compete may be cities rather than countries. In fact, the most striking feature of today’s economic geography is the concentration of economic activities in cities. At present, in most developed countries and many developing countries, more than 70 per cent of the population reside in cities (United Nations, 1993). In addition, given the recent progress towards a borderless world economy, cities have been enhancing their importance as basic units of international economic systems (as well as of domestic systems). Given the increasing prominence of cities in national and world economies, this paper focuses on the question of how to formally analyse the formation and evolution of the spatial organization of an economy dominated by cities. In this paper, we propose an evolutionary approach to the study of economic geography, in which a general spatial equilibrium model is combined with an adjustment dynamics. Specifically, we consider a spatial economy in which the agglomeration force is generated through product variety in manufactured consumption goods, while the expansion of the agricultural hinterland induces the dispersion of the location of manufacturing production. It has been demonstrated that when the economy contains multiple groups of the manufactured goods (characterized by different degree of product differentiation and/or different transport costs), as the population size of the economy increases gradually, a Christaller-type hierarchical urban system emerges in a self-organizing manner. Although the main stream economics has almost entirely neglected the topics related to space, the study of cities has, in fact, a rich history in which three different traditions have been developed almost independently. The first tradition stems from the Alonso (1964)—Mills (1967)—Muth (1969) model of monocentric city. This is a rebirth of the von Thu¨nen (1826) model of land use, in which the ‘isolated town’ is replaced by the central business district (CBD), and farmers by commuters. Although it remains to this day the basis for an extensive theoretical and empirical literature, it has an important limitation. That is, when our question is not simply how land use is determined given a pre-existing town or CBD, but rather how land use is determined when the location of towns or CBDs are themselves endogenous, this approach offers little help.
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The second tradition was introduced a generation ago by Henderson (1974), which models the economy as an urban system-that is, as a collection of cities. This remains the workhorse approach into the actual distributions of sizes and types of urban areas (see in particular his own later work (Henderson, 1987, 1988)). It has, however, one disturbing aspect. Although Henderson-type models deal with an essentially spatial issue, they are themselves aspatial. In particular, they certainly have nothing to say about where cities themselves are located, relative to each other or anything else. For many purposes this may not be an important question; but if our intention is to bring space back into economics, explaining where cities are and why becomes a central concern. There is a third tradition, the focus of this paper, which at first sight seems to offer an answer to the question of how economies of scale and transport costs interact to produce a spatial economy: the central-place theory of Christaller (1933) and Lo¨sch (1940). The basic ideas of central place theory seem powerfully intuitive. Imagine a featureless plain, inhabited by an evenly spread population of farmers. Imagine also that there are some activities that serve the farmers, but which cannot be evenly spread because they are subject to economies of scale — manufacturing, administration, and so on. Then it seems obvious that the tradeoff between scale economies and transportation costs will lead to the emergence of a lattice of ‘central places’, each serving the surrounding farmers. Less obvious, but still intuitively persuasive once presented, are the refinements introduced by Christaller and Lo¨sch. Christaller argued, and produced evidence in support, that central places, or cities, form a hierarchy: the city at the top would produce the entire range of urban products and lower order cities successively fewer products. Lo¨sch pointed out that if a lattice is going to minimize transportation costs for a given density of central places, the market areas must be hexagonal. And thus every textbook on location theory contains a picture of an idealized central place system in which a hierarchy of central places occupy a set of nested hexagons. The original central place theory story was applied to towns serving as a rural market. But it is obvious that a similar story can be applied to business districts within a metropolitan area. Small neighbourhood shopping districts are scattered across the basins that surround larger districts with more specialized The basic idea of Henderson’s analysis is beautifully clear and simple: there is a tension between ‘external economies’ associated with geographic concentration of industry within a city, on the one hand, and diseconomies such as commuting costs associated with large cities on the other. The net effect of this tension is that the relationship between the size of a city and the utility of a representative resident is an inverted U-curve. With the added assumption of the forward-looking behaviour of large agents or ‘city corporations’, the equilibrium size of each city equals the optimal sized determined at the top of that U-curve. Since external economies tend to be specific to particular industries while diseconomies (such as commuting costs) tend to depend on the overall size of a city, this approach can explain the formation of a large variety of specialized cities having quite different sizes.
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stores, all eventually cantering on the downtown with its great department stores and high-end boutiques. Indeed, the hierarchical image is so natural that it is hard to avoid describing things that way. Unfortunately, as soon as one begins to think hard about central place theory one realizes that it does not quite hang together as an economic model. In economic modelling we try to show how a phenomenon emerges from the interaction of decisions by individual families or firms; the most satisfying models are those in which the emergent behaviour is most surprising given the ‘micro motives’ of the players. What is therefore deeply disappointing about central place theory is that it does not give any account along these lines. Lo¨sch showed that a hexagonal lattice is efficient; he did not show that it would tend to emerge out of any decentralized process. Christaller suggested the plausibility of a hierarchical structure; he did not give an account of how individual actions would produce such a hierarchy (or even sustain one once it had been somehow created). In this paper, we try to resurrect the central place theory as an economic model in space: it is a powerful idea too good for being left as an obscure theory. To do so, the first step is, obviously, to develop a general equilibrium model of micro-economics which could yield a hierarchical central place pattern as an equilibrium solution. We achieve this, in Section 2, by developing a general spatial-equilibrium model as a multi-city, multi-industry extension of the monopolistic competition model by Fujita and Krugman (1995). As in the model of single-city economy, agglomeration forces are created through the interaction of product variety in manufactured consumption-goods and scale economies in their production, while the demand of consumption goods by dispersed agricultural population induces the dispersion of industries. The tension between these forces for and against agglomeration underlies an equilibrium distribution of urban industries. However, such a general equilibrium model by itself fails to explain the emergence of surprisingly regular urban hierarchies that have been frequently observed in the real world. This is because the model also would yield a continuum of possible equilibria with different spatial configurations, thus providing little insight to why such a regular central place pattern is so often observed in actual economies. Multiple solutions arise because, by the very nature of spatial agglomeration forces, 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 equilibrium model. Furthermore, when such lock-in effects of many cities interact together, more indeterminacy to the equilibrium spatial structure will occur. Hence, in general, there may exist a continuum of equilibria that could be sustained once established. Therefore, to discuss the structure of multiple-city economies in general is to risk becoming lost in an endlessly complex taxonomy. How can we avoid such a risk? Here, we propose an evolutionary approach in which the analysis of
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multiple-city economies proceed in tandem with the analysis of the process of city formation. Only by telling some kind of story about the process of city formation, in fact, we can reduce the taxonomy to manageable size. While it is possible to imagine a discussion of city formation from various initial conditions — for example, from the hypothetical Flat Earth that was used elsewhere (e.g. Krugman, 1996, Chapter 8; Krugman and Venables, 1995) — we find it most natural to think of new cities as emerging as an economy which already has an urban structure grows over time. This means, however, that our discussion of city formation must, necessarily, take place in a model of a multiplecity system. In Fujita and Mori (1997), we experimented this approach in the context of an economy with a single manufacturing industry. In this paper, we extend it to a multi-industry context, and study the evolution of central place hierarchy. To formalize our evolutionary approach in Section 3, we imagine an economy with an existing city or cities, and allow its population to grow steadily. We then consider when and where new cities emerge, using a dynamic adjustment process for the location of urban industries and their workers. This evolutionary approach offers, we believe, a valuable perspective on how economies evolve in space. In particular, one gets an exciting new way of thinking about spatial economics in terms of the coevolution of two ‘landscapes’: the landscape defined by the current distribution of economic activity, and the implied landscape of ‘market potential’ which determines the future evolution of that distribution. To be more specific, imagine a long, narrow economy — effectively one dimensional — that stretches sufficiently far that we can disregard boundary conditions. Along this line lies land of homogeneous quality, with one unit of land per unit distance. Suppose that the initial landscape of the economy is the ‘isolated state’ of von Thu¨nen: all manufacturing industries located in a single city, while the city surrounded by agricultural area. Given this economic landscape, if we take a representative firm belonging to an industry, and draw the relative profitability (normalized to be unity at the existing city) of the firm at each location, then we obtain a curve, or the market potential of that industry. As shown in Section 4, this potential curve has the shape of a lain brace (refer to Fig. 4), with a cusp at the location of the city, and a wave in each side (extending to the edge of the cultivation). The cusp reflects, of course, the lock-in effects of the city. The potential curve is specific for each industry. As the economy’s population grows, the agricultural frontier expands, and the potential curve of each industry shifts upward (except at the city location where the potential of each industry is normalized to be one). As long as the potential curves of all industries are strictly less than 1 everywhere (except the city location), the monocentric configuration of this Von Thu¨nen landscape is a stable equilibrium. For the self-organization of a hierarchical urban system, the first key observation is that each industry has a different shape of critical potential curve. That is,
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given the monocentric configuration of the economy, as the economy’s population increases, the top of the ‘wing’ in each side of the market potential of any industry eventually hits 1 at a certain ‘critical distance’ when the population reaches a certain ‘critical size’. The critical distance and critical population systematically vary across industries. For example, if all industries have the same characteristics except the price-elasticity of their products, the industry supplying the products with the highest price-elasticity has the shortest critical distance (and the smallest critical population), the industry supplying the products with the second-highest price-elasticity has the second-shortest critical distance, and so on. Thus, industries can be ordered naturally by the length of their critical distance as in Christaller (1933). But, here, this critical distance is endogenous. The industry having the shortest critical-distance is called the level 1 industry, the industry with the next shortest critical-distance the level 2, and so on. When the economy’s population reaches the critical population of level 1 industry, the monocentric spatial configuration becomes unstable, leading to the formation of a pair of new cities at the location of critical-distance in each side; each new city contains, of course, level 1 industry only. Numerical simulations in Section 5 indicate that as the agricultural frontier shifts outward, several such ‘lowest-order’ cities form before it becomes profitable for the level 2 industry to start production at a new location. When it does become profitable for some firms of level 2 industry to start production at new location, it is not only to serve the agricultural population but also to get closer to the consumers in these lowest-order cities. And the demand-pull of the preexisting lowest-order cities generate cusps, at their locations, in the market potential curve for the level 2 industry. This makes it most likely (though not certain) that when firms of level 2 industry start production at new location at which this industry’s potential takes the maximum, they will do so at the location of a cusp that contains a large concentration of level 1 firms. Thus, we have a process in which population growth generates a pattern of several small cities containing only level 1 industry, followed by a larger city that contains both industries, followed by several more small cities, and so on. In general, when firms find it profitable to establish a new location for the production of higher-level goods, they tend to choose an existing lowerorder city, due to the demand-pull of the consumers in such cities; so when a higher-order city emerges, it normally does so via the ‘upgrading’ of an existing lower-order city. Repetition of the process eventually evolves a
This is for the case when the original critical-distances of the two industries are significantly different. When they are very close to each other, the location of the two industries tends to become identical.
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central-place hierarchy. We will show an example of a three-order central-place hierarchy. Although the main concern of this paper is on the methodological approach for the study of economic geography, it is interesting to note that the urbanization process similar to the one presented above did happen in history. For instance, Fig. 1 depicts the evolution of the U.S. urban system during the period from 1830 to 1870. Over this period, the U.S. population increased threefold from approximately 13 million to 39 million, which resulted in the major expansion of the agricultural area towards the Great West, well beyond Chicago and St. Louis. The figure shows the location of major cities (with population more than 30 thousand) in 1870. These cities in 1870 are classified into three orders such that New York, indicated by the largest circle, is the single first-order (or the highest-order) city with the population of more than 1300 thousand, those cities denoted by a middle-size circle represents the second-order cities with the population between 1300 thousand and 130 thousand, and those cities shown by the smallest circle belong to the third-order (or the lowest-order) cities with the population between 130 thousand and 30 thousand. This same figure also shows how this urban system in 1870 had evolved from that in 1830 by indicating the change of the size-order of each city from that in 1830. That is, the cities which were upgraded in one or more orders are denoted by shaded circles, those which were downgraded in one or more orders but were still higher than or equal to the third-order by , those which dropped out of the third-order by white circles, and those which stayed in the same order as that in 1830 by black circles. Fig. 1 reveals several interesting facts. First, not surprisingly, most old cities established well by 1830 (except the steel town, Pittsburgh) located along the northern part of Atlantic coast or navigable rivers, reflecting the importance of
The Great West is the 19th century naming of the area which represents the vast interior region of the U.S. in the west of Ohio River or Lake Michigan (Cronon, 1991). Fig. 1 has been created by adapting Figs. 5—8 in Borchert (1967). Based on the rank size distribution of 178 cities of the U.S. in 1960, Borchert defined four thresholds of population size (at which the slope of the rank size distribution changed noticeably). Then, for the years before 1960, each ith size-order threshold, ¹ , in year s (s"1830 or 1870) was defined by relationship, GQ ¹ "¹ (N /N ), where ¹ is the ith threshold population in 1960, and N (resp., N ) is the U.S. GQ G Q G Q population in year s (resp., 1960). In developing Fig. 1, for simplicity, we combined the original second-order and third-order cities, while the cities of the original fourth and fifth order were combined to form the new third order. Based on the same rule in footnote 5, the threshold population for each size-order in 1830 is given as follows. The first-order cities are those with population beyond 530 thousand (which actually did not exist), the second-order cities with population between 90 and 530 thousand, and the third-order cities with population between 15 and 90 thousand.
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Fig. 1. Evolution of the U.S. urban system from 1830 to 1870.
sea- and river transportation for trade with Europe as well as within the U.S. Second, in the Mid West and Great West, a large number of new third-order cities emerged by 1870. This is due to the expansion of the agricultural area towards the Great West resulting from the threefold increase of the U.S. population between 1830 and 1870. In order to provide farmers with ordinary consumption goods and farming tools, new small cities appeared by keeping an appropriate area in the Mid West and Great West. Third, several old thirdorder cities in the Mid West (such as St. Louis, Chicago, Cleveland and Detroit) upgraded themselves to second-order cities. These second-order cities were not only larger in population than third-order cities, but also played the role of regional center by supplying higher-order goods and services (such as
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business/trade services and sophisticated farming machines) to larger hinterlands. Fourth, a previously second-order city, New York, upgraded itself to the unparalleled first-order city of the U.S., providing the entire U.S. the highest-order goods and services (such as major financial services and national newspapers). Finally, we can also see in the figure that although most cities existed in 1830 still existed in 1870, several old frontier cities had disappeared (i.e., their population became smaller than the third-order threshold). In this way, as the population size of the U.S. increased rapidly (combined with the westward expansion of canal and railway network), a highly systematic hierarchical system of cities had been established in the U.S. by 1870. Given that no agent, in practice, had fully controlled the development process of spatial organization of the entire U.S., this example provides a clear case for the self-organization of a spatial economy towards a complex system.
2. The model The underlying structure of this paper’s model is closely related to that of the models in Krugman (1991), Fujita and Krugman (1995), and Fujita and Mori (1997). We will, therefore, be brief in describing its formal structure. We consider a boundless, one-dimensional location space of the economy, X, along which lies land of homogeneous quality, with one unit of land per unit distance. The economy has an agricultural sector (A-sector) and H types of manufacturing industries (M-industries). The A-sector provides a single, homogeneous good to consumers, while each M-industry supplies a continuum of differentiated goods to consumers. The agricultural production uses land and labor, and hence it is necessarily dispersed in space. In contrast, the production activity of M-industries takes place under an increasing returns technology, using labor only, and we assume that at each point of time, M-industries are concentrated in a finite number of locations, called ‘cities’. (The number and location of cities are determined endogenously through evolutionary process.) We also assume the ‘iceberg’ form of transport costs such that if a unit of the A-good [any variety of goods produced by h the industry] is shipped from
These second-order cities, of course, also provided nearby hinterlands most of the goods and services that were provided by third-order cities. In other words, they upgraded themselves to second-order cities by adding additional functions to those of third-order cities. The same note applies to the highest-order city. Hence, the entire system has, roughly speaking, a hierarchical structure in which a higher-order city supplies its own goods as well as most goods supplied in lower-order cities. This story is not unique. For another example, the urbanization process similar to this case took place in Europe in the 12th century as its population increased rapidly. For a comprehensive study of urban development in Europe and the U.S., see, for example, Bairoch (1985) and Marshall (1989).
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a location r3X to another location s3X, only a fraction, e\O Q\P [e\OF Q\P ], of the original unit actually arrives, while the rest melts away en route. At a given point of time, suppose the economy is endowed with a homogeneous labor force of N workers, who are free to choose both the location/city and the sector/industry in which they work. In this section, we assume a priori that the economy has K cities, each at location r (k"1, 2,2, K), and obtain the I conditions for the spatial equilibrium of the economy. Although each consumption and production activity takes place at a specific location, first we describe each type of activity without explicitly referring to the location. The consumers of the economy consist of N workers plus a class of landlords, who for simplicity are assumed to live on their land holdings — that is, land rents are consumed where they are accrued. Every consumer shares the same Cobb—Douglas utility tastes for the (1#H) groups of goods: & º"AI (CF)IF, k # kF"1, (2.1) F F where A is the consumption of the agricultural good (A-good), CF represents the composite index of the manufactured goods by industry h (MF-goods), and kF is a constant representing the expenditure share on MF-goods. When the industry h supplies a continuum of differentiated goods of size nF, CF is given by
LF MF (mF(i))MFdi , 0(oF(1, (2.2) where mF(i) represents the consumption of each variety i3[0, nF ]. In this specification, due to Spence (1976) and Dixit and Stiglitz (1977), the parameter oF represents the intensity of the preference for variety in MF-goods. When oF is close to 1, differentiated MF-goods are close to perfect substitutes; as oF decreases towards 0, the desire to consume a greater variety of MF-goods increases. If we set pF,1/(1!oF), then pF represents the elasticity of substitution between any two varieties. In the self-organization of urban hierarchy later, the composition of oF’s or pF’s for H industries plays the central role. Given income ½, and a set of prices, p for the A-good and pF(i) for each variety i of MF-goods, the budget constraint of a consumer is p A# LFpF(i)mF(i) di"½. Maximizing utility (2.1) subject to this budget constraint, F we can derive the following demand functions: CF"
A"k ½/p ,
(2.3)
mF(i)"kF½(pF(i))\NF(GF)NF\ for i3[0, nF], h"1, 2,2, H,
(2.4)
In a general equilibrium model with land, the question of where land rents are spent is a nuisance issue that unfortunately must be dealt with one way or another.
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where GF is the price index for MF-goods given by
GF"
LF \ NF\ pF(i)\NF\di .
(2.5)
which together yield the following indirect utility function:
(2.6) º" (kF)IF (k )I ½ (GF)\IF (p )\I . F F Next, the A good is produced under a constant-return technology, using labor and land with fixed proportions: c units of labor and one unit of land are required to produce one unit of output. In contrast, each M-good is produced under an increasing-return technolgy, using labor only. All products of the same h-industry have the same production technology such that the production of quantity qF(i) of any variety of MF-goods at any given location requires labor input l(i), given by lF(i)"FF#cFqF(i), h"1, 2,2, H,
(2.7)
where FF and cF are, respectively, the fixed and mariginal labor requirements. Through the appropriate normalization of the units of the output, qF(i), and the size, nF, of each M-industry, we can assume without the loss of generality that cF"oF and FF"kF/pF for h"1, 2,2, H.
(2.8)
Due to scale economies in specialized production of M-goods, any variety of M-goods is assumed to be produced by a single firm only that chooses its location and f.o.b. (mill) price in a nonstrategic manner in the spirit of Chamberlin. Hence, in each industry h, the ‘number’ of active firms equals the ‘number’ of varieties being produced in that industry. This also implies that if pF(r) is the f.o.b. price of an MF-variety produced at location r, due to the assumption of iceberg transport technology, its delivered price at each location s is given by pF(r, s)"pF(r)eOF Q\P .
(2.9)
Now, with explicit consideration of the spatial distribution of activity, let nF be I the number of MF-varieties being produced in city k, ¸F be the size of labor force I in industry h in city k, and define ¸ , ¸F. Let X be the agricultural area of I F I the economy, a subset of X in which land is actually used for agriculture. And, let p (r) be the A-good price at each r3X, w(r) be the wage rate at each r, and let w ,w(r ) be the wage rate in city k which is the same regardless of the sector for I I which an individual works. Furthermore, let pF,pF(r ) be the f.o.b. price of each I I MF-variety produced in city k.
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In this context, first, the price index for MF-goods at each location r can be obtained by using Eq. (2.5) as follows:
\ NF\ GF(r)" nF [pFeOF P\PI ]\NF\ , (2.10) I I I and we define GF(r ),GF for each city k. Next, suppose that a firm in the I I h-industry locates at r and chooses an f.o.b. price, pF(r), for its product. Then, since the total income in each city k equals ¸ w , and the income from I I agriculture per unit distance at each rOr equals p (r), the firm’s total sales can I be obtained by using Eq. (2.4) as follows: qF(r, pF(r))" kF¸ w [pF(r)eOF PI\P ]\NF(GF)NF\eOF PI\P I I I I
kFp (s)[pF(r)eOF Q\P ]\NF(GF(s))NF\eOF Q\P ds,
(2.11) 6 where on each term on the right-hand side above, the multiplication at the end by exp(qF"r !r") or exp(qF"s!r") reflects the consumption of the firm’s product I in transportation. Rewriting (2.11), we have #
qF(r, pF(r))"(pF(r))\NFuF(r),
(2.12)
where uF(r), kF¸ w e\N\OF PI\P (GF)NF\ I I I I
kFp (s)e\N\OF Q\P (GF(s))NF\ ds.
(2.13) 6 Since the firm is assumed to take all components in Eq. (2.13) as given, Eq. (2.12) implies that the price elasticity of the aggregate demand of a firm belonging to industry h equals pF. Thus, under the normalization in Eq. (2.8), the equality of the marginal revenue and marginal cost leads to #
pF(r)"w(r).
(2.14)
Under this pricing rule, the firm’s profit equals nF(r)"w(r)qF(r, w(r)) !w(r)(FF#cFqF(r, w(r))). Using Eq. (2.8), this leads to nF(r)"w(r)[qF(r, w(r))!kF]/pF.
(2.15)
Hence, if the firm actually operates at r, then by the zero-profit condition under free entry and exit, qF(r, w(r))"kF,
(2.16)
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must hold under which the firm’s labor requirement equals FF#cFkF"kF, using Eq. (2.8). Thus, the equilibrium output, qF*, and the equilibrium labor requirement, lF*, of any active firm in industry h equal a constant given by qF*"lF*"kF.
(2.17)
In each city k, then, the equilibrium number of firms in each industry h is given by nF"¸F/lF*"¸F/kF. I I I
(2.18)
Since Eq. (2.14) implies that pF"w for each city k, the price index Eq. (2.10) I I can be rewritten, using also Eq. (2.18), as
\ NF\ . (2.19) GF(r)" (¸F/kF)w\NF\e\NF\OF P\PI I I I Next, we characterize the equilibrium of labor markets. If we solve relation Eq. (2.16) for w(r) and denote the solution by wF(r), then it represents the wage rate that industry h can bear under zero-profit condition. Using Eqs. (2.12) and (2.13), this zero-profit wage rate for industry h at each location r can be obtained as
wF(r)" ¸ w e\NF\OF PI\P (GF)NF\ I I I I
#
p (s)e\NF\OF Q\P (GF(s))NF\ds
6
NF .
(2.20)
Setting ½"wF(r) in Eq. (2.6) and neglecting constants, the associated real wage rate is given by
uF(r)"wF(r) (GF(r))\IF (p (r))\I . F For convenience, for each city k we define
(2.21)
wF(r ),wF and uF(r ),uF. I I I I
(2.22)
Now, for the spatial equilibrium of the K-city economy, the five sets of conditions below must be satisfied. First, given that every worker is free to choose his job and location, all workers should enjoy the same equilibrium utility, or the same real income, which is denoted by u*. Under the zero-profit condition of all active firms, this implies that in each city k, uF"u* for every industry h such that ¸F'0, I I
(2.23)
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which in turn implies that there exists a common wage rate w in each city k such I that
wF"w "u* (GF)IF (p (r ))I if ¸F'0. (2.24) I I I I I F Second, for all agricultural workers (A-workers) to attain the same real income u*, the wage rate for A-workers, w (r), at each location r in the agricultural area X must be such that
w (r)"u* (GF(r))IF (p (r))I . (2.25) F Also at each city location r , for convenience, we define the potential value of I w (r ) by Eq. (2.25). I Third, the clearing of labor market requires ¸F#c "X ""N, (2.26) I I F where "X " represents the size of the agricultural area. Fourth, the A-good market must be cleared. In general, the full description of this condition is complex because the agricultural area of the economy, X , and the flow pattern of the A-good in X need to be determined endogenously. Hence, the clearing condition of the A-good market is provided later for each specific case. Here, it suffices to say that at each agricultural location r, the land rent is the value of output from each unit of land minus the wage rate bill for the c workers needed to farm that land: R(r)"p (r)!c w (r).
(2.27)
Finally, we have to see if the K-city configuration of the economy is really sustainable — so no firm in any industy should be able to attain a positive profit at any possible location. Since condition (2.23) assures that all workers in K cities enjoy the real wage rate u*, the configuration is sustainable if the zero-profit real wage rate, uF(r), of each industry h, defined by (2.21), exceeds u* nowhere. It actually turns out to be convenient to work with a monotonic transformation of real wages; we define the (market) potential function of each industry h as the ratio, [uF(r)]NF XF(r)" * F . [u ]N
(2.28)
The K-city configuration, then, is sustainable if and only if XF(r)41 for all r,
(2.29)
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for every industry h"1, 2,2, H. As explained in the next section, this potential function also allows us to determine the location of new cities. To rewrite the potential function in an explicit form, we use Eq. (2.25) to obtain that u*"w (r)+ (GF(r))\IF,(p (r))\I . Substitution of this relation and F Eq. (2.21) into Eq. (2.28) leads to XF(r)"[wF(r)]NF/[w (r)]NF. Hence, using Eq. (2.20), the potential function takes the following form:
XF(r)" ¸ w e\NF\OF PI\P (GF)NF\ I I I I
#
6
p (s)e\NF\OF Q\P (GF(s))NF\ ds [w (r)]\NF.
(2.30)
3. Adjustment dynamics Given the model in the previous section, our task is now to examine how the spatial configuration of the economy changes in the long-run as the economy’s population increases steadily over time. Since the population growth is exogenously specified over time, the task of tracing the change of equilibrium spatial configuration is essentially the problem of comparative statics in terms of population size. Here, however, we must face the difficult issue of multiple equilibria mentioned in the Introduction. To resolve this problem, we adopt a sort of evolutionary approach in which we consider ad hoc dynamics of labor migration among cities and the agricultural area, and trace a sequence of stable equilibria. Specifically, we imagine an economy in which there are two sources of change over time. First, there are ‘extrinsic’ dynamics arising from a steady process of population growth, which we regard as exogenous. Second, there are ‘intrinsic’ dynamics as workers move toward locations that offer higher real wages, and by so doing in turn alter the wages offered at different locations. In general, we should think of these sources of change as operating simultaneously. For the sake of simplicity, however, we will instead imagine that the extrinsic change in the economy moves very slowly compared with the intrinsic adjustment process. Or to be a bit less cryptic, we will think of this economy as evolving by a sort of two-step process. We start from an equilibrium spatial configuration, then increase the population a bit and hold it there; let the economy settle into a new equilibrium; then repeat. The dynamic process we use for the adjustment of urban population is mathematically analogous to the ‘replicator dynamics’ used in evolutionary game theory (see, for example, Freedman, 1991), except one difference: here, we must always consider the possible emergence of new cities at any possible set of locations in a continuous location space. To be specific, suppose that given the
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economy’s population N, we have obtained the associated spatial equilibrium, in which K cities exist respectively at r (k"1, 2,2, K). Choose any finite set of I hypothetical new cities, k"K#1, K#2,2, DK, at a set of locations, r , ,r . Including these DK new (potential) cities, suppose that the )> 2 )>D) economy experiences a small perturbation in population distribution such that the population of each city k changes from ¸ to ¸ #e , where ¸ "0 and I I I I e '0 for k"K#1,2,DK. Given this small perturbation of population, the I economy is then assumed to go through an adjustment process following the migration dynamics defined by, ¸ "¸ (u !uN )N, k"1, 2,2, K#DK. (3.1) I I I Here, u is the instantaneous-equilibrium real wage common to all workers in I city k, uN ,(¸ u # ¸ u )/N is the average real wage, and u is the real wage I I I common to all agricultural workers. By definition, ¸ "N! ¸ . These I I instantaneous real wages, u and u , are to be determined by using the I instanteneous equilibrium conditions in which population of each city is temporarily fixed. Here, for simplicity, we assume that the agricultural population moves instantaneously to equalize agricultural real wages, this giving u , the real wage common to all agricultural workers. Dynamics (3.1) indicates that labor migration is directed towards higher real-wage cities. There may seem to be an arbitrariness in the choice of which hypothetical new cities to include in the system. We need not agonize over this question, however, because there are never more than a few interesting potential new cities to consider. To identify these potential new cities, we turn to the market potential functions introduced in Section 2, and use the analysis developed there. Consider an urban system in which one or more cities already exist. If this system is in spatial equilibrium, then the real wages of all agricultural workers and of all manufacturing workers in existing cities must be the same. We can therefore assess the attractiveness of a location to each industry h using the same market potential function defined by Eq. (2.28). Let XF(r) be the market potential curve of each industry h in this economy. Now consider two possible small perturbations of the system. First, suppose that the populations of the existing cities are changed by small amounts, but no new cities are ‘seeded’. This perturbation will be eliminated if this system is stable; and let us suppose that it is (otherwise we would not observe this configuration of cities in the first place). Next, suppose that we also create some small urban populations at new locations. Will these new potential cities take hold and grow? Clearly, if they are at locations where
The instantaneous equilibrium conditions are the same as those in Section 2 except that the population size of each city and that of the farmers are temporarily fixed at each instant. Within the same city, workers are assumed to be free to choose the industry to work in, and hence the wage rate is the same for all workers there.
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the potential curve of every industry is below 1, they will not: the real wage they will offer will be less than that in agriculture or in existing cities. And so if the potential curve of every industry is strictly below 1 everywhere except the locations of existing cities, no new cities will emerge. But suppose that population growth has just pushed the potential curve of one industry up to the point where it lies slightly above 1 in some locations. Then a perturbation that puts some manufacturing workers in those locations will tend to grow, not die away, over time — and the existing urban system will become unstable. In short, as soon as the market potential curve of one industry humps itself above 1, we can expect new cities to emerge where it does so. In the following sections, we will demonstrate how this works. 4. The growth of the monocentric economy As noted in the Introduction, in this paper we think of multi-city systems as emerging from an imaginary history in which a growing economy add cities over time: the starting point for such a history must, obviously, be an economy with only one city, i.e., the monocentric economy of von Thu¨nen. In this section, first we determine the spatial equilibrium of the monocentric economy under a given population size of labor force, N. Then, assuming that N increases gradally (from near zero), we investigate when it eventually becomes unsustainable. As shown below, the study of the potential curves associated with the monocentric system reveals much about the characteristics of the evolutionary path of the urban system that will be manifested later. 4.1. The von Thu¨ nen economy Consider the ‘von Thu¨nen’ spatial configuration depicted in Fig. 2. In this figure, the production of all manufactures is assumed to take place in a single city; we relabel locations if necessary so as to make that urban site location 0. The agricultural area extends from !f to f, where f represents the (endogenous) agricultural frontier. The city exports manufactures to its agricultural hinterland and imports agricultural goods in return. In the context of the general spatial system in the previous section, this implies that K"1, and we denote the only city by k"1. For the moment, let us simply assume that this is the economy’s spatial structure, and use that assumption to determine equilibrium goods prices, factor prices, and land use. Each location in the agricultural hinterland produces one unit of agricultural output and exports the surplus over local production to the city. Let p ,p (0) be the price of the agricultural good at the city. Agricultural transport costs mean that the price farmers receive is lower the further they are from the city: p (r)"p e\O P .
(4.1)
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Fig. 2. The monocentric spatial structure.
Let R(r) and w (r) be the land rent and wage rate for agricultural workers at location r. Substitution of Eq. (4.1) into Eq. (2.27) yields that R(r)" p e\O P !c w (r). Land rents will be zero at the edge of the cultivation, so it must be true that p e\O D w ( f )" . c
(4.2)
Turning to manufacturing, to normalize prices we set the wage rate at the city equal 1. Then, by Eq. (2.14) pF(0)"w(0)"1 for h"1,2,2, H.
(4.3)
The price index GF(r) takes a very simple form because of the assumption that manufacturing occurs only at the central location. Let ¸F be the size of labor force in each industry h at the city. Then, using the definition of GF(r) by Eq. (2.19),
GF(r)"
¸F \ NF\ F eO P . kF
(4.4)
The presence of manufacturing trade costs means that this index is increasing as we move to locations further away from the center. We now have all the information we need to determine equilibrium — again, simply assuming for the moment that manufacturing is concentrated in the city. We can think of equilibrium as determined by two conditions: market clearing
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in the market for agricultural output, and equality of real wages between farmers and workers. Let us consider these in turn. First, income earned in the city is w(0)¸+"¸+ by Eq. (4.3) where ¸+, ¸F; a share k of that income will be spent on A, so food consumption F at the city will be D "k ¸+/p . Meanwhile, each rural location will spend a share k of its own income on food, leaving 1!k units to be shipped to the city. Only fraction e\O Q of the units shipped from location s arrive at the city, so the supply of food to the city will be S "2(1!k )De\O Q ds. But the urban labor force is the total labor force less the number of farmers, ¸+"N!2c f, so we can summarize the market-clearing condition for agriculture as a relationship between the number of farmers and the price of food: k (N!2c f ) . (4.5) p " 2(1!k )De\O Q ds Looking at real wages, Eq. (4.2) gives us the nominal wage received by the frontier farmer. His real wage is
u ( f )"w ( f ) GF( f )\IF p ( f )\I F
1 " (p )\I (GF)\IF e\ RFIFOF>\I O D, c F where GF,GF(0), while the real wage of a worker in the city is
u+" (GF)\IF (p )\I . F
(4.6)
(4.7)
Thus, equality of real wages requires that p "c e FIFOF>\I O D"c eI+ O+>O D,
(4.8)
where we have defined & kFqF k+, kF"1!k , and qN +, F . k+ F
(4.9)
Fig. 3 shows how the price of A and the size of the agricultural hinterland are simultaneously determined by the market-clearing condition (4.5) and the equal-real-wage condition (4.8). It is immediately apparent that an increase in the population will require, other things being equal, a rise in p to clear the market; as N increases the market-clearing curve will shift up, and in equilibrium the frontier will move out.
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Fig. 3. The determination of the equilibrium p and f.
To be specific, by eliminating p from Eqs. (4.5) and (4.8), we have
D (4.10) e\O Q ds, which determines f uniquely as an increasing function of N. Once the frontier distance f is determined, the labor size of each industry h can be obtained as k (N/c !2f )e\I+ ON +>O D"2(1!k )
kF kF ¸F" ¸+" (N!2c f ). k+ k+
(4.11)
Given relative prices and the allocation of labor between manufacturing and agriculture, everything else can be determined. 4.2. Market potential functions and the sustainability of the von Thu¨ nen structure So far, we have simply assumed that manufacturing production takes place exclusively at the city. To claim, however, that this monocentric configuration is an equilibrium, we must make sure that no firm has an incentive to ‘defect’ from the city. That is, the sustainablility condition (2.29), must hold for every industry. To check this sustainability condition, first we obtain the potential function of each industry in the present context of the monocentric economy. By successive substitutions into the general form of potential function (2.30), we can obtain the
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specific form of the potential function of each industry as follows (refer to Appendix A for derivation): For r50,
XF(r)"eNF I O \I+ON + P
1!k+ 1#k+ eNF\OFPtF(r, f )# e\NF\OFP , 2 2 (4.12)
where P e\O Q[1!e\NF\OFP\Q] ds tF(r, f )"1! . (4.13) De\O Q ds For the monocentric configuration to be sustainable, it must hold that for every industry h, Eq. (4.12) exceeds unity nowhere. To find the circumstances under which this condition is satisfied, we next examine the properties of function (4.12). First, it is obvious by Eqs. (4.12) and (4.13) that XF(0)"1 for every industry h. Second, the gradient of the potential function (4.12) in the neighbourhood of the city is easily found by differentiating to give (4.14)
dXF(0)/dr"pF[k q !k+(q+#oFqF)],
which indicates that all potential curves for the same industry have the same slope at the city. Having a negative gradient as we move away from the center (to the right, increasing r) is a necessary condition for the monocentric structure to be sustainable, and this condition must hold for all H industries. Third, since the function tF(r, f ) is increasing in f, so is XF(r). Hence, as population increases, the potential functions of all industries shift upwards (except at r "0), and it is helpful to look at their limiting values. Letting f"R in Eq. (4.13), we obtain
P tF(r, R)"e\O P#q e\NF\OFP e NF\OF\O Q ds. (4.15) Setting f"R in Eq. (4.12) and using Eq. (4.15), the limiting potential curve for each industry can be expressed as
XM F(r)"eNF MFO >OF\I+O >O+ P
P 1!k+ 1!k+ # q e\ NF\OF\O P\Qds 2 2
1#k+ # e\ NF\O+\O P . 2
(4.16)
Since the potential function of the monocentric economy is symmetric with respect to the city, here we focus on the right-hand side of the city.
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The limiting behaviour of these curves depends on the exponent of the term outside the curly brackets in Eq. (4.16). If the exponent is negative then XF(r) is strictly decreasing in r and hence it is less than 1 for all rO0, and if this holds for all industries, then the monocentric structure is always sustainable. But if the exponent is positive for any industry h, then this term goes to infinity as rPR; meanwhile, the term inside the curly brackets is always greater than (1!k+)/2. Hence, as rPR, X F(r) tends to infinity. This implies that for large enough values of f and r, the potential function XF(r) eventually exceeds 1, meaning that at some value of N the monocentric structure will break down. Pulling this together, we can summarize the results as in Table 1. First, in the right-hand column, if k q 'k+(q+#oFqF) for any h, then the potential curve XF(r) has a positive slope at the edge of the city, and hence the firms in industry h will certainly move out of the city, meaning that the monocentric system is never an equilibrium. Second, in the left-hand column, if k q (k+(q+#oFqF) and k+(q #q+)/(q #qF)5oF for every h, then the limiting potential curve of every industry is strictly decreasing at every r, meaning that the monocentric system is always an equilibrium however large N becomes. That is, the centripetal force created by the agglomeration of all industries at the city is so strong that no new city can emerge, however large N becomes. Therefore, since we are interested in the case in which the spatial system of the economy becomes increasingly complex as N increases gradually, hereafter we focus on the case corresponding to the middle column of Table 1. That is, we assume that, we can divide industries into two groups, h4hI and h5hI #1, for which the following conditions are satisfied: k q (k+(q+#oFqF) for every h, and
(4.17a)
k+(q #q+)/(q #qF)(oF for h"1,2,2, hI , while
(4.17b)
k+(q #q+)/(q #qF)5oF for h"hI #1,2, H,
where 14hI (H. Condition (4.17a) means that the A-good transport cost is not too high, while (4.17b) implies that there exist industries (h"1, 2,2, hI ) for which substitution parameters, oF are significantly large, and hence their products have high price elasticities. Given these two conditions, for each industry h5hI #1 such that k+(q #q+)/(q #qF)5oF, the associated potential curves are always below 1 everywhere under any value of N, as explained before. In contrast, for each industry h4hI such that k+(q #q+)/(q #qF)(oF, the behaviour of its potential curve is more interesting.
This can be shown essentially the same way as in Appendix B of Fujita and Krugman (1995).
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Table 1 The possibility of a monocentric equilibrium k q (k+(q+#oFqF), for every h
k+
q #q+ 5oF, q #qF
k q 'k+(q+#oFqF), for any h
k+
q #q+ (oF, q #qF
for every h
for at least one h
Always
For small N
Never
For a numerical example, let us set k "k+"0.5, q "0.8, qF"qN +"1 for all h, and c "0.5
(4.18)
and focusing on a specific industry, say h, and assume that oF"0.75 (i.e., pF"4). Then, since k q "0.4(0.5 (1#oHqH) always, condition (4.17a) is satisfied. And, since k+(q #q+)/(q #qF)"k+"0.5(0.75"oF, we have that h4hI . [Notice by Eq. (4.12) that the potential function of any one industry is indepent of the values of parameter o’s for the rest of industries.] Fig. 4 depicts the potential curve of this specific industry h under various values of N. We can see from the figure that when N is sufficiently small, the potential curve is below 1 for all rO0. As N continues to increase, the potential curve gradually shifts upward everywhere except at r"0; and it eventually exceeds 1 in the periphery. For each industry h4hI , the potential curves behave in essentially the same manner as in Fig. 4. In particular, for each industry h4hI , there exits a critical population, NI F, and a critical distance, rJ F, such that the critical potential curve associated with NI F just hits 1 at rJ F. Notice also by Eq. (4.14) that all potential curves belonging to the same industry, say h, have the same negative slope at the edge of the city, of which absolute value is given by hI F, pF[k+(q+#oFqF)!k q ]. Thus far, we have examined the behavior of potential curves separately for each industry. As we can see by Eqs. (4.12) and (4.13), however, the movements of the potential curves of all H industries are synchronized through the common parameter f which increases monotonically with the economy’s population, N. In particular, suppose that the initial value of N is sufficiently small so that the economy’s spatial structure is monocentric. Then, as N increases gradually with time, all industries’ potential curves will gradually move upward everywhere (except at r"0), as explained before. In this context, the first new city (or the first pair of new cities) will emerge when one industry’s potential curve reaches 1 in the periphery for the first time (among all industries). Therefore, the crucial
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Fig. 4. Potential curves for the monocentric system under various N.
question is: Which industry’s potential curve will reach 1 first? One might suspect that an industry will ‘spin out’ from the city sooner if it has either a high elasticity of substitution (meaning lower scale economies in equilibrium) or high transport costs (meaning that it tends to follow the agricultural frontier). And this is indeed the case. In Appendix B we show the following: Given assumptions (4.17a) and (4.17b) suppose further that k q 4k+q+. ¹hen, given any pair of industries, h and g, such that h4hI and g4hI , if it is the case that +oF'oE and qF5qE, or +oF5oE and qF'qE,,
(4.19)
then, in the context of a monocentric spatial system, we have that NI F(NI E, rJ F(rJ E,
and hJ F'hJ E.
(4.20)
In general, given a pair of industries, g and h, if g has either a lower o and a q that is no higher than h, or vice versa, we say that g is of higher order than h. Clearly, there is nothing that says that industries must be rankable by order: an industry could have a low elasticity of susbtitution and high transport costs, or vice versa. But if industries can be ranked, and we imagine gradually increasing the population of a monocentric economy, the potential curve of a lower-order industry will always hump itself above 1 sooner than that of a higher-order industry. Fig. 5 illustrates the impact of oF on the shape of the critical potential curve, XF(r; NI F). (The impact of qF is essentially the same.) Three hypothetical
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Fig. 5. Examples of critical potential curves for the monocentric system.
industries are shown, with o"0.90'o"0.75'o"0.20,
(4.21)
while all other parameters are fixed as follows: k "0.5, k"k"0.1, k"0.3, q "0.8, qF"1 for all h, and c "0.5,
(4.22)
Thus, industry 3 is of the highest order, and industry 1 is of the lowest order. We can readily confirm that these parameters satisfy both conditions in Eqs. (4.17a) and (4.17b) with hI "2. The associated critical potential curves can be depicted as in Fig. 5, which are accompanied with the following characteristic values: NI "0.88(NI "4.36, fI "0.40(fI "1.40, rJ "0.32(rJ "1.10, and hI "5.5'hI "1.90'hI "0.25.
(4.23)
Therefore, among the three industries, the critical potential curve of industry 1 hits 1 first at the nearest location, rJ "0.32, when the economy’s population reaches the smallest critical value, NI "0.88.
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5. Self-organization towards a hierarchical system The discussion in the previous section suggests that the growth of an economy containing many industries of different orders will naturally lead over time to the formation of a hierarchical urban system. Due to the complexity of the issue, we cannot offer any analytical confirmation. We can, however, illustrate the point with numerical simulations. In order to make our simulation analysis tractable, we choose parameters such that & (kF/oF)51, (5.1) F which ensures that the spatial structure of the economy remains monopolar; that is, the economy has a unique highest order city in which all groups of M-goods are produced. Lower order cities can still emerge, however; as we will show, they do so through a series of bifurcations associated with the birth of new cities, relocation and merger of existing cities, and changes of the industrial compositions of cities. Given that the long-run evolutionary process of the spatial system is qualitatively similar as long as the parameters satisfy conditions (4.17a), (4.17b) and (5.1), in the following we present the results of a representative simulation example. Suppose that the economy has three groups of manufactured goods, h "1, 2, 3. We fix parameters at those values specified in Eqs. (4.21) and (4.22), which yield the characteristic values in Eq. (4.23). These parameters satisfy both conditions (4.17a) and (4.17b) as well as the condition (5.1), where hI "2; and the critical potential curves of the three M-industries (associated with the monocentric configuration) can be depicted as in Fig. 5. Since Eqs. (4.21) and (4.22) imply that +o'o and q"q, and +o'o and q"q,, as noted before, Mindustry is of the highest-order, M -industry the second-order, and M -industry the third-order. In this context, we assume that the population size of the economy, N(t), increases gradually over the time, and examines the evolutionary process of the spatial system in the long-run. In the first subsection, we examine in detail what happens when N(t) reaches the first critical value, NI . In the second subsection, then, we examine the long-run evolutionary process of the spatial system.
We can show that if condition (5.1) holds, then when the economy is kept to be monocentric (i.e., of a single-city economy), then the equilibrium real wage of each worker keeps increasing for all N. Our preliminary simulation study suggests that if condition (5.1) is not met, then the initial monopolar economy will be transformed later into a multipolar system, through catastrophic bifurcations, in which new highest-order cities emerge. The analytical proof of this point and a more complete study of multipolar spatial system are left as future tasks.
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5.1. From one city to three Recall from the discussion at the end of the previous section that when N(t) is sufficiently small so that N(t)(NI "0.88, all the three potential curves associated with the monocentric system are strictly less than 1 everywhere outside the city at r"0. Therefore, the monocentric system is in stable equilibrium, with all three groups of M-goods being produced exclusively in the single city at r"0. However, when N(t) reaches the smallest critical value, NI "0.88, the potential curve of M-industry hits 1 at rJ "0.32 (refer to Fig. 4), resulting in the breakdown of the stability of the monocentric system. What happens then? A pair of new cities emerge at rJ and !rJ and, under our dynamics, the population of these new cities grows continuously from zero as N increases gradually beyond NI . Hence, the new cities emerge in the form of a continuous bifurcation. The reason for this continuous bifurcation is that the bifurcation only occurs for industry 1, and one industry (with a small expenditure share, k"0.1) alone does not create sufficient forward and backward linkages to lead to the discontinuous behaviour in our previous works (e.g. Krugman, 1991; Fujita and Mori, 1997). The new cities are completely specialized in industry 1, as can be checked by noting that after the bifurcation the potential curves of M- and M-industries are strictly below 1 at rO0. The size of the industry in each frontier city can grow only gradually as the local demand for that industry’s products increases gradually in association with the expansion of the agricultural frontier. The fact that we have a continuous bifurcation has two implications. First, as N(t) reaches NI , a pair of new cities must be created at rJ and !rJ . Second, during the immediate period after this bifurcation time, each frontier city is so small that it does not have an enough lock-in force to stay at the same location. If each frontier city remained at the same location, then as the agricultural area expands further, the potential curve would have positive gradient on the frontier side of the city, thus violating the location equilibrium condition. Therefore, in order for the spatial system to remain in stable equilibrium, each frontier city will move continuously outwards until it gains a sufficient lock-in force. 5.2. The long-run evolution Given the preliminary analysis above, now we present the long-run evolutionary process of the spatial system. The economy’s population size, N(t), is More precisely, at this moment the monocentric system becomes structurally unstable, implying that any small increase of N beyond NI makes the monocentric system unstable (in the usual sense). This continuity of the initial population size of these lowest-order cities seems to be of a qualitative nature, not of a numerical illusion. This is in contrast with the case of single-industry economy (and hence single-order urban system) studied by Fujita and Mori (1997), in which new cities always emerge as a result of discontinuous (i.e. catastrophic) bifurcations.
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assumed to increase gradually over time, starting with a small value less than NI . All other parameters are fixed at those values specified in (4.21) and (4.22). Fig. 6 summarizes the associated evolutionary process of the spatial system, in which all bifurcations that occur when N(t) increases from NI " 0.88 to 9.79 are presented, while Fig. 7 depicts the changing shapes of potential curves along the evolutionary path. Recall that there are three possible city types — the highest-order, with all three industries; a middle-order with industries 1 and 2; and the lowest-order, with industry 1. (Parameters are chosen for industry 3 such that it never leaves the original central city.) In Fig. 6, given each population N(t), the associated spatial structure of the economy is depicted on the horizontal line. (Symmetry in the spatial structure of the economy means we need only describe its right side.) For example, line 20 in Fig. 6 indicates that when N(t)"5.83, the equilibrium spatial system contains a unique highest-order city at r"0, a unique middleorder city (on the right half of the economy) at r"1.37, and four lowest-order cities respectively at r"0.49, 0.65, 0.98 and 1.70, while the right side A-fringe is at r"1.82. First, in terms of Figs. 6 and 7, we briefly summarize what have been explained in the previous subsection. As is illustrated in Fig. 7a, when N(t) is sufficiently small so that N(t)(NI "0.88, then all three potential curves associated with the monocentric system are strictly less than 1 everywhere outside the city at r"0. Therefore, the monocentric system is in stable equilibrium, with all three groups of M-goods being produced exclusively at the (highest-order) city at r"0. When N(t) reaches the smallest critical value, NI "0.88, as depicted in Fig. 7b, the potential curve of industry 1 hits 1 at r"$rJ "$0.32, resulting in the emergence of a pair of frontier cities there. Since these frontier cities emerge as a result of a continuous bifurcation, during the immediate period after their birth, each frontier city is so small that it does not have an enough lock-in effect to stay at the same location. This point is illustrated in Fig. 7c which indicates that even when N"0.95, the slope of the potential curve of industry 1 at the right edge of the frontier city is zero. Therefore, as indicated by the dotted curve between lines (1) and (2) in Fig. 6, in order for the spatial system to remain in stable spatial equilibrium, the frontier city must move continuously outward from r"0.32 to 0.41 as N(t) increases form 0.88 to 1.23. Only when N reaches 1.23, the frontier city has gathered a sufficient size so that its lock-in effect is strong enough to be able to remain at the same location (refer to Fig. 7d). Next, as N increases further (beyond 1.23) and hence as the frontier area keeps expanding, the frontier potential curve moves upward; it eventually hits 1 at r"0.47 when N(t)"1.37 (Fig. 7e). At this moment, as the result of a (small) catastrophic bifurcation, the existing frontier city at r"0.41 relocates to r"0.47 (line (3) in Fig. 6 and Fig. 7f ). At N(t)"1.61, as a result of a continuous bifurcation, a new frontier city starts growing at r"0.58, while the old frontier
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Fig. 6. The evolutionary process of the hierachical urban system.
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Fig. 7. Changes of potential curves along the evolutionary path.
M. Fujita et al. / European Economic Review 43 (1999) 209—251
Fig. 7. Continued.
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Fig. 7. Continued.
city remains there (line (4) in Fig. 6 and Fig. 7g and h). The old frontier city, however, is soon absorbed by the new frontier city and disappears when N reaches 1.84 (line (5) in Fig. 6). When N reaches 2.01, again as a result of a continuous bifurcation, a new frontier city starts growing at r"0.72, while the old frontier city remains there (line (6) in Fig. 6). Then, after a series of maneuvers, involving a creation of another lowest-order city at r"0.43 (line (7) in Fig. 6), and then through the merger of two existing cities located at r"0.43 and r"0.58 at the new location, r"0.49 ( line (8) in Fig. 6), the lowest-order city at r"0.58 eventually relocates to r"0.49. As can be seen in Fig. 6, this lowest-order city at r"0.49 continues to remain there hereafter. At N"2.42, as the result of another continuous bifurcation, the new frontier city starts growing at r"0.85 (line (9) in Fig. 6 and Fig. 7i), while at N"2.77 the old frontier city (at r"0.72) relocates to r"0.65 (line (10) in Fig. 6). (As can be seen in Fig. 6, this lowest-order city at r"0.65 continues to exist there hereafter.) Unlike the first frontier city, this second frontier city (on the right side of r"0) stays at the same location, although it emerges as a result of a continuous bifurcation. In general, when a new frontier city emerges next to a large (highest- or medium-order) city, it relocates continuously outward for a while. In contrast, if a new frontier city emerges next to a lowest-order city (which is always small), it grows very rapidly and remains at the same location from the beginning.
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Furthermore, at N"2.87, again as a result of a continuous bifurcation, the new frontier city starts growing at r"0.98 (line (11) in Fig. 6). Meanwhile, observe from the past diagrams in Fig. 7 that the potential curve of industry 2 has been steadily moving upward, with kinks at the location of existing lowest-order cities. In particular, as indicated by Fig. 7(k), at N"3.09, the potential curve of industry 2 first reaches 1 at the location of the frontier city (at r"0.98). Hence, a lowest-order frontier city becomes transformed into a middle-order city, producing both M-goods and M-goods. This arises because the industry 2 potential curve is bent and pulled upwards by the demand of M-goods by the workers in the lowest-order cities. In particular, since the frontier city has the largest population among all existing lowest-order cities, the potential curve of industry 2 is most shaply kinked and uplifted toward the frontier city (refer to Fig. 7k). Consequently, the potential curve of industry 2 reaches 1 first at the frontier city, resulting in the transformation of the frontier city to a middle-order city. Since this middle-order city has been created by adding a new industry to the second-order city, it naturally satisfies the hierarchical principle by Christaller (1933). In other words, this middleorder city has been created as a result of mutual entrainment of the location of industries 1 and 2. Fig. 8 demonstrates how strong is the effect of the demand-pull of existing third-order cities on the shape of the potential curves of higher-order industries. This figure is an enlargement of Fig. 7k, with addition of one new curve: the +X(r) of Mono,-curve represents the potential curve of industry 2 associated with the hypothetical monocentric urban system (in which all M-firms are forced to locate in the city at r"0) under the same population, N"3.09. We can see from this figure that the potential curve of industry 2 has been greatly pulled-up by the demand of workers in the existing lowest-order cities (in particular, by the demand of the large lowest-order city located at the frontier r"0.98), while the potential curve of industry 3 has been affected little. Since M-goods have a very low price elasticity (p"1.25), it is still much more profitable for industry 3’s firms to serve the demand of these small cities from the central city than to relocate there. The frontier city at r"0.98 was transformed to a middle-order city as the result of a continuous bifurcation. However, since it grows rapidly, soon it absorbs the nearby lowest-order city at r"0.85 (Fig. 7l). As N increases further (beyond 3.10), the frontier area expands and the new frontier city starts growing at r"1.15 when N reaches 3.43 (line (13) in Fig. 6). However, in comparison with the middle-order city at r"0.98, this new frontier city is too small to
Christaller’s hierarchical principle states that a higher-order central place shall provide its own order of consumption goods as well as all consumption goods which are provided at any lowerorder central places.
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Fig. 8. Potential curves of the 9-city equilibrium at N"3.09.
gather a sufficient lock-in effect there. Hence, as the frontier area further expands, the frontier city keeps relocating (discretely) until it settles down at r"1.37 (lines (13)—(17) in Fig. 6). Observe by Fig. 7m that as N increases further, the potential curve of industry 2 has been gradually moving upward again in the frontier area. In particular, as shown in Fig. 7n, at N"4.70, the potential curve of industry 2 reaches 1 again at the location of the frontier city (at r"1.37). At this moment, the existing middle-order city (at r"0.98) becomes less attractive for industry 2 (than the frontier city). Hence, as a consequence of a large catastrophic bifurcation, the entire industry 2 at r"0.98 moves to the frontier city, degrading the past middle-order city at r"0.98 to a lowest-order city, while upgrading the frontier city to a middle-order city (line (18) in Fig. 6, and Fig. 7n and o). This represents another instance of the mutual entraiment of the location of industries 1 and 2. We can see by lines (18)—(30) in Fig. 6 that after this change in city-order, the spatial structure in the area between the central city (at r"0) and the new middle-order city (at r"1.37) has been stabilized. In the frontier area, however, the spatial structure keeps changing (lines (19)—(25) in Fig. 6, and Fig. 7p—r). In particular, when N reaches 7.60, the frontier city at r"1.92 is upgraded to a middle-order city. Thus, now two middle-order cities exist (in the right side of r"0), with a lowest-order city locating in the middle (line (24) in Fig. 6. and Fig. 7r). After the formation of this middle-order city at r"1.92, the spatial configuration in the frontier area keeps changing as before (through the birth of new cities and relocation of existing cities), while the lowest-order city in the middle of the two middle-order cities moves to the center of them (lines (25)—(30) in Fig. 6 and Fig. 7s and t). The simulation ends when N reaches 9.79, with the emergence of a quite systematic hierarchical urban system a la Christaller (line (30) in Figs. 6 and 7t)).
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Fig. 9. Wage curve of the 15-city equilibrium at N"9.30.
Next, in Fig. 9, the real line depicts the equilibrium (nominal) wage curve when N"9.30, while the broken line represents the corresponding curve associated with the (forced) monocentric system under the same population size. We can see that although the sizes of all middle- and lowest-order cities are much smaller than the size of the central city, the shape of the wage curve is much affected by the presence of these relatively small lower-order cities. In particular, middle-order cities (located at r"1.37 and r"1.92) affect greatly the local wage rates, for each of them satisfies a large proportion of local demands of M-goods and M-goods in its vicinity. The last point above can be understood more clearly by Fig. 10 which depicts the trade patterns of M-goods in the context of 15-city equilibrium at N"9.30. (In this figure, each city number coincides with that in the bottom of Fig. 6.) Hence, city 1 is the highest order city, cities 5 and 7 are the middle-order, and the rest are the lowest order. For each industry h at each consumption location r, the market-share, MSF(r), of MF-goods produced in city k (in terms of I delivered price at r) can be obtained [by using Eq. (2.4) in the present context] as follows:
MSF(r)"¸Fw\NF\e\NF\OF P\PI ¸Fw\NF\e\NF\OF P\PH . I I I H H H
(5.2)
In terms of the city-index, k"1, 2,2, 7, indicated at the bottom of Fig. 6, when N"9.30, for example, the population of each city is as follows: N "6.42, N "0.00691, N "0.0722, N "0.0478, N "0.167, N "0.0007673, N "0.138.
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Fig. 10. The market share curves of M-goods produced in each city in the 15-city equilibrium at N"9.30.
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Setting h"1 and 2, respectively, Eq. (5.2) yields the market share curve of each group of goods produced in each city. [For h"3, by definition, the highestorder city at r"0 has 100% share at every r.] In the middle of diagram (a), for example, the bell-shaped real-line curve shows the market share at each r of M-goods produced by city 5. We can see from diagram (a) that except the smallest two cities, 2 and 6, every city is almost self-sufficient in the production and consumption of M-goods, while these two small cities receive most Mgoods from their neighboring cities. More interestingly, diagram (b) shows that for M-goods, each of cities 1, 5 and 7 has a bell-shaped market share curve covering a large number of neighboring cities. That is, not only the highest-order city, 1, but also both middle-order cities, 5 and 7, have the largest market shares of M-goods in their own markets, and they export significant amounts of M-goods to many neighboring cities. Therefore, the present hierarchical urban system also exhibits a rich spatial structure in terms of M-good trade. Finally, we examine the trends of two welfare measures along the evolutionary path. First, the numerical result shows that the equilibrium real wage of each worker increases almost proportionally with the economy’s population size. Second, the total real rent (TRR) of the economy, defined by
¹RR"2
D
R(r) (GF(r))\IF (p (r))\I dr, F
(5.3)
also increases more than proportionally with the population size. Thus, the economy is in the phase of increasing returns forever. Since we have chosen a parameter set that satisfies condition (5.1), the economy maintains forever monopolar spatial structure, and the population growth of large cities (in particular, that of the highest-order city) continues to propel the engine of the economy’s growth through the expansion of M-good varieties.
6. Conclusion Our study remains preliminary, and a great deal of work is left for the future. First, we need to get a firmer analytical grip on the emergence of hierarchical spatial structures. Second, although this paper focused on the evolutionary process of an urban system associated with a gradual increase of the population size of an 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). Third, in order to make the preliminary model here more realistic, a variety of extensions is needed. In this respect, although the agglomeration forces in the present model were created through product variety in consumption goods, the
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variety in intermediate inputs is equally important in explaining the formation of specialized cities in reality. Another interesting extension is to consider multi-location firms. Eventually, by introducing into such basic models various realistic features (such as national borders, political restrictions, natural geographic features, and transport networks, one will be able to develop a realistic spatial economy model which would be useful in studying the future economic geography of nation or group of nations, and in designing effective regional development policies.
Acknowledgements The authors are grateful to David Bernstein, Vernon Henderson, Tony E. Smith, Takatoshi Tabuchi, Jacques Thisse, Anthony Venables, and an anonymous referee for their valuable comments on earlier versions of the paper.
Appendix A: Derivation of the potential function (4.12) As was explained in the derivation of Eq. (2.30), the original form of the potential function for industry h is given by XF(r)"[wF(r)]NF[w (r)]\NF.
(A.1)
To make this equation more explicit, we obtain several preliminary functions. First, using Eq. (4.4) we have GF,GF(0)"(¸F/kF)\ NF\.
(A.2)
Eqs. (4.1) and (4.8) lead to p (r)"c eI+ O +>O De\O P .
(A.3)
In equilibrium, u*"u+. Hence, by substituting Eqs. (A.2) and (4.8) into Eq. (4.7), and using Eqs. (4.10) and (4.11), we have that
u*"
2(1!e\O D) k q
F
IF NF\
[c eD F IFO >OF ] F IF NF\ \I .
(A.4)
The substitution of Eqs. (A.2), (A.3) and (A.4) into Eq. (2.25) leads to [w (r)]\NF"eNF I O \I+ O + P .
(A.5)
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Next, by setting in Eq. (2.20) that ¸ "¸+, ¸ "0 for all kO1, w "1 and I GF "GF, and using Eqs. (4.1) and (A.2), we have that
kFp D kF¸+ [uF(r)]NF" e\O Q eNF\OF Q \ Q\P ds. e\NF\OF P # ¸F ¸F \D
(A.6)
After the calculation of the integral above by dividing into the three intervals, (!f, 0), (0, r) and (r, f), if we use Eqs. (4.8) and (4.10), we can obtain that for r50,
[wF(r)]NF"
1!k+ 1#k+ eNF\OFPt(r, f )# e\NF\OFP, 2 2
(A.7)
where the function t(r, f ) is defined by Eq. (4.13). Finally, the substitution of Eqs. (A.5) and (A.7) into Eq. (A.1) leads to Eq. (4.12).
Appendix B: The proof that (4.19) implies (4.20) Since the complete description of the proof is very lengthy and tedious, here we provide a brief sketch. [For a complete proof, refer to Appendix 6 of Fujita et al. (1995).] We summarize below the proof in several steps. (a) First, we rewrite the potential function (4.12) in a more convenient form for the present purpose. Taking the negative exponential term outside the brackets of Eq. (4.12), and expressing the potential explicitly as a function of two parameters, r and f, we have that XF(r; f )"e\EFP
1!k+ 1#k+ eNF\OFPtFF(r, f )# , 2 2
(B.1)
where gF,pF[k+q+!k q ]#(pF!1)qF. After the substitution of Eq. (4.13) into Eq. (B.1), we use the following identity:
P e NF\OF\O P"1#[2(pF!1)qF!q ] e NF\OF\O Q ds,
(B.2)
and arrange terms appropriately. Then, we can eventually obtain the new expression of the potential function as follows:
P 1!e\O Q XF(r; f )"e\EFP 1#k (pF!1)qF eNF\OFQ 1! ds . (B.3) 1!e\O D (b) Next, in the context of Section 4.2, take any industry h4hI , and let fI F be the critical fringe distance associated with the critical population, NI F. Then, the
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critical potential curve, XF(r; fI F), is tangent to the horizontal line having the height 1 at the critical distance rJ F (refer to Fig. 4). Hence, the pair, (rJ F, fI F), must satisfy the following two relations: XF(rJ F; fI F)"1,
(B.4)
*XF(rJ F; fI F)/*r"0,
(B.5)
and
where rJ F'0 and fI F'0. For convenience, let us define v (s),1!e\O Q
gF and vF(s),1! e\ NF\OF\EF Q. k (pF!1)qF
(B.6)
By solving (B.4) for the term, 1!e\O DI F, we have that
1!e\O DI F"
PJ F
eNF\OFQv (s) ds
while the condition (B.5) yields that
PJ F
eNF\OFQvF(s) ds,
(B.7)
1!e\O DI F"v (rJ F)/vF(rJ F).
(B.8)
Hence, if we define
P P QF(r), eNF\OFQvF(s) ds vF(r)! eNF\OFQv (s) ds v (r) P v (s) vF(s) vF(r) " eNF\OFQ ! ds, vF(r) v (s) v (r) then, by Eqs. (B.7) and (B.8), rJ F must be such that
QF(rJ F)"0,
(B.9)
(B.10)
and rJ F'0. It is not difficult to show that Eq. (B.10) has a unique positive solution. (c) To be explicit about the parameters involved, let us rewrite (B.9) as follows:
P v (s) vF(s) vF(r) Q(r; k , q , D¹, pF, qF), eNF\OFQ ! ds, vF(r) v (s) v (r) where D¹,k+q+!k q " kFqF!k q , F
gF,pFD¹#(pF!1)qF,
(B.11)
(B.12)
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Then, as noted above, there exists a unique positive rJ F,rJ (k , q , D¹, pF,qF) such that Q(rJ F; k ,q , D¹, pF, qF)"0.
(B.13)
(d) Next, focusing on the parameter pF[,1/(1!oF)], we take the total derivative of Eq. (B.13) with respect to rJ F and pF such that drJ F *Q/*pF "! . dpF *Q/*rJ F
(B.14)
It is not difficult to show that *Q/*rJ F is always positive. We can also show that +D¹50N*Q/*pF'0,, and hence we can conclude by Eq. (B.14) that drJ F *rJ (k , q , D¹, pF, qF) , (0. D¹50N *pF dpF
(B.15)
(e) Next, if we substitute rJ F,rJ (kF, q , D¹, pF, qF) into Eq. (B.8), fI F can be uniquely determined, which is denoted by fI (k , q , D¹, pF, qF). It is not difficult to show by using Eq. (B.15) that dfI F *fI (k , q , D¹, pF, qF) D¹50N , (0. dpF *pF
(B.16)
If we replace f in Eq. (4.10) by fI F,fI (k , q , D¹, pF, qF), then NI F, NI (k , q , D¹, pF, qF) is uniquely determined. Then, since the relation (4.10) determines f as an increasing function of N, we can conclude by Eq. (B.16) that D¹50N
dNI F *NI (k , q , D¹, pF, qF) , (0. *pF dpF
(B.17)
(f ) Next, although D¹(, kFqF!k q ) contains each qF, here we treat D¹ F as an independent parameter. Then, in similar manners as in (e) above, we can show that
drJ F D¹50N dqF D
*rJ (k , q , D¹, pF, qF) , (0, *qF 2
(B.18)
dfI F *fI (k , q , D¹, pF, qF) D¹50N , (0, *qF dqF D 2
(B.19)
dNI F D¹50N dqF D
*NI (k , q , D¹, pF, qF) , (0, *qF 2
(B.20)
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(g) Now, we fix all parameters of the model (and hence, the value of D¹ is also fixed), and assume that the assumption (4.17) holds and that D¹50. In this context, take any h4hI and g4hI , and suppose that oF'oE (i.e., pF'pE)
and qF5qE.
(B.21)
Then, it follows that rJ F"rJ (k , q , D¹, pF, qF)(rJ (k , q , D¹, pE, qF) by (B.17) 4rJ (k , q , D¹, pE, qE)
by (B.18)
"rJ E. We can show similarly that Eq. (B.21) also implies that fI F(fI E and NI F(NI E. In the same way, we can also show that if oF5oE (i.e., pF5pE) and qF'qE, then rJ F(rJ E, fI F(fI E, and NI F(NI E. Finally, it follows immediately by definition that Eq. (4.19) implies hI F'hI E. Therefore, sumarizing the results, we can conclude that Eq. (4.19) implies Eq. (4.20).
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On the evolution of hierarchical urban systems Masahisa Fujita *, Paul Krugman, Tomoya Mori Institute of Economic Research, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan Department of Economics, Massachusetts Institute of Technology, Cambridge, MA 02139-4309, USA Accepted 1 June 1998
Abstract The rapid urbanization trend of the world economy implies an increasing importance of cities as basic units of national and international trade. Given that the cities within an economy constitute some form of hierarchical structure, we model the endogenous formation of a hierarchical urban system. To overcome the multiplicity of equilibria, we propose an evolutionary approach which combines a general equilibrium model with an adjustment dynamics. It is demonstrated that as the economy’s population size increases gradually, the urban system self-organizes into a highly regular hierarchical system a la Christaller. 1999 Elsevier Science B.V. All rights reserved. JEL classification: R11; F12; D5 Keywords: Urban system; Self-organization; Evolution
1. Introduction Given that national boundaries no longer provide the most natural unit of economic analysis, what could replace them? The next citation from * Corresponding author. Tel.: #81 75 753 7122; fax: #81 75 753 7198; e-mail: fujita@kier. kyoto-u.ac.jp. The first version of the paper was presented at the 41st North American Meetings of Regional Science International, Niagara Falls, Ontario, Canada, 17—20 November, 1994. 0014-2921/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 4 - 2 9 2 1 ( 9 8 ) 0 0 0 6 6 - X
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¹he Economist (1995, 29 July, survey p. 18) provides a clue: The liberalization of world trade and the influence of regional trading groups such as NAFTA and the EU will not only reduce the powers of national governments, but also increase those of cities. This is because an open trading system will have the effect of making national economies converge, thus evening out the competitive advantages of countries, while leaving those of cities largely untouched. So in future the arenas in which companies will compete may be cities rather than countries. In fact, the most striking feature of today’s economic geography is the concentration of economic activities in cities. At present, in most developed countries and many developing countries, more than 70 per cent of the population reside in cities (United Nations, 1993). In addition, given the recent progress towards a borderless world economy, cities have been enhancing their importance as basic units of international economic systems (as well as of domestic systems). Given the increasing prominence of cities in national and world economies, this paper focuses on the question of how to formally analyse the formation and evolution of the spatial organization of an economy dominated by cities. In this paper, we propose an evolutionary approach to the study of economic geography, in which a general spatial equilibrium model is combined with an adjustment dynamics. Specifically, we consider a spatial economy in which the agglomeration force is generated through product variety in manufactured consumption goods, while the expansion of the agricultural hinterland induces the dispersion of the location of manufacturing production. It has been demonstrated that when the economy contains multiple groups of the manufactured goods (characterized by different degree of product differentiation and/or different transport costs), as the population size of the economy increases gradually, a Christaller-type hierarchical urban system emerges in a self-organizing manner. Although the main stream economics has almost entirely neglected the topics related to space, the study of cities has, in fact, a rich history in which three different traditions have been developed almost independently. The first tradition stems from the Alonso (1964)—Mills (1967)—Muth (1969) model of monocentric city. This is a rebirth of the von Thu¨nen (1826) model of land use, in which the ‘isolated town’ is replaced by the central business district (CBD), and farmers by commuters. Although it remains to this day the basis for an extensive theoretical and empirical literature, it has an important limitation. That is, when our question is not simply how land use is determined given a pre-existing town or CBD, but rather how land use is determined when the location of towns or CBDs are themselves endogenous, this approach offers little help.
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The second tradition was introduced a generation ago by Henderson (1974), which models the economy as an urban system-that is, as a collection of cities. This remains the workhorse approach into the actual distributions of sizes and types of urban areas (see in particular his own later work (Henderson, 1987, 1988)). It has, however, one disturbing aspect. Although Henderson-type models deal with an essentially spatial issue, they are themselves aspatial. In particular, they certainly have nothing to say about where cities themselves are located, relative to each other or anything else. For many purposes this may not be an important question; but if our intention is to bring space back into economics, explaining where cities are and why becomes a central concern. There is a third tradition, the focus of this paper, which at first sight seems to offer an answer to the question of how economies of scale and transport costs interact to produce a spatial economy: the central-place theory of Christaller (1933) and Lo¨sch (1940). The basic ideas of central place theory seem powerfully intuitive. Imagine a featureless plain, inhabited by an evenly spread population of farmers. Imagine also that there are some activities that serve the farmers, but which cannot be evenly spread because they are subject to economies of scale — manufacturing, administration, and so on. Then it seems obvious that the tradeoff between scale economies and transportation costs will lead to the emergence of a lattice of ‘central places’, each serving the surrounding farmers. Less obvious, but still intuitively persuasive once presented, are the refinements introduced by Christaller and Lo¨sch. Christaller argued, and produced evidence in support, that central places, or cities, form a hierarchy: the city at the top would produce the entire range of urban products and lower order cities successively fewer products. Lo¨sch pointed out that if a lattice is going to minimize transportation costs for a given density of central places, the market areas must be hexagonal. And thus every textbook on location theory contains a picture of an idealized central place system in which a hierarchy of central places occupy a set of nested hexagons. The original central place theory story was applied to towns serving as a rural market. But it is obvious that a similar story can be applied to business districts within a metropolitan area. Small neighbourhood shopping districts are scattered across the basins that surround larger districts with more specialized The basic idea of Henderson’s analysis is beautifully clear and simple: there is a tension between ‘external economies’ associated with geographic concentration of industry within a city, on the one hand, and diseconomies such as commuting costs associated with large cities on the other. The net effect of this tension is that the relationship between the size of a city and the utility of a representative resident is an inverted U-curve. With the added assumption of the forward-looking behaviour of large agents or ‘city corporations’, the equilibrium size of each city equals the optimal sized determined at the top of that U-curve. Since external economies tend to be specific to particular industries while diseconomies (such as commuting costs) tend to depend on the overall size of a city, this approach can explain the formation of a large variety of specialized cities having quite different sizes.
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stores, all eventually cantering on the downtown with its great department stores and high-end boutiques. Indeed, the hierarchical image is so natural that it is hard to avoid describing things that way. Unfortunately, as soon as one begins to think hard about central place theory one realizes that it does not quite hang together as an economic model. In economic modelling we try to show how a phenomenon emerges from the interaction of decisions by individual families or firms; the most satisfying models are those in which the emergent behaviour is most surprising given the ‘micro motives’ of the players. What is therefore deeply disappointing about central place theory is that it does not give any account along these lines. Lo¨sch showed that a hexagonal lattice is efficient; he did not show that it would tend to emerge out of any decentralized process. Christaller suggested the plausibility of a hierarchical structure; he did not give an account of how individual actions would produce such a hierarchy (or even sustain one once it had been somehow created). In this paper, we try to resurrect the central place theory as an economic model in space: it is a powerful idea too good for being left as an obscure theory. To do so, the first step is, obviously, to develop a general equilibrium model of micro-economics which could yield a hierarchical central place pattern as an equilibrium solution. We achieve this, in Section 2, by developing a general spatial-equilibrium model as a multi-city, multi-industry extension of the monopolistic competition model by Fujita and Krugman (1995). As in the model of single-city economy, agglomeration forces are created through the interaction of product variety in manufactured consumption-goods and scale economies in their production, while the demand of consumption goods by dispersed agricultural population induces the dispersion of industries. The tension between these forces for and against agglomeration underlies an equilibrium distribution of urban industries. However, such a general equilibrium model by itself fails to explain the emergence of surprisingly regular urban hierarchies that have been frequently observed in the real world. This is because the model also would yield a continuum of possible equilibria with different spatial configurations, thus providing little insight to why such a regular central place pattern is so often observed in actual economies. Multiple solutions arise because, by the very nature of spatial agglomeration forces, 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 equilibrium model. Furthermore, when such lock-in effects of many cities interact together, more indeterminacy to the equilibrium spatial structure will occur. Hence, in general, there may exist a continuum of equilibria that could be sustained once established. Therefore, to discuss the structure of multiple-city economies in general is to risk becoming lost in an endlessly complex taxonomy. How can we avoid such a risk? Here, we propose an evolutionary approach in which the analysis of
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multiple-city economies proceed in tandem with the analysis of the process of city formation. Only by telling some kind of story about the process of city formation, in fact, we can reduce the taxonomy to manageable size. While it is possible to imagine a discussion of city formation from various initial conditions — for example, from the hypothetical Flat Earth that was used elsewhere (e.g. Krugman, 1996, Chapter 8; Krugman and Venables, 1995) — we find it most natural to think of new cities as emerging as an economy which already has an urban structure grows over time. This means, however, that our discussion of city formation must, necessarily, take place in a model of a multiplecity system. In Fujita and Mori (1997), we experimented this approach in the context of an economy with a single manufacturing industry. In this paper, we extend it to a multi-industry context, and study the evolution of central place hierarchy. To formalize our evolutionary approach in Section 3, we imagine an economy with an existing city or cities, and allow its population to grow steadily. We then consider when and where new cities emerge, using a dynamic adjustment process for the location of urban industries and their workers. This evolutionary approach offers, we believe, a valuable perspective on how economies evolve in space. In particular, one gets an exciting new way of thinking about spatial economics in terms of the coevolution of two ‘landscapes’: the landscape defined by the current distribution of economic activity, and the implied landscape of ‘market potential’ which determines the future evolution of that distribution. To be more specific, imagine a long, narrow economy — effectively one dimensional — that stretches sufficiently far that we can disregard boundary conditions. Along this line lies land of homogeneous quality, with one unit of land per unit distance. Suppose that the initial landscape of the economy is the ‘isolated state’ of von Thu¨nen: all manufacturing industries located in a single city, while the city surrounded by agricultural area. Given this economic landscape, if we take a representative firm belonging to an industry, and draw the relative profitability (normalized to be unity at the existing city) of the firm at each location, then we obtain a curve, or the market potential of that industry. As shown in Section 4, this potential curve has the shape of a lain brace (refer to Fig. 4), with a cusp at the location of the city, and a wave in each side (extending to the edge of the cultivation). The cusp reflects, of course, the lock-in effects of the city. The potential curve is specific for each industry. As the economy’s population grows, the agricultural frontier expands, and the potential curve of each industry shifts upward (except at the city location where the potential of each industry is normalized to be one). As long as the potential curves of all industries are strictly less than 1 everywhere (except the city location), the monocentric configuration of this Von Thu¨nen landscape is a stable equilibrium. For the self-organization of a hierarchical urban system, the first key observation is that each industry has a different shape of critical potential curve. That is,
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given the monocentric configuration of the economy, as the economy’s population increases, the top of the ‘wing’ in each side of the market potential of any industry eventually hits 1 at a certain ‘critical distance’ when the population reaches a certain ‘critical size’. The critical distance and critical population systematically vary across industries. For example, if all industries have the same characteristics except the price-elasticity of their products, the industry supplying the products with the highest price-elasticity has the shortest critical distance (and the smallest critical population), the industry supplying the products with the second-highest price-elasticity has the second-shortest critical distance, and so on. Thus, industries can be ordered naturally by the length of their critical distance as in Christaller (1933). But, here, this critical distance is endogenous. The industry having the shortest critical-distance is called the level 1 industry, the industry with the next shortest critical-distance the level 2, and so on. When the economy’s population reaches the critical population of level 1 industry, the monocentric spatial configuration becomes unstable, leading to the formation of a pair of new cities at the location of critical-distance in each side; each new city contains, of course, level 1 industry only. Numerical simulations in Section 5 indicate that as the agricultural frontier shifts outward, several such ‘lowest-order’ cities form before it becomes profitable for the level 2 industry to start production at a new location. When it does become profitable for some firms of level 2 industry to start production at new location, it is not only to serve the agricultural population but also to get closer to the consumers in these lowest-order cities. And the demand-pull of the preexisting lowest-order cities generate cusps, at their locations, in the market potential curve for the level 2 industry. This makes it most likely (though not certain) that when firms of level 2 industry start production at new location at which this industry’s potential takes the maximum, they will do so at the location of a cusp that contains a large concentration of level 1 firms. Thus, we have a process in which population growth generates a pattern of several small cities containing only level 1 industry, followed by a larger city that contains both industries, followed by several more small cities, and so on. In general, when firms find it profitable to establish a new location for the production of higher-level goods, they tend to choose an existing lowerorder city, due to the demand-pull of the consumers in such cities; so when a higher-order city emerges, it normally does so via the ‘upgrading’ of an existing lower-order city. Repetition of the process eventually evolves a
This is for the case when the original critical-distances of the two industries are significantly different. When they are very close to each other, the location of the two industries tends to become identical.
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central-place hierarchy. We will show an example of a three-order central-place hierarchy. Although the main concern of this paper is on the methodological approach for the study of economic geography, it is interesting to note that the urbanization process similar to the one presented above did happen in history. For instance, Fig. 1 depicts the evolution of the U.S. urban system during the period from 1830 to 1870. Over this period, the U.S. population increased threefold from approximately 13 million to 39 million, which resulted in the major expansion of the agricultural area towards the Great West, well beyond Chicago and St. Louis. The figure shows the location of major cities (with population more than 30 thousand) in 1870. These cities in 1870 are classified into three orders such that New York, indicated by the largest circle, is the single first-order (or the highest-order) city with the population of more than 1300 thousand, those cities denoted by a middle-size circle represents the second-order cities with the population between 1300 thousand and 130 thousand, and those cities shown by the smallest circle belong to the third-order (or the lowest-order) cities with the population between 130 thousand and 30 thousand. This same figure also shows how this urban system in 1870 had evolved from that in 1830 by indicating the change of the size-order of each city from that in 1830. That is, the cities which were upgraded in one or more orders are denoted by shaded circles, those which were downgraded in one or more orders but were still higher than or equal to the third-order by , those which dropped out of the third-order by white circles, and those which stayed in the same order as that in 1830 by black circles. Fig. 1 reveals several interesting facts. First, not surprisingly, most old cities established well by 1830 (except the steel town, Pittsburgh) located along the northern part of Atlantic coast or navigable rivers, reflecting the importance of
The Great West is the 19th century naming of the area which represents the vast interior region of the U.S. in the west of Ohio River or Lake Michigan (Cronon, 1991). Fig. 1 has been created by adapting Figs. 5—8 in Borchert (1967). Based on the rank size distribution of 178 cities of the U.S. in 1960, Borchert defined four thresholds of population size (at which the slope of the rank size distribution changed noticeably). Then, for the years before 1960, each ith size-order threshold, ¹ , in year s (s"1830 or 1870) was defined by relationship, GQ ¹ "¹ (N /N ), where ¹ is the ith threshold population in 1960, and N (resp., N ) is the U.S. GQ G Q G Q population in year s (resp., 1960). In developing Fig. 1, for simplicity, we combined the original second-order and third-order cities, while the cities of the original fourth and fifth order were combined to form the new third order. Based on the same rule in footnote 5, the threshold population for each size-order in 1830 is given as follows. The first-order cities are those with population beyond 530 thousand (which actually did not exist), the second-order cities with population between 90 and 530 thousand, and the third-order cities with population between 15 and 90 thousand.
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Fig. 1. Evolution of the U.S. urban system from 1830 to 1870.
sea- and river transportation for trade with Europe as well as within the U.S. Second, in the Mid West and Great West, a large number of new third-order cities emerged by 1870. This is due to the expansion of the agricultural area towards the Great West resulting from the threefold increase of the U.S. population between 1830 and 1870. In order to provide farmers with ordinary consumption goods and farming tools, new small cities appeared by keeping an appropriate area in the Mid West and Great West. Third, several old thirdorder cities in the Mid West (such as St. Louis, Chicago, Cleveland and Detroit) upgraded themselves to second-order cities. These second-order cities were not only larger in population than third-order cities, but also played the role of regional center by supplying higher-order goods and services (such as
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business/trade services and sophisticated farming machines) to larger hinterlands. Fourth, a previously second-order city, New York, upgraded itself to the unparalleled first-order city of the U.S., providing the entire U.S. the highest-order goods and services (such as major financial services and national newspapers). Finally, we can also see in the figure that although most cities existed in 1830 still existed in 1870, several old frontier cities had disappeared (i.e., their population became smaller than the third-order threshold). In this way, as the population size of the U.S. increased rapidly (combined with the westward expansion of canal and railway network), a highly systematic hierarchical system of cities had been established in the U.S. by 1870. Given that no agent, in practice, had fully controlled the development process of spatial organization of the entire U.S., this example provides a clear case for the self-organization of a spatial economy towards a complex system.
2. The model The underlying structure of this paper’s model is closely related to that of the models in Krugman (1991), Fujita and Krugman (1995), and Fujita and Mori (1997). We will, therefore, be brief in describing its formal structure. We consider a boundless, one-dimensional location space of the economy, X, along which lies land of homogeneous quality, with one unit of land per unit distance. The economy has an agricultural sector (A-sector) and H types of manufacturing industries (M-industries). The A-sector provides a single, homogeneous good to consumers, while each M-industry supplies a continuum of differentiated goods to consumers. The agricultural production uses land and labor, and hence it is necessarily dispersed in space. In contrast, the production activity of M-industries takes place under an increasing returns technology, using labor only, and we assume that at each point of time, M-industries are concentrated in a finite number of locations, called ‘cities’. (The number and location of cities are determined endogenously through evolutionary process.) We also assume the ‘iceberg’ form of transport costs such that if a unit of the A-good [any variety of goods produced by h the industry] is shipped from
These second-order cities, of course, also provided nearby hinterlands most of the goods and services that were provided by third-order cities. In other words, they upgraded themselves to second-order cities by adding additional functions to those of third-order cities. The same note applies to the highest-order city. Hence, the entire system has, roughly speaking, a hierarchical structure in which a higher-order city supplies its own goods as well as most goods supplied in lower-order cities. This story is not unique. For another example, the urbanization process similar to this case took place in Europe in the 12th century as its population increased rapidly. For a comprehensive study of urban development in Europe and the U.S., see, for example, Bairoch (1985) and Marshall (1989).
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a location r3X to another location s3X, only a fraction, e\O Q\P [e\OF Q\P ], of the original unit actually arrives, while the rest melts away en route. At a given point of time, suppose the economy is endowed with a homogeneous labor force of N workers, who are free to choose both the location/city and the sector/industry in which they work. In this section, we assume a priori that the economy has K cities, each at location r (k"1, 2,2, K), and obtain the I conditions for the spatial equilibrium of the economy. Although each consumption and production activity takes place at a specific location, first we describe each type of activity without explicitly referring to the location. The consumers of the economy consist of N workers plus a class of landlords, who for simplicity are assumed to live on their land holdings — that is, land rents are consumed where they are accrued. Every consumer shares the same Cobb—Douglas utility tastes for the (1#H) groups of goods: & º"AI (CF)IF, k # kF"1, (2.1) F F where A is the consumption of the agricultural good (A-good), CF represents the composite index of the manufactured goods by industry h (MF-goods), and kF is a constant representing the expenditure share on MF-goods. When the industry h supplies a continuum of differentiated goods of size nF, CF is given by
LF MF (mF(i))MFdi , 0(oF(1, (2.2) where mF(i) represents the consumption of each variety i3[0, nF ]. In this specification, due to Spence (1976) and Dixit and Stiglitz (1977), the parameter oF represents the intensity of the preference for variety in MF-goods. When oF is close to 1, differentiated MF-goods are close to perfect substitutes; as oF decreases towards 0, the desire to consume a greater variety of MF-goods increases. If we set pF,1/(1!oF), then pF represents the elasticity of substitution between any two varieties. In the self-organization of urban hierarchy later, the composition of oF’s or pF’s for H industries plays the central role. Given income ½, and a set of prices, p for the A-good and pF(i) for each variety i of MF-goods, the budget constraint of a consumer is p A# LFpF(i)mF(i) di"½. Maximizing utility (2.1) subject to this budget constraint, F we can derive the following demand functions: CF"
A"k ½/p ,
(2.3)
mF(i)"kF½(pF(i))\NF(GF)NF\ for i3[0, nF], h"1, 2,2, H,
(2.4)
In a general equilibrium model with land, the question of where land rents are spent is a nuisance issue that unfortunately must be dealt with one way or another.
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where GF is the price index for MF-goods given by
GF"
LF \ NF\ pF(i)\NF\di .
(2.5)
which together yield the following indirect utility function:
(2.6) º" (kF)IF (k )I ½ (GF)\IF (p )\I . F F Next, the A good is produced under a constant-return technology, using labor and land with fixed proportions: c units of labor and one unit of land are required to produce one unit of output. In contrast, each M-good is produced under an increasing-return technolgy, using labor only. All products of the same h-industry have the same production technology such that the production of quantity qF(i) of any variety of MF-goods at any given location requires labor input l(i), given by lF(i)"FF#cFqF(i), h"1, 2,2, H,
(2.7)
where FF and cF are, respectively, the fixed and mariginal labor requirements. Through the appropriate normalization of the units of the output, qF(i), and the size, nF, of each M-industry, we can assume without the loss of generality that cF"oF and FF"kF/pF for h"1, 2,2, H.
(2.8)
Due to scale economies in specialized production of M-goods, any variety of M-goods is assumed to be produced by a single firm only that chooses its location and f.o.b. (mill) price in a nonstrategic manner in the spirit of Chamberlin. Hence, in each industry h, the ‘number’ of active firms equals the ‘number’ of varieties being produced in that industry. This also implies that if pF(r) is the f.o.b. price of an MF-variety produced at location r, due to the assumption of iceberg transport technology, its delivered price at each location s is given by pF(r, s)"pF(r)eOF Q\P .
(2.9)
Now, with explicit consideration of the spatial distribution of activity, let nF be I the number of MF-varieties being produced in city k, ¸F be the size of labor force I in industry h in city k, and define ¸ , ¸F. Let X be the agricultural area of I F I the economy, a subset of X in which land is actually used for agriculture. And, let p (r) be the A-good price at each r3X, w(r) be the wage rate at each r, and let w ,w(r ) be the wage rate in city k which is the same regardless of the sector for I I which an individual works. Furthermore, let pF,pF(r ) be the f.o.b. price of each I I MF-variety produced in city k.
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In this context, first, the price index for MF-goods at each location r can be obtained by using Eq. (2.5) as follows:
\ NF\ GF(r)" nF [pFeOF P\PI ]\NF\ , (2.10) I I I and we define GF(r ),GF for each city k. Next, suppose that a firm in the I I h-industry locates at r and chooses an f.o.b. price, pF(r), for its product. Then, since the total income in each city k equals ¸ w , and the income from I I agriculture per unit distance at each rOr equals p (r), the firm’s total sales can I be obtained by using Eq. (2.4) as follows: qF(r, pF(r))" kF¸ w [pF(r)eOF PI\P ]\NF(GF)NF\eOF PI\P I I I I
kFp (s)[pF(r)eOF Q\P ]\NF(GF(s))NF\eOF Q\P ds,
(2.11) 6 where on each term on the right-hand side above, the multiplication at the end by exp(qF"r !r") or exp(qF"s!r") reflects the consumption of the firm’s product I in transportation. Rewriting (2.11), we have #
qF(r, pF(r))"(pF(r))\NFuF(r),
(2.12)
where uF(r), kF¸ w e\N\OF PI\P (GF)NF\ I I I I
kFp (s)e\N\OF Q\P (GF(s))NF\ ds.
(2.13) 6 Since the firm is assumed to take all components in Eq. (2.13) as given, Eq. (2.12) implies that the price elasticity of the aggregate demand of a firm belonging to industry h equals pF. Thus, under the normalization in Eq. (2.8), the equality of the marginal revenue and marginal cost leads to #
pF(r)"w(r).
(2.14)
Under this pricing rule, the firm’s profit equals nF(r)"w(r)qF(r, w(r)) !w(r)(FF#cFqF(r, w(r))). Using Eq. (2.8), this leads to nF(r)"w(r)[qF(r, w(r))!kF]/pF.
(2.15)
Hence, if the firm actually operates at r, then by the zero-profit condition under free entry and exit, qF(r, w(r))"kF,
(2.16)
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must hold under which the firm’s labor requirement equals FF#cFkF"kF, using Eq. (2.8). Thus, the equilibrium output, qF*, and the equilibrium labor requirement, lF*, of any active firm in industry h equal a constant given by qF*"lF*"kF.
(2.17)
In each city k, then, the equilibrium number of firms in each industry h is given by nF"¸F/lF*"¸F/kF. I I I
(2.18)
Since Eq. (2.14) implies that pF"w for each city k, the price index Eq. (2.10) I I can be rewritten, using also Eq. (2.18), as
\ NF\ . (2.19) GF(r)" (¸F/kF)w\NF\e\NF\OF P\PI I I I Next, we characterize the equilibrium of labor markets. If we solve relation Eq. (2.16) for w(r) and denote the solution by wF(r), then it represents the wage rate that industry h can bear under zero-profit condition. Using Eqs. (2.12) and (2.13), this zero-profit wage rate for industry h at each location r can be obtained as
wF(r)" ¸ w e\NF\OF PI\P (GF)NF\ I I I I
#
p (s)e\NF\OF Q\P (GF(s))NF\ds
6
NF .
(2.20)
Setting ½"wF(r) in Eq. (2.6) and neglecting constants, the associated real wage rate is given by
uF(r)"wF(r) (GF(r))\IF (p (r))\I . F For convenience, for each city k we define
(2.21)
wF(r ),wF and uF(r ),uF. I I I I
(2.22)
Now, for the spatial equilibrium of the K-city economy, the five sets of conditions below must be satisfied. First, given that every worker is free to choose his job and location, all workers should enjoy the same equilibrium utility, or the same real income, which is denoted by u*. Under the zero-profit condition of all active firms, this implies that in each city k, uF"u* for every industry h such that ¸F'0, I I
(2.23)
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which in turn implies that there exists a common wage rate w in each city k such I that
wF"w "u* (GF)IF (p (r ))I if ¸F'0. (2.24) I I I I I F Second, for all agricultural workers (A-workers) to attain the same real income u*, the wage rate for A-workers, w (r), at each location r in the agricultural area X must be such that
w (r)"u* (GF(r))IF (p (r))I . (2.25) F Also at each city location r , for convenience, we define the potential value of I w (r ) by Eq. (2.25). I Third, the clearing of labor market requires ¸F#c "X ""N, (2.26) I I F where "X " represents the size of the agricultural area. Fourth, the A-good market must be cleared. In general, the full description of this condition is complex because the agricultural area of the economy, X , and the flow pattern of the A-good in X need to be determined endogenously. Hence, the clearing condition of the A-good market is provided later for each specific case. Here, it suffices to say that at each agricultural location r, the land rent is the value of output from each unit of land minus the wage rate bill for the c workers needed to farm that land: R(r)"p (r)!c w (r).
(2.27)
Finally, we have to see if the K-city configuration of the economy is really sustainable — so no firm in any industy should be able to attain a positive profit at any possible location. Since condition (2.23) assures that all workers in K cities enjoy the real wage rate u*, the configuration is sustainable if the zero-profit real wage rate, uF(r), of each industry h, defined by (2.21), exceeds u* nowhere. It actually turns out to be convenient to work with a monotonic transformation of real wages; we define the (market) potential function of each industry h as the ratio, [uF(r)]NF XF(r)" * F . [u ]N
(2.28)
The K-city configuration, then, is sustainable if and only if XF(r)41 for all r,
(2.29)
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for every industry h"1, 2,2, H. As explained in the next section, this potential function also allows us to determine the location of new cities. To rewrite the potential function in an explicit form, we use Eq. (2.25) to obtain that u*"w (r)+ (GF(r))\IF,(p (r))\I . Substitution of this relation and F Eq. (2.21) into Eq. (2.28) leads to XF(r)"[wF(r)]NF/[w (r)]NF. Hence, using Eq. (2.20), the potential function takes the following form:
XF(r)" ¸ w e\NF\OF PI\P (GF)NF\ I I I I
#
6
p (s)e\NF\OF Q\P (GF(s))NF\ ds [w (r)]\NF.
(2.30)
3. Adjustment dynamics Given the model in the previous section, our task is now to examine how the spatial configuration of the economy changes in the long-run as the economy’s population increases steadily over time. Since the population growth is exogenously specified over time, the task of tracing the change of equilibrium spatial configuration is essentially the problem of comparative statics in terms of population size. Here, however, we must face the difficult issue of multiple equilibria mentioned in the Introduction. To resolve this problem, we adopt a sort of evolutionary approach in which we consider ad hoc dynamics of labor migration among cities and the agricultural area, and trace a sequence of stable equilibria. Specifically, we imagine an economy in which there are two sources of change over time. First, there are ‘extrinsic’ dynamics arising from a steady process of population growth, which we regard as exogenous. Second, there are ‘intrinsic’ dynamics as workers move toward locations that offer higher real wages, and by so doing in turn alter the wages offered at different locations. In general, we should think of these sources of change as operating simultaneously. For the sake of simplicity, however, we will instead imagine that the extrinsic change in the economy moves very slowly compared with the intrinsic adjustment process. Or to be a bit less cryptic, we will think of this economy as evolving by a sort of two-step process. We start from an equilibrium spatial configuration, then increase the population a bit and hold it there; let the economy settle into a new equilibrium; then repeat. The dynamic process we use for the adjustment of urban population is mathematically analogous to the ‘replicator dynamics’ used in evolutionary game theory (see, for example, Freedman, 1991), except one difference: here, we must always consider the possible emergence of new cities at any possible set of locations in a continuous location space. To be specific, suppose that given the
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economy’s population N, we have obtained the associated spatial equilibrium, in which K cities exist respectively at r (k"1, 2,2, K). Choose any finite set of I hypothetical new cities, k"K#1, K#2,2, DK, at a set of locations, r , ,r . Including these DK new (potential) cities, suppose that the )> 2 )>D) economy experiences a small perturbation in population distribution such that the population of each city k changes from ¸ to ¸ #e , where ¸ "0 and I I I I e '0 for k"K#1,2,DK. Given this small perturbation of population, the I economy is then assumed to go through an adjustment process following the migration dynamics defined by, ¸ "¸ (u !uN )N, k"1, 2,2, K#DK. (3.1) I I I Here, u is the instantaneous-equilibrium real wage common to all workers in I city k, uN ,(¸ u # ¸ u )/N is the average real wage, and u is the real wage I I I common to all agricultural workers. By definition, ¸ "N! ¸ . These I I instantaneous real wages, u and u , are to be determined by using the I instanteneous equilibrium conditions in which population of each city is temporarily fixed. Here, for simplicity, we assume that the agricultural population moves instantaneously to equalize agricultural real wages, this giving u , the real wage common to all agricultural workers. Dynamics (3.1) indicates that labor migration is directed towards higher real-wage cities. There may seem to be an arbitrariness in the choice of which hypothetical new cities to include in the system. We need not agonize over this question, however, because there are never more than a few interesting potential new cities to consider. To identify these potential new cities, we turn to the market potential functions introduced in Section 2, and use the analysis developed there. Consider an urban system in which one or more cities already exist. If this system is in spatial equilibrium, then the real wages of all agricultural workers and of all manufacturing workers in existing cities must be the same. We can therefore assess the attractiveness of a location to each industry h using the same market potential function defined by Eq. (2.28). Let XF(r) be the market potential curve of each industry h in this economy. Now consider two possible small perturbations of the system. First, suppose that the populations of the existing cities are changed by small amounts, but no new cities are ‘seeded’. This perturbation will be eliminated if this system is stable; and let us suppose that it is (otherwise we would not observe this configuration of cities in the first place). Next, suppose that we also create some small urban populations at new locations. Will these new potential cities take hold and grow? Clearly, if they are at locations where
The instantaneous equilibrium conditions are the same as those in Section 2 except that the population size of each city and that of the farmers are temporarily fixed at each instant. Within the same city, workers are assumed to be free to choose the industry to work in, and hence the wage rate is the same for all workers there.
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the potential curve of every industry is below 1, they will not: the real wage they will offer will be less than that in agriculture or in existing cities. And so if the potential curve of every industry is strictly below 1 everywhere except the locations of existing cities, no new cities will emerge. But suppose that population growth has just pushed the potential curve of one industry up to the point where it lies slightly above 1 in some locations. Then a perturbation that puts some manufacturing workers in those locations will tend to grow, not die away, over time — and the existing urban system will become unstable. In short, as soon as the market potential curve of one industry humps itself above 1, we can expect new cities to emerge where it does so. In the following sections, we will demonstrate how this works. 4. The growth of the monocentric economy As noted in the Introduction, in this paper we think of multi-city systems as emerging from an imaginary history in which a growing economy add cities over time: the starting point for such a history must, obviously, be an economy with only one city, i.e., the monocentric economy of von Thu¨nen. In this section, first we determine the spatial equilibrium of the monocentric economy under a given population size of labor force, N. Then, assuming that N increases gradally (from near zero), we investigate when it eventually becomes unsustainable. As shown below, the study of the potential curves associated with the monocentric system reveals much about the characteristics of the evolutionary path of the urban system that will be manifested later. 4.1. The von Thu¨ nen economy Consider the ‘von Thu¨nen’ spatial configuration depicted in Fig. 2. In this figure, the production of all manufactures is assumed to take place in a single city; we relabel locations if necessary so as to make that urban site location 0. The agricultural area extends from !f to f, where f represents the (endogenous) agricultural frontier. The city exports manufactures to its agricultural hinterland and imports agricultural goods in return. In the context of the general spatial system in the previous section, this implies that K"1, and we denote the only city by k"1. For the moment, let us simply assume that this is the economy’s spatial structure, and use that assumption to determine equilibrium goods prices, factor prices, and land use. Each location in the agricultural hinterland produces one unit of agricultural output and exports the surplus over local production to the city. Let p ,p (0) be the price of the agricultural good at the city. Agricultural transport costs mean that the price farmers receive is lower the further they are from the city: p (r)"p e\O P .
(4.1)
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Fig. 2. The monocentric spatial structure.
Let R(r) and w (r) be the land rent and wage rate for agricultural workers at location r. Substitution of Eq. (4.1) into Eq. (2.27) yields that R(r)" p e\O P !c w (r). Land rents will be zero at the edge of the cultivation, so it must be true that p e\O D w ( f )" . c
(4.2)
Turning to manufacturing, to normalize prices we set the wage rate at the city equal 1. Then, by Eq. (2.14) pF(0)"w(0)"1 for h"1,2,2, H.
(4.3)
The price index GF(r) takes a very simple form because of the assumption that manufacturing occurs only at the central location. Let ¸F be the size of labor force in each industry h at the city. Then, using the definition of GF(r) by Eq. (2.19),
GF(r)"
¸F \ NF\ F eO P . kF
(4.4)
The presence of manufacturing trade costs means that this index is increasing as we move to locations further away from the center. We now have all the information we need to determine equilibrium — again, simply assuming for the moment that manufacturing is concentrated in the city. We can think of equilibrium as determined by two conditions: market clearing
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in the market for agricultural output, and equality of real wages between farmers and workers. Let us consider these in turn. First, income earned in the city is w(0)¸+"¸+ by Eq. (4.3) where ¸+, ¸F; a share k of that income will be spent on A, so food consumption F at the city will be D "k ¸+/p . Meanwhile, each rural location will spend a share k of its own income on food, leaving 1!k units to be shipped to the city. Only fraction e\O Q of the units shipped from location s arrive at the city, so the supply of food to the city will be S "2(1!k )De\O Q ds. But the urban labor force is the total labor force less the number of farmers, ¸+"N!2c f, so we can summarize the market-clearing condition for agriculture as a relationship between the number of farmers and the price of food: k (N!2c f ) . (4.5) p " 2(1!k )De\O Q ds Looking at real wages, Eq. (4.2) gives us the nominal wage received by the frontier farmer. His real wage is
u ( f )"w ( f ) GF( f )\IF p ( f )\I F
1 " (p )\I (GF)\IF e\ RFIFOF>\I O D, c F where GF,GF(0), while the real wage of a worker in the city is
u+" (GF)\IF (p )\I . F
(4.6)
(4.7)
Thus, equality of real wages requires that p "c e FIFOF>\I O D"c eI+ O+>O D,
(4.8)
where we have defined & kFqF k+, kF"1!k , and qN +, F . k+ F
(4.9)
Fig. 3 shows how the price of A and the size of the agricultural hinterland are simultaneously determined by the market-clearing condition (4.5) and the equal-real-wage condition (4.8). It is immediately apparent that an increase in the population will require, other things being equal, a rise in p to clear the market; as N increases the market-clearing curve will shift up, and in equilibrium the frontier will move out.
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Fig. 3. The determination of the equilibrium p and f.
To be specific, by eliminating p from Eqs. (4.5) and (4.8), we have
D (4.10) e\O Q ds, which determines f uniquely as an increasing function of N. Once the frontier distance f is determined, the labor size of each industry h can be obtained as k (N/c !2f )e\I+ ON +>O D"2(1!k )
kF kF ¸F" ¸+" (N!2c f ). k+ k+
(4.11)
Given relative prices and the allocation of labor between manufacturing and agriculture, everything else can be determined. 4.2. Market potential functions and the sustainability of the von Thu¨ nen structure So far, we have simply assumed that manufacturing production takes place exclusively at the city. To claim, however, that this monocentric configuration is an equilibrium, we must make sure that no firm has an incentive to ‘defect’ from the city. That is, the sustainablility condition (2.29), must hold for every industry. To check this sustainability condition, first we obtain the potential function of each industry in the present context of the monocentric economy. By successive substitutions into the general form of potential function (2.30), we can obtain the
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specific form of the potential function of each industry as follows (refer to Appendix A for derivation): For r50,
XF(r)"eNF I O \I+ON + P
1!k+ 1#k+ eNF\OFPtF(r, f )# e\NF\OFP , 2 2 (4.12)
where P e\O Q[1!e\NF\OFP\Q] ds tF(r, f )"1! . (4.13) De\O Q ds For the monocentric configuration to be sustainable, it must hold that for every industry h, Eq. (4.12) exceeds unity nowhere. To find the circumstances under which this condition is satisfied, we next examine the properties of function (4.12). First, it is obvious by Eqs. (4.12) and (4.13) that XF(0)"1 for every industry h. Second, the gradient of the potential function (4.12) in the neighbourhood of the city is easily found by differentiating to give (4.14)
dXF(0)/dr"pF[k q !k+(q+#oFqF)],
which indicates that all potential curves for the same industry have the same slope at the city. Having a negative gradient as we move away from the center (to the right, increasing r) is a necessary condition for the monocentric structure to be sustainable, and this condition must hold for all H industries. Third, since the function tF(r, f ) is increasing in f, so is XF(r). Hence, as population increases, the potential functions of all industries shift upwards (except at r "0), and it is helpful to look at their limiting values. Letting f"R in Eq. (4.13), we obtain
P tF(r, R)"e\O P#q e\NF\OFP e NF\OF\O Q ds. (4.15) Setting f"R in Eq. (4.12) and using Eq. (4.15), the limiting potential curve for each industry can be expressed as
XM F(r)"eNF MFO >OF\I+O >O+ P
P 1!k+ 1!k+ # q e\ NF\OF\O P\Qds 2 2
1#k+ # e\ NF\O+\O P . 2
(4.16)
Since the potential function of the monocentric economy is symmetric with respect to the city, here we focus on the right-hand side of the city.
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The limiting behaviour of these curves depends on the exponent of the term outside the curly brackets in Eq. (4.16). If the exponent is negative then XF(r) is strictly decreasing in r and hence it is less than 1 for all rO0, and if this holds for all industries, then the monocentric structure is always sustainable. But if the exponent is positive for any industry h, then this term goes to infinity as rPR; meanwhile, the term inside the curly brackets is always greater than (1!k+)/2. Hence, as rPR, X F(r) tends to infinity. This implies that for large enough values of f and r, the potential function XF(r) eventually exceeds 1, meaning that at some value of N the monocentric structure will break down. Pulling this together, we can summarize the results as in Table 1. First, in the right-hand column, if k q 'k+(q+#oFqF) for any h, then the potential curve XF(r) has a positive slope at the edge of the city, and hence the firms in industry h will certainly move out of the city, meaning that the monocentric system is never an equilibrium. Second, in the left-hand column, if k q (k+(q+#oFqF) and k+(q #q+)/(q #qF)5oF for every h, then the limiting potential curve of every industry is strictly decreasing at every r, meaning that the monocentric system is always an equilibrium however large N becomes. That is, the centripetal force created by the agglomeration of all industries at the city is so strong that no new city can emerge, however large N becomes. Therefore, since we are interested in the case in which the spatial system of the economy becomes increasingly complex as N increases gradually, hereafter we focus on the case corresponding to the middle column of Table 1. That is, we assume that, we can divide industries into two groups, h4hI and h5hI #1, for which the following conditions are satisfied: k q (k+(q+#oFqF) for every h, and
(4.17a)
k+(q #q+)/(q #qF)(oF for h"1,2,2, hI , while
(4.17b)
k+(q #q+)/(q #qF)5oF for h"hI #1,2, H,
where 14hI (H. Condition (4.17a) means that the A-good transport cost is not too high, while (4.17b) implies that there exist industries (h"1, 2,2, hI ) for which substitution parameters, oF are significantly large, and hence their products have high price elasticities. Given these two conditions, for each industry h5hI #1 such that k+(q #q+)/(q #qF)5oF, the associated potential curves are always below 1 everywhere under any value of N, as explained before. In contrast, for each industry h4hI such that k+(q #q+)/(q #qF)(oF, the behaviour of its potential curve is more interesting.
This can be shown essentially the same way as in Appendix B of Fujita and Krugman (1995).
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Table 1 The possibility of a monocentric equilibrium k q (k+(q+#oFqF), for every h
k+
q #q+ 5oF, q #qF
k q 'k+(q+#oFqF), for any h
k+
q #q+ (oF, q #qF
for every h
for at least one h
Always
For small N
Never
For a numerical example, let us set k "k+"0.5, q "0.8, qF"qN +"1 for all h, and c "0.5
(4.18)
and focusing on a specific industry, say h, and assume that oF"0.75 (i.e., pF"4). Then, since k q "0.4(0.5 (1#oHqH) always, condition (4.17a) is satisfied. And, since k+(q #q+)/(q #qF)"k+"0.5(0.75"oF, we have that h4hI . [Notice by Eq. (4.12) that the potential function of any one industry is indepent of the values of parameter o’s for the rest of industries.] Fig. 4 depicts the potential curve of this specific industry h under various values of N. We can see from the figure that when N is sufficiently small, the potential curve is below 1 for all rO0. As N continues to increase, the potential curve gradually shifts upward everywhere except at r"0; and it eventually exceeds 1 in the periphery. For each industry h4hI , the potential curves behave in essentially the same manner as in Fig. 4. In particular, for each industry h4hI , there exits a critical population, NI F, and a critical distance, rJ F, such that the critical potential curve associated with NI F just hits 1 at rJ F. Notice also by Eq. (4.14) that all potential curves belonging to the same industry, say h, have the same negative slope at the edge of the city, of which absolute value is given by hI F, pF[k+(q+#oFqF)!k q ]. Thus far, we have examined the behavior of potential curves separately for each industry. As we can see by Eqs. (4.12) and (4.13), however, the movements of the potential curves of all H industries are synchronized through the common parameter f which increases monotonically with the economy’s population, N. In particular, suppose that the initial value of N is sufficiently small so that the economy’s spatial structure is monocentric. Then, as N increases gradually with time, all industries’ potential curves will gradually move upward everywhere (except at r"0), as explained before. In this context, the first new city (or the first pair of new cities) will emerge when one industry’s potential curve reaches 1 in the periphery for the first time (among all industries). Therefore, the crucial
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Fig. 4. Potential curves for the monocentric system under various N.
question is: Which industry’s potential curve will reach 1 first? One might suspect that an industry will ‘spin out’ from the city sooner if it has either a high elasticity of substitution (meaning lower scale economies in equilibrium) or high transport costs (meaning that it tends to follow the agricultural frontier). And this is indeed the case. In Appendix B we show the following: Given assumptions (4.17a) and (4.17b) suppose further that k q 4k+q+. ¹hen, given any pair of industries, h and g, such that h4hI and g4hI , if it is the case that +oF'oE and qF5qE, or +oF5oE and qF'qE,,
(4.19)
then, in the context of a monocentric spatial system, we have that NI F(NI E, rJ F(rJ E,
and hJ F'hJ E.
(4.20)
In general, given a pair of industries, g and h, if g has either a lower o and a q that is no higher than h, or vice versa, we say that g is of higher order than h. Clearly, there is nothing that says that industries must be rankable by order: an industry could have a low elasticity of susbtitution and high transport costs, or vice versa. But if industries can be ranked, and we imagine gradually increasing the population of a monocentric economy, the potential curve of a lower-order industry will always hump itself above 1 sooner than that of a higher-order industry. Fig. 5 illustrates the impact of oF on the shape of the critical potential curve, XF(r; NI F). (The impact of qF is essentially the same.) Three hypothetical
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Fig. 5. Examples of critical potential curves for the monocentric system.
industries are shown, with o"0.90'o"0.75'o"0.20,
(4.21)
while all other parameters are fixed as follows: k "0.5, k"k"0.1, k"0.3, q "0.8, qF"1 for all h, and c "0.5,
(4.22)
Thus, industry 3 is of the highest order, and industry 1 is of the lowest order. We can readily confirm that these parameters satisfy both conditions in Eqs. (4.17a) and (4.17b) with hI "2. The associated critical potential curves can be depicted as in Fig. 5, which are accompanied with the following characteristic values: NI "0.88(NI "4.36, fI "0.40(fI "1.40, rJ "0.32(rJ "1.10, and hI "5.5'hI "1.90'hI "0.25.
(4.23)
Therefore, among the three industries, the critical potential curve of industry 1 hits 1 first at the nearest location, rJ "0.32, when the economy’s population reaches the smallest critical value, NI "0.88.
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5. Self-organization towards a hierarchical system The discussion in the previous section suggests that the growth of an economy containing many industries of different orders will naturally lead over time to the formation of a hierarchical urban system. Due to the complexity of the issue, we cannot offer any analytical confirmation. We can, however, illustrate the point with numerical simulations. In order to make our simulation analysis tractable, we choose parameters such that & (kF/oF)51, (5.1) F which ensures that the spatial structure of the economy remains monopolar; that is, the economy has a unique highest order city in which all groups of M-goods are produced. Lower order cities can still emerge, however; as we will show, they do so through a series of bifurcations associated with the birth of new cities, relocation and merger of existing cities, and changes of the industrial compositions of cities. Given that the long-run evolutionary process of the spatial system is qualitatively similar as long as the parameters satisfy conditions (4.17a), (4.17b) and (5.1), in the following we present the results of a representative simulation example. Suppose that the economy has three groups of manufactured goods, h "1, 2, 3. We fix parameters at those values specified in Eqs. (4.21) and (4.22), which yield the characteristic values in Eq. (4.23). These parameters satisfy both conditions (4.17a) and (4.17b) as well as the condition (5.1), where hI "2; and the critical potential curves of the three M-industries (associated with the monocentric configuration) can be depicted as in Fig. 5. Since Eqs. (4.21) and (4.22) imply that +o'o and q"q, and +o'o and q"q,, as noted before, Mindustry is of the highest-order, M -industry the second-order, and M -industry the third-order. In this context, we assume that the population size of the economy, N(t), increases gradually over the time, and examines the evolutionary process of the spatial system in the long-run. In the first subsection, we examine in detail what happens when N(t) reaches the first critical value, NI . In the second subsection, then, we examine the long-run evolutionary process of the spatial system.
We can show that if condition (5.1) holds, then when the economy is kept to be monocentric (i.e., of a single-city economy), then the equilibrium real wage of each worker keeps increasing for all N. Our preliminary simulation study suggests that if condition (5.1) is not met, then the initial monopolar economy will be transformed later into a multipolar system, through catastrophic bifurcations, in which new highest-order cities emerge. The analytical proof of this point and a more complete study of multipolar spatial system are left as future tasks.
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5.1. From one city to three Recall from the discussion at the end of the previous section that when N(t) is sufficiently small so that N(t)(NI "0.88, all the three potential curves associated with the monocentric system are strictly less than 1 everywhere outside the city at r"0. Therefore, the monocentric system is in stable equilibrium, with all three groups of M-goods being produced exclusively in the single city at r"0. However, when N(t) reaches the smallest critical value, NI "0.88, the potential curve of M-industry hits 1 at rJ "0.32 (refer to Fig. 4), resulting in the breakdown of the stability of the monocentric system. What happens then? A pair of new cities emerge at rJ and !rJ and, under our dynamics, the population of these new cities grows continuously from zero as N increases gradually beyond NI . Hence, the new cities emerge in the form of a continuous bifurcation. The reason for this continuous bifurcation is that the bifurcation only occurs for industry 1, and one industry (with a small expenditure share, k"0.1) alone does not create sufficient forward and backward linkages to lead to the discontinuous behaviour in our previous works (e.g. Krugman, 1991; Fujita and Mori, 1997). The new cities are completely specialized in industry 1, as can be checked by noting that after the bifurcation the potential curves of M- and M-industries are strictly below 1 at rO0. The size of the industry in each frontier city can grow only gradually as the local demand for that industry’s products increases gradually in association with the expansion of the agricultural frontier. The fact that we have a continuous bifurcation has two implications. First, as N(t) reaches NI , a pair of new cities must be created at rJ and !rJ . Second, during the immediate period after this bifurcation time, each frontier city is so small that it does not have an enough lock-in force to stay at the same location. If each frontier city remained at the same location, then as the agricultural area expands further, the potential curve would have positive gradient on the frontier side of the city, thus violating the location equilibrium condition. Therefore, in order for the spatial system to remain in stable equilibrium, each frontier city will move continuously outwards until it gains a sufficient lock-in force. 5.2. The long-run evolution Given the preliminary analysis above, now we present the long-run evolutionary process of the spatial system. The economy’s population size, N(t), is More precisely, at this moment the monocentric system becomes structurally unstable, implying that any small increase of N beyond NI makes the monocentric system unstable (in the usual sense). This continuity of the initial population size of these lowest-order cities seems to be of a qualitative nature, not of a numerical illusion. This is in contrast with the case of single-industry economy (and hence single-order urban system) studied by Fujita and Mori (1997), in which new cities always emerge as a result of discontinuous (i.e. catastrophic) bifurcations.
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assumed to increase gradually over time, starting with a small value less than NI . All other parameters are fixed at those values specified in (4.21) and (4.22). Fig. 6 summarizes the associated evolutionary process of the spatial system, in which all bifurcations that occur when N(t) increases from NI " 0.88 to 9.79 are presented, while Fig. 7 depicts the changing shapes of potential curves along the evolutionary path. Recall that there are three possible city types — the highest-order, with all three industries; a middle-order with industries 1 and 2; and the lowest-order, with industry 1. (Parameters are chosen for industry 3 such that it never leaves the original central city.) In Fig. 6, given each population N(t), the associated spatial structure of the economy is depicted on the horizontal line. (Symmetry in the spatial structure of the economy means we need only describe its right side.) For example, line 20 in Fig. 6 indicates that when N(t)"5.83, the equilibrium spatial system contains a unique highest-order city at r"0, a unique middleorder city (on the right half of the economy) at r"1.37, and four lowest-order cities respectively at r"0.49, 0.65, 0.98 and 1.70, while the right side A-fringe is at r"1.82. First, in terms of Figs. 6 and 7, we briefly summarize what have been explained in the previous subsection. As is illustrated in Fig. 7a, when N(t) is sufficiently small so that N(t)(NI "0.88, then all three potential curves associated with the monocentric system are strictly less than 1 everywhere outside the city at r"0. Therefore, the monocentric system is in stable equilibrium, with all three groups of M-goods being produced exclusively at the (highest-order) city at r"0. When N(t) reaches the smallest critical value, NI "0.88, as depicted in Fig. 7b, the potential curve of industry 1 hits 1 at r"$rJ "$0.32, resulting in the emergence of a pair of frontier cities there. Since these frontier cities emerge as a result of a continuous bifurcation, during the immediate period after their birth, each frontier city is so small that it does not have an enough lock-in effect to stay at the same location. This point is illustrated in Fig. 7c which indicates that even when N"0.95, the slope of the potential curve of industry 1 at the right edge of the frontier city is zero. Therefore, as indicated by the dotted curve between lines (1) and (2) in Fig. 6, in order for the spatial system to remain in stable spatial equilibrium, the frontier city must move continuously outward from r"0.32 to 0.41 as N(t) increases form 0.88 to 1.23. Only when N reaches 1.23, the frontier city has gathered a sufficient size so that its lock-in effect is strong enough to be able to remain at the same location (refer to Fig. 7d). Next, as N increases further (beyond 1.23) and hence as the frontier area keeps expanding, the frontier potential curve moves upward; it eventually hits 1 at r"0.47 when N(t)"1.37 (Fig. 7e). At this moment, as the result of a (small) catastrophic bifurcation, the existing frontier city at r"0.41 relocates to r"0.47 (line (3) in Fig. 6 and Fig. 7f ). At N(t)"1.61, as a result of a continuous bifurcation, a new frontier city starts growing at r"0.58, while the old frontier
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Fig. 6. The evolutionary process of the hierachical urban system.
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Fig. 7. Changes of potential curves along the evolutionary path.
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Fig. 7. Continued.
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Fig. 7. Continued.
city remains there (line (4) in Fig. 6 and Fig. 7g and h). The old frontier city, however, is soon absorbed by the new frontier city and disappears when N reaches 1.84 (line (5) in Fig. 6). When N reaches 2.01, again as a result of a continuous bifurcation, a new frontier city starts growing at r"0.72, while the old frontier city remains there (line (6) in Fig. 6). Then, after a series of maneuvers, involving a creation of another lowest-order city at r"0.43 (line (7) in Fig. 6), and then through the merger of two existing cities located at r"0.43 and r"0.58 at the new location, r"0.49 ( line (8) in Fig. 6), the lowest-order city at r"0.58 eventually relocates to r"0.49. As can be seen in Fig. 6, this lowest-order city at r"0.49 continues to remain there hereafter. At N"2.42, as the result of another continuous bifurcation, the new frontier city starts growing at r"0.85 (line (9) in Fig. 6 and Fig. 7i), while at N"2.77 the old frontier city (at r"0.72) relocates to r"0.65 (line (10) in Fig. 6). (As can be seen in Fig. 6, this lowest-order city at r"0.65 continues to exist there hereafter.) Unlike the first frontier city, this second frontier city (on the right side of r"0) stays at the same location, although it emerges as a result of a continuous bifurcation. In general, when a new frontier city emerges next to a large (highest- or medium-order) city, it relocates continuously outward for a while. In contrast, if a new frontier city emerges next to a lowest-order city (which is always small), it grows very rapidly and remains at the same location from the beginning.
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Furthermore, at N"2.87, again as a result of a continuous bifurcation, the new frontier city starts growing at r"0.98 (line (11) in Fig. 6). Meanwhile, observe from the past diagrams in Fig. 7 that the potential curve of industry 2 has been steadily moving upward, with kinks at the location of existing lowest-order cities. In particular, as indicated by Fig. 7(k), at N"3.09, the potential curve of industry 2 first reaches 1 at the location of the frontier city (at r"0.98). Hence, a lowest-order frontier city becomes transformed into a middle-order city, producing both M-goods and M-goods. This arises because the industry 2 potential curve is bent and pulled upwards by the demand of M-goods by the workers in the lowest-order cities. In particular, since the frontier city has the largest population among all existing lowest-order cities, the potential curve of industry 2 is most shaply kinked and uplifted toward the frontier city (refer to Fig. 7k). Consequently, the potential curve of industry 2 reaches 1 first at the frontier city, resulting in the transformation of the frontier city to a middle-order city. Since this middle-order city has been created by adding a new industry to the second-order city, it naturally satisfies the hierarchical principle by Christaller (1933). In other words, this middleorder city has been created as a result of mutual entrainment of the location of industries 1 and 2. Fig. 8 demonstrates how strong is the effect of the demand-pull of existing third-order cities on the shape of the potential curves of higher-order industries. This figure is an enlargement of Fig. 7k, with addition of one new curve: the +X(r) of Mono,-curve represents the potential curve of industry 2 associated with the hypothetical monocentric urban system (in which all M-firms are forced to locate in the city at r"0) under the same population, N"3.09. We can see from this figure that the potential curve of industry 2 has been greatly pulled-up by the demand of workers in the existing lowest-order cities (in particular, by the demand of the large lowest-order city located at the frontier r"0.98), while the potential curve of industry 3 has been affected little. Since M-goods have a very low price elasticity (p"1.25), it is still much more profitable for industry 3’s firms to serve the demand of these small cities from the central city than to relocate there. The frontier city at r"0.98 was transformed to a middle-order city as the result of a continuous bifurcation. However, since it grows rapidly, soon it absorbs the nearby lowest-order city at r"0.85 (Fig. 7l). As N increases further (beyond 3.10), the frontier area expands and the new frontier city starts growing at r"1.15 when N reaches 3.43 (line (13) in Fig. 6). However, in comparison with the middle-order city at r"0.98, this new frontier city is too small to
Christaller’s hierarchical principle states that a higher-order central place shall provide its own order of consumption goods as well as all consumption goods which are provided at any lowerorder central places.
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Fig. 8. Potential curves of the 9-city equilibrium at N"3.09.
gather a sufficient lock-in effect there. Hence, as the frontier area further expands, the frontier city keeps relocating (discretely) until it settles down at r"1.37 (lines (13)—(17) in Fig. 6). Observe by Fig. 7m that as N increases further, the potential curve of industry 2 has been gradually moving upward again in the frontier area. In particular, as shown in Fig. 7n, at N"4.70, the potential curve of industry 2 reaches 1 again at the location of the frontier city (at r"1.37). At this moment, the existing middle-order city (at r"0.98) becomes less attractive for industry 2 (than the frontier city). Hence, as a consequence of a large catastrophic bifurcation, the entire industry 2 at r"0.98 moves to the frontier city, degrading the past middle-order city at r"0.98 to a lowest-order city, while upgrading the frontier city to a middle-order city (line (18) in Fig. 6, and Fig. 7n and o). This represents another instance of the mutual entraiment of the location of industries 1 and 2. We can see by lines (18)—(30) in Fig. 6 that after this change in city-order, the spatial structure in the area between the central city (at r"0) and the new middle-order city (at r"1.37) has been stabilized. In the frontier area, however, the spatial structure keeps changing (lines (19)—(25) in Fig. 6, and Fig. 7p—r). In particular, when N reaches 7.60, the frontier city at r"1.92 is upgraded to a middle-order city. Thus, now two middle-order cities exist (in the right side of r"0), with a lowest-order city locating in the middle (line (24) in Fig. 6. and Fig. 7r). After the formation of this middle-order city at r"1.92, the spatial configuration in the frontier area keeps changing as before (through the birth of new cities and relocation of existing cities), while the lowest-order city in the middle of the two middle-order cities moves to the center of them (lines (25)—(30) in Fig. 6 and Fig. 7s and t). The simulation ends when N reaches 9.79, with the emergence of a quite systematic hierarchical urban system a la Christaller (line (30) in Figs. 6 and 7t)).
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Fig. 9. Wage curve of the 15-city equilibrium at N"9.30.
Next, in Fig. 9, the real line depicts the equilibrium (nominal) wage curve when N"9.30, while the broken line represents the corresponding curve associated with the (forced) monocentric system under the same population size. We can see that although the sizes of all middle- and lowest-order cities are much smaller than the size of the central city, the shape of the wage curve is much affected by the presence of these relatively small lower-order cities. In particular, middle-order cities (located at r"1.37 and r"1.92) affect greatly the local wage rates, for each of them satisfies a large proportion of local demands of M-goods and M-goods in its vicinity. The last point above can be understood more clearly by Fig. 10 which depicts the trade patterns of M-goods in the context of 15-city equilibrium at N"9.30. (In this figure, each city number coincides with that in the bottom of Fig. 6.) Hence, city 1 is the highest order city, cities 5 and 7 are the middle-order, and the rest are the lowest order. For each industry h at each consumption location r, the market-share, MSF(r), of MF-goods produced in city k (in terms of I delivered price at r) can be obtained [by using Eq. (2.4) in the present context] as follows:
MSF(r)"¸Fw\NF\e\NF\OF P\PI ¸Fw\NF\e\NF\OF P\PH . I I I H H H
(5.2)
In terms of the city-index, k"1, 2,2, 7, indicated at the bottom of Fig. 6, when N"9.30, for example, the population of each city is as follows: N "6.42, N "0.00691, N "0.0722, N "0.0478, N "0.167, N "0.0007673, N "0.138.
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Fig. 10. The market share curves of M-goods produced in each city in the 15-city equilibrium at N"9.30.
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Setting h"1 and 2, respectively, Eq. (5.2) yields the market share curve of each group of goods produced in each city. [For h"3, by definition, the highestorder city at r"0 has 100% share at every r.] In the middle of diagram (a), for example, the bell-shaped real-line curve shows the market share at each r of M-goods produced by city 5. We can see from diagram (a) that except the smallest two cities, 2 and 6, every city is almost self-sufficient in the production and consumption of M-goods, while these two small cities receive most Mgoods from their neighboring cities. More interestingly, diagram (b) shows that for M-goods, each of cities 1, 5 and 7 has a bell-shaped market share curve covering a large number of neighboring cities. That is, not only the highest-order city, 1, but also both middle-order cities, 5 and 7, have the largest market shares of M-goods in their own markets, and they export significant amounts of M-goods to many neighboring cities. Therefore, the present hierarchical urban system also exhibits a rich spatial structure in terms of M-good trade. Finally, we examine the trends of two welfare measures along the evolutionary path. First, the numerical result shows that the equilibrium real wage of each worker increases almost proportionally with the economy’s population size. Second, the total real rent (TRR) of the economy, defined by
¹RR"2
D
R(r) (GF(r))\IF (p (r))\I dr, F
(5.3)
also increases more than proportionally with the population size. Thus, the economy is in the phase of increasing returns forever. Since we have chosen a parameter set that satisfies condition (5.1), the economy maintains forever monopolar spatial structure, and the population growth of large cities (in particular, that of the highest-order city) continues to propel the engine of the economy’s growth through the expansion of M-good varieties.
6. Conclusion Our study remains preliminary, and a great deal of work is left for the future. First, we need to get a firmer analytical grip on the emergence of hierarchical spatial structures. Second, although this paper focused on the evolutionary process of an urban system associated with a gradual increase of the population size of an 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). Third, in order to make the preliminary model here more realistic, a variety of extensions is needed. In this respect, although the agglomeration forces in the present model were created through product variety in consumption goods, the
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variety in intermediate inputs is equally important in explaining the formation of specialized cities in reality. Another interesting extension is to consider multi-location firms. Eventually, by introducing into such basic models various realistic features (such as national borders, political restrictions, natural geographic features, and transport networks, one will be able to develop a realistic spatial economy model which would be useful in studying the future economic geography of nation or group of nations, and in designing effective regional development policies.
Acknowledgements The authors are grateful to David Bernstein, Vernon Henderson, Tony E. Smith, Takatoshi Tabuchi, Jacques Thisse, Anthony Venables, and an anonymous referee for their valuable comments on earlier versions of the paper.
Appendix A: Derivation of the potential function (4.12) As was explained in the derivation of Eq. (2.30), the original form of the potential function for industry h is given by XF(r)"[wF(r)]NF[w (r)]\NF.
(A.1)
To make this equation more explicit, we obtain several preliminary functions. First, using Eq. (4.4) we have GF,GF(0)"(¸F/kF)\ NF\.
(A.2)
Eqs. (4.1) and (4.8) lead to p (r)"c eI+ O +>O De\O P .
(A.3)
In equilibrium, u*"u+. Hence, by substituting Eqs. (A.2) and (4.8) into Eq. (4.7), and using Eqs. (4.10) and (4.11), we have that
u*"
2(1!e\O D) k q
F
IF NF\
[c eD F IFO >OF ] F IF NF\ \I .
(A.4)
The substitution of Eqs. (A.2), (A.3) and (A.4) into Eq. (2.25) leads to [w (r)]\NF"eNF I O \I+ O + P .
(A.5)
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Next, by setting in Eq. (2.20) that ¸ "¸+, ¸ "0 for all kO1, w "1 and I GF "GF, and using Eqs. (4.1) and (A.2), we have that
kFp D kF¸+ [uF(r)]NF" e\O Q eNF\OF Q \ Q\P ds. e\NF\OF P # ¸F ¸F \D
(A.6)
After the calculation of the integral above by dividing into the three intervals, (!f, 0), (0, r) and (r, f), if we use Eqs. (4.8) and (4.10), we can obtain that for r50,
[wF(r)]NF"
1!k+ 1#k+ eNF\OFPt(r, f )# e\NF\OFP, 2 2
(A.7)
where the function t(r, f ) is defined by Eq. (4.13). Finally, the substitution of Eqs. (A.5) and (A.7) into Eq. (A.1) leads to Eq. (4.12).
Appendix B: The proof that (4.19) implies (4.20) Since the complete description of the proof is very lengthy and tedious, here we provide a brief sketch. [For a complete proof, refer to Appendix 6 of Fujita et al. (1995).] We summarize below the proof in several steps. (a) First, we rewrite the potential function (4.12) in a more convenient form for the present purpose. Taking the negative exponential term outside the brackets of Eq. (4.12), and expressing the potential explicitly as a function of two parameters, r and f, we have that XF(r; f )"e\EFP
1!k+ 1#k+ eNF\OFPtFF(r, f )# , 2 2
(B.1)
where gF,pF[k+q+!k q ]#(pF!1)qF. After the substitution of Eq. (4.13) into Eq. (B.1), we use the following identity:
P e NF\OF\O P"1#[2(pF!1)qF!q ] e NF\OF\O Q ds,
(B.2)
and arrange terms appropriately. Then, we can eventually obtain the new expression of the potential function as follows:
P 1!e\O Q XF(r; f )"e\EFP 1#k (pF!1)qF eNF\OFQ 1! ds . (B.3) 1!e\O D (b) Next, in the context of Section 4.2, take any industry h4hI , and let fI F be the critical fringe distance associated with the critical population, NI F. Then, the
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critical potential curve, XF(r; fI F), is tangent to the horizontal line having the height 1 at the critical distance rJ F (refer to Fig. 4). Hence, the pair, (rJ F, fI F), must satisfy the following two relations: XF(rJ F; fI F)"1,
(B.4)
*XF(rJ F; fI F)/*r"0,
(B.5)
and
where rJ F'0 and fI F'0. For convenience, let us define v (s),1!e\O Q
gF and vF(s),1! e\ NF\OF\EF Q. k (pF!1)qF
(B.6)
By solving (B.4) for the term, 1!e\O DI F, we have that
1!e\O DI F"
PJ F
eNF\OFQv (s) ds
while the condition (B.5) yields that
PJ F
eNF\OFQvF(s) ds,
(B.7)
1!e\O DI F"v (rJ F)/vF(rJ F).
(B.8)
Hence, if we define
P P QF(r), eNF\OFQvF(s) ds vF(r)! eNF\OFQv (s) ds v (r) P v (s) vF(s) vF(r) " eNF\OFQ ! ds, vF(r) v (s) v (r) then, by Eqs. (B.7) and (B.8), rJ F must be such that
QF(rJ F)"0,
(B.9)
(B.10)
and rJ F'0. It is not difficult to show that Eq. (B.10) has a unique positive solution. (c) To be explicit about the parameters involved, let us rewrite (B.9) as follows:
P v (s) vF(s) vF(r) Q(r; k , q , D¹, pF, qF), eNF\OFQ ! ds, vF(r) v (s) v (r) where D¹,k+q+!k q " kFqF!k q , F
gF,pFD¹#(pF!1)qF,
(B.11)
(B.12)
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Then, as noted above, there exists a unique positive rJ F,rJ (k , q , D¹, pF,qF) such that Q(rJ F; k ,q , D¹, pF, qF)"0.
(B.13)
(d) Next, focusing on the parameter pF[,1/(1!oF)], we take the total derivative of Eq. (B.13) with respect to rJ F and pF such that drJ F *Q/*pF "! . dpF *Q/*rJ F
(B.14)
It is not difficult to show that *Q/*rJ F is always positive. We can also show that +D¹50N*Q/*pF'0,, and hence we can conclude by Eq. (B.14) that drJ F *rJ (k , q , D¹, pF, qF) , (0. D¹50N *pF dpF
(B.15)
(e) Next, if we substitute rJ F,rJ (kF, q , D¹, pF, qF) into Eq. (B.8), fI F can be uniquely determined, which is denoted by fI (k , q , D¹, pF, qF). It is not difficult to show by using Eq. (B.15) that dfI F *fI (k , q , D¹, pF, qF) D¹50N , (0. dpF *pF
(B.16)
If we replace f in Eq. (4.10) by fI F,fI (k , q , D¹, pF, qF), then NI F, NI (k , q , D¹, pF, qF) is uniquely determined. Then, since the relation (4.10) determines f as an increasing function of N, we can conclude by Eq. (B.16) that D¹50N
dNI F *NI (k , q , D¹, pF, qF) , (0. *pF dpF
(B.17)
(f ) Next, although D¹(, kFqF!k q ) contains each qF, here we treat D¹ F as an independent parameter. Then, in similar manners as in (e) above, we can show that
drJ F D¹50N dqF D
*rJ (k , q , D¹, pF, qF) , (0, *qF 2
(B.18)
dfI F *fI (k , q , D¹, pF, qF) D¹50N , (0, *qF dqF D 2
(B.19)
dNI F D¹50N dqF D
*NI (k , q , D¹, pF, qF) , (0, *qF 2
(B.20)
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(g) Now, we fix all parameters of the model (and hence, the value of D¹ is also fixed), and assume that the assumption (4.17) holds and that D¹50. In this context, take any h4hI and g4hI , and suppose that oF'oE (i.e., pF'pE)
and qF5qE.
(B.21)
Then, it follows that rJ F"rJ (k , q , D¹, pF, qF)(rJ (k , q , D¹, pE, qF) by (B.17) 4rJ (k , q , D¹, pE, qE)
by (B.18)
"rJ E. We can show similarly that Eq. (B.21) also implies that fI F(fI E and NI F(NI E. In the same way, we can also show that if oF5oE (i.e., pF5pE) and qF'qE, then rJ F(rJ E, fI F(fI E, and NI F(NI E. Finally, it follows immediately by definition that Eq. (4.19) implies hI F'hI E. Therefore, sumarizing the results, we can conclude that Eq. (4.19) implies Eq. (4.20).
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