Regional Specialization, Urban Hierarchy, and Commuting Costs

Transcription

1 Regional Specialization, Urban Hierarchy, and Commuting Costs Takatoshi Tabuchi and Jacques-François Thisse 6 January 2005 Abstract We consider an economic geography model of a new genre: all firms and workers are mobile and their agglomeration within a city generates rising urban costs through competition on a land market. When commuting costs are low (high), the industry tends to be agglomerated (dispersed). With two sectors, the same tendencies prevail for extreme commuting cost values, but richer patterns arise for intermediate values. When one good is perfectly mobile, the corresponding industry is partially dispersed and the other industry is agglomerated, thus showing regional specialization. When one sector supplies a nontradeable consumption good, this sector is more agglomerated than the other. The corresponding equilibrium involves an urban hierarchy: a larger array of varieties of each good is produced within the same city. Keywords: interregional mobility, intersectional mobility, agglomeration, commuting costs. J.E.L. Classification: F12, F16, J60, L13, R12. We wish to thank K. Behrens, M. Brülhart, E. Glaeser, V. Henderson, S. Mun, G. Ottaviano, P. Wang, and the audience of a workshop held at University of British Columbia for helpful comments. This research was supported by the Japanese Ministry of Eduction and Science (Grant-in-Aid for Science Research 09CE2002 and ) and by the Ministère de l éducation, de la recherche et de la formation (Communauté française de Belgique), Convention 00/ Faculty of Economics, University of Tokyo (Japan) CORE, Université catholique de Louvain (Belgium), CERAS, Ecole nationale des ponts et chaussées (France), INRA (France) and CEPR. 1

2 1 Introduction The main distinctive feature of the modern space-economy is the existence of urban systems formed by large and diversified cities as well as by small and specialized cities, all involved in trade (Henderson, 1988). What are the main economic factors explaining such a pattern of production and consumption is the main goal of the so-called new economic geography (in short NEG). The canonical model of this field, developed by Krugman (1991) and known as the core-periphery model, is such that the spatial concentration of the industry is produced by the interplay between forces pushing in opposite directions. The agglomeration force finds its origin in the presence of increasing returns, consumers love for variety, and the mobility of industrial workers, whereas the origin of the dispersion force lies in positive transport costs and the immobility of farmers. Under monopolistic competition in the industry and perfect competition in agriculture, Krugman shows that industrial agglomeration arises when transport costs are sufficiently low. The degree of urbanization that characterizes our economies is such that housing and commuting costs, which we call urban costs, standforalarge (in fact, the main) share in consumers expenditure. In developed countries, they account for more than one third of individual incomes and may even reach one half of them. To take an example, in France the share of housing and transport costs have increased from 23.4 to 42% in consumers budget from 1960 to 2000 (Rignols, 2002). In addition, both urban economics and empirical evidence show that such costs rapidly increase with city size. For example, French data show that in 2000 urban costs stand for about 45% of individual incomes in large cities, but for 34% in the small ones. Similar data couldbepresentedforothercountriesandexplainwhyweconsiderurban costs as one of the main forces pushing toward the geographical dispersion of activities in modern economies. 1 The primary purpose of this paper is, therefore, to show how preference for variety on the demand side and increasing returns on the supply side interact with urban costs to shape the space-economy. Thus, our thought experiment will be on commuting costs (per unit of distance) because they are critical in understanding how modern space-economies are organized, especially when transport costs have reached the low level that now prevails 1 Mano and Otsuka (2000) empirically demonstrate that urban costs have acted as a strong dispersion force in the Japanese manufacturing activities. 2

3 (Glaeser and Kohlhane, 2004). To be sure, unit commuting costs have also vastly decreased during the 20th century, due mainly to the development of rapid transits (Mills and Hamilton, 1994, pp.21-30) and the growing adoption of individual cars (Glaeser and Kahn, 2004), but they remain substantial when we account for the value of time spent in commuting. 2 In order to achieve our goal, we use the framework of NEG but modify it in several important respects with the aim of embodying more relevant forces in the process of city formation and intercity trade. Following a well-established tradition in regional economics (Borts and Stein, 1964; Henderson, 1988), we disregard the agricultural sector and assume that all workers and firms are mobile. As in Helpman (1998) and Tabuchi (1998), the dispersion force rests on urban costs that rise with the size of the population established within the same city. Such a setting, which combines the mobility of industry and the existence of urban costs, strikes us as fitting well modern economies. In this respect, we want to stress the fact that one of the distinctive features of the standard core-periphery model is the role ascribed to the agricultural sector (or, more generally, to any land-intensive sector). 3 First, farmers are not allowed to move between regions and sectors, whereas such moves are at the origin of the urbanization of industrialized countries. Second, even if agriculture is just a label for land-intensive activities, the share of such sectors has sharply declined in modern economies (for example, the share of agricultural employment in the United States has decreased from 12.2% in 1950 to 1.8% in 2000). Third, in Krugman s model, the immobile sector must be sufficiently large for dispersion to arise as an equilibrium outcome (otherwise there is always agglomeration). Thus, if the immobile sector is crucial in explaining the process of urbanization, it plays a somewhat disproportionate role for the core-periphery model to describe well the working of urbanized and developed economies. One of the most exciting aspects of NEG is that it has triggered a large number of empirical studies, which aim at testing the main conclusions of 2 According to Mokyr (2002), the commuting costs in terms of time in the United States can be roughly estimated at about $356 billion in However they are measured, the importance of urban costs is not disputable. 3 **According to Pines in his review of the book by Fujita, Krugman and Venables, which has been published in the Journal of Economic Geography in 2001, the coreperiphery model is anachronical because it neglects internal urban structure and land markets. The same point is made by Anas (2004) who addresses an issue different from ours.** 3

4 the core-periphery model in the presence of several industries (see the survey by Head and Mayer, 2004). Even though NEG focuses on a single industry, 4 there is no reason to believe that what is true for one industry still holds in the case of an economy involving several of them, simply because a result holding for I =1industry need not to hold for larger value of I. In addition, the one-industry model is unable to cope with the well-documented fact that different industries typically display different spatial patterns. The secondary purpose of this paper is thus to study the location of several industries and to investigate what the prediction of the one-industry model becomes in this context. It is important to know whether a result valid for one industry holds true in the case of several industries, for otherwise we run the risk of drawing policy implications that are not founded. To this end, we consider two industries that interact in consumers demands through the relative size of their range of varieties, and differ in the cost of shipping their output and show how such a difference may affect their location. Though restrictive, this modeling strategy is in the spirit of new trade theory, while being at thesametimeverymuchinthetraditionofweber s(1909)andhoover s (1948) location theory. Recall that, for these authors, different transport costs explain why firms belonging to different industries obey different locational patterns. 5 We show that the two-industry model generates results that differ from thoseweobtainwithasingleindustry in that our setting yields a much broader array of results. Although our results are similar to those obtained in the one-industry model in the two polar cases of large or small commuting costs, by allowing for several industries to seek location, we open the door to new and richer spatial configurations for intermediate values of commuting costs as the industrial composition of regions is now endogenous. Among other things, extending our basic model to two industries allows us to get rid of some of the exotica of NEG, such as catastrophic changes in the urban system. Specifically, our analysis suggests that other outcomes such as the permanence of urban system as well as gradual shifts in cities industrial composition are more likely to be observed. Such findings shed new light on the empirical evidence put forward by Eaton and Eckstein (1997) and Davis and Weinstein (2002). 4 For noticeable exceptions, see Fujita et al. (1999a) and Venables (1999). 5 In passing, note that our paper may then be viewed as an attempt to connect classical location theory and new economic geography. 4

5 Our two-industry setting also allows us to study some new issues, such as regional specialization and urban hierarchy, which cannot be properly addressed in a model involving a single mobile sector. This is an important point because, as said in the foregoing, one of the most striking features of the space-economy istheexistenceofcitiesofdifferent sizes showing contrasted internal production patterns. More precisely, we observe large and small cities together with diversified and specialized cities, each producing some goods for local consumption as well as other goods whose consumption is spread over several cities (Henderson, 1988). This problem is very complex, however, because workers must now choose the industry in which they work as well as the place where they live. Thisinturnimpliesthatthesizeofeach industry, not just the size of each city, is endogenous. In order to make our model comparable to the existing literature, we begin with the study of the location of one industry. The general pattern of one industry against declining commuting costs is similar to the one obtained in the core-periphery model when transport costs decrease. In the case of two industries, commodity-specific transport costs may lead to equilibria in which different sectors display different spatial configurations, ranging from full dispersion to full agglomeration, as in the foregoing case. Unfortunately, allowing for both regional and sectoral mobility of workers renders the analysis especially complex and prevents a full characterization of the equilibrium configurations. This leads us to investigate the special, but meaningful, case of an industry with negligible (zero) transport costs, whereas the other faces positive costs. We show that the imperfectly mobile good sector is agglomerated whereas the footloose sector is partially dispersed. To us, this is indication of regional specialization, even though full specialization never arises as a stable equilibrium outcome. To evaluate the robustness of this result, we also investigate the case in which the two industries sell products characterized by different degrees of substitutability but identical transport costs. In order to gain further insights, we consider another extreme case in which one good is perfectly mobile while the other good is nontradeable. Indeed, despite strong decreases in transport and communication costs, many consumer services are supplied locally often in large cities (Daniels, 1993). In such a context, the market outcome involves an urban hierarchy in that one city supplies more varieties of each good than the other. In addition, our equilibrium pattern also implies that the large city supplies a relatively larger share of the nontradeable good, which allows it to attract a larger share of the perfectly mobile good industry. This can be viewed as a comparative 5

6 advantage à la Ricardo, the intensity of which is endogenous instead of being given a priori. Finally, although we assume no technological linkage between industries, the market equilibrium involves diversified cities in that each one attracts firms belonging to both industries. However, despite the fact that our setting is a priori symmetric, cities have different degrees of diversity as the relative share of each industry varies across regions. The remainder of the paper is organized as follows. The model with one industry is introduced in section 2. Instead of using the Dixit-Stiglitz-iceberg framework, we retain the alternative model developed by Ottaviano et al. (2002)becauseitleadstoanalyticalresultsthatmaybederivedwithpaper and pencil. The case of two industries is considered in section 3, whereas section 4 concludes. Related literature. Even though the pioneering works of Christaller (1933) and Lösch (1940) are now fairly old and despite the importance of the questions these authors address for trade and development, it is only recently that such issues have attracted attention in the economics profession. This has been accomplished in the context of systems of cities (see Abdel-Rahman and Anas, 2004, for a detailed survey of the existing literature). However, the corresponding models often fail to stress their trade and factor mobility implications. One prominent exception is the work of Henderson (1974, 1988), who considers a setting in which returns to scale are constant at the firm s level but increasing in the aggregate once they are located together. Workers consume land and face positive commuting costs within each city. Put together, these two forces imply that each city has a positive and finite size. Because of increasing returns, it is efficient for each city to be specialized in the production of a particular good. Accordingly, Henderson ends up with an urban system having cities of different sizes and types according to the good they produce. However, shipping a city s output is costless in his setting, which amounts to treating cities like floating islands. Hence, we do not know where cities are on the map, nor does Henderson provide an explanation for a system involving both diversified and specialized cities. Our model is also connected to that developed by Fujita and Krugman (1995) as well as by Fujita et al. (1999a). These authors use monopolistic competition (but of the Dixit-Stiglitz genre) to explain the emergence and location of cities in a continuous space-economy; the industrial good is produced in cities that have no spatial extension and the agricultural good in the country by means of land and labor. It should be clear that those models do not allow for the study of the question we address in this paper: transport 6

7 costs between cities/regions are zero in Henderson-like models, whereas there are no urban costs in the model developed by Fujita et al. However,asthe intercity distance is assumed to be given here, our setting and the one proposed by Fujita et al. may be viewed as being complementary. They are also complementary in that they seem to fit well different stages of the process of economic development. Note also that our approach may be interpreted as an extension of the literature initiated by Rosen (1979) and Roback (1982) in which differentials across cities are explained by differences in the amount of urban amenities. Unlike this literature, however, we endogenize the amenity index by allowing the number of varieties of the nontradeable good produced in a city to be variable. 2 The one-industry economy 2.1 The model Consider an economy formed by a continuum of mobile workers whose mass is 1, by two regions, denoted H and F, and by three goods. The first good is homogenous and available as an endowment; it can be shipped costlessly between the two regions and is chosen as the numéraire. The second one is a differentiated good made available under the form of a continuum of varieties. The utility function of a worker definedonthesetwogoodsisas follows: 6 Z N U(q 0 ; q(v),v [0,N]) = α q(v)dv β γ Z N [q(v)] 2 dv γ Z N 2 q(v)dv + q 0 (1) 2N 0 in which N stands for the number of varieties of the differentiated good, q(v) for the quantity of variety v and q 0 for the quantity of the numéraire. 7 6 This utility is the one proposed by Vives (1990), which slightly differs from that used by Ottaviano et al. (2002). It has been chosen because the coefficients a, b and c defined below are independent of the number of varieties supplied by a sector, which is variable in the two-sector model. 7 The third term in the RHS of (1) is divided by N, as in the case where all varieties of the differentiated good are consumed (q(j) > 0 for all j). When this does not hold, N 7

8 Both α and β are positive, whereas γ is positive (resp., negative) if varieties are substitutes (resp., complements). For the utility to be quasi-concave, we assume β > γ. Because a quasilinear utility abstracts from income effect in the demand for the differentiated good, our modeling strategy has a partial equilibrium flavor. However, it does not remove the interaction between product and labor markets, thus allowing us to develop a full-fledged model of agglomeration formation. Note also that our assumption on preferences does not impose any restriction on the share of the industry in consumers expenditure. We will see that our sole assumption is that the equilibrium consumption of the homogenous good is strictly positive. The third good is land (or housing). In what follows, we consider a onedimensional continuous space in which each location has a unit amount of land and where the opportunity cost of land is zero. Regions have an internal structure in that each of them involves a linear and monocentric city whose size is variable. As in standard urban economics, the city center is a central business district (CBD) in which all firmslocateoncetheyhavechosen to set up in the corresponding region. 8 Likewise, firms are assumed not to consume land whereas workers, when they choose to live in a certain region, are urban residents who consume land and commute to the regional CBD in which jobs and varieties of the differentiated good are available. Hence, unlike Krugman (1991) but like Tabuchi (1998), each agglomeration has a spatial extension that imposes commuting and land costs on the corresponding workers. 9 For simplicity, workers consume a fixed lot size normalized to unity, 10 while commuting costs are linear in distance, the commuting cost per unit of distance being given by θ > 0 units of the numéraire. Because local varieties are supplied at the CBD whereas interregional trade flows go from one CBD to the other, shipping costs are naturally included in workers commuting costs. Finally, the two CBDs are two remote points of the location space. Shipping one unit of each variety costs τ units of the numéraire so that both cities/regions are physically separated as regions are in NEG. Assume that λ r workers live in region r = H, F, whereλ r denotes the must be replaced by the measure of the support of the function q(j). The same restriction applies to (3) definedinsection3. 8 See Fujita and Thisse (2002) for various arguments explaining why firms want to be agglomerated in a CBD. 9 To be precise, Helpman (1998) assumes a fixed supply of housing that has no spatial extension so that his model does not include commuting costs. 10 Hence, there is no income effect in the demand for housing. 8

9 share of workers residing in region r with λ H + λ F =1. At the land market equilibrium, all workers in region r are equally distributed around the r-cbd and, since they earn the same wage, reach the same utility level. Furthermore, since each of them consumes one unit of land, the equal utility condition within region r together with the assumption of zero opportunity land cost implies that the equilibrium land rent at distance x λ r /2 from the r-cbd is given by µ λr R(x) =θ 2 x When the land rents go to absentee landlords, individual urban costs, defined bycommutingcostpluslandrentateachdistancex, aregivenbyθλ r /2. In order to close the model, we assume that land is publicly owned and that the aggregate land rent is equally redistributed among the r-city workers. Consequently, the individual urban costs after redistribution are equal to θλ r /4. Hence, the budget constraint of a worker residing in city r is given by Z N 0 p r (v)q r (v)dv + θ 4 λ r + q 0 = q 0 + w r where p r (v) is the consumer price of variety v in region r, q r (v) the individual demand for variety v of a worker residing in region r, andw r thewageshe earns in this region. The initial endowment q 0 is supposed to be sufficiently large for its equilibrium consumption to be positive for each individual. Let a α/β, b 1/(β γ) and c γ/[β(β γ)], withb>c. The individual demand q r (v) is given by q r (v) =a bp r (v)+c P r N where the price index P r in region r is defined as follows: P r Z N 0 p r (y)dy Each variety is supplied by a single firm producing under increasing returns so that N is also the number of firms. The common fixed requirement of labor is denoted by φ > 0, whereas the marginal requirement, as in Forslid and Ottaviano (2003), is expressed in terms of the numéraire. Without loss of generality, this constant can be normalized to zero by rescaling the intercept 9

10 of the inverse demand function of each variety. The total number of firms N is thus equal to 1/φ. Markets are segmented, that is, each firm is able to set a price specific to the market in which its output is sold. This assumption is supported by empirical investigations (Greenhut, 1981; Haskel and Wolf, 2001) and can be given theoretical justification (Thisse and Vives, 1988). Because firms located in region r are symmetric, they charge the same price in equilibrium. Hence, the profits made by a firm located in region r = H, F may be written as follows: Π r = p rr q r (p rr )λ r +(p rs τ)q s (p rs )λ s φw r where p rr and p rs are respectively the prices quoted by a firm located in r and selling in r and in s 6= r. The labor market clearing condition in region r implies that any change in the population of workers in this region must be accompanied by a corresponding change in the number of firms. As in Krugman (1991), entry and exit are free so that profits are zero in equilibrium. The equilibrium wage w r in region r is then obtained from the zero-profit condition evaluated at the equilibrium prices. These two assumptions have a major implication for our analysis: it is sufficient to describe the migration of workers because the supply of entrepreneurs in each region is supposed to be large enough for the zero-profit condition to be satisfied regardless of the number of workers. Prices and wages are determined instantaneously once workers have made their locational decisions. Stated differently, equilibrium prices and wages depend on the interregional distribution of workers (λ r ). This distribution depends itself on the transport costs τ and the commuting costs θ. Itisa well-documented fact that both types of costs have dramatically decreased since the beginning of the Industrial Revolution (Bairoch, 1985), although the commuting costs are often neglected in the general economics literature. As argued in the introduction, we focus instead on commuting costs because transport costs of many commodities are now very low. 2.2 The spatial pattern of the industry For any given value of λ r, maximizing Π r with respect to p rr and p rs yields the equilibrium prices as follows: p rr = 2a + τcλ s 2(2b c) 10 p rs = p ss + τ 2 s 6= r

11 Plugging the resulting expressions into Π r and solving for w r gives the equilibrium wage prevailing in region r: w r = bn[(p rr) 2 λ r +(p rs τ) 2 λ s ] Thus, equilibrium prices and wages depend on the distribution of firms between regions, but not on the total number of firms. Note that two-way trade occurs for any λ r if τ < τ trade 2a 2b c The welfare of a worker in region r is given by her indirect utility V r evaluated at the prevailing equilibrium prices and wages, which depend themselves on the distribution λ r of workers. A spatial equilibrium thus arises when no worker has a unilateral incentive to move from her location (Nash). More precisely, such an equilibrium arises at 0 < λ r < 1 when V (λ r) V r (λ r) V s (λ r)=0,oratλ r =1when V (1) 0, oratλ r =0when V (0) 0. We follow a well-established tradition in migration modeling by assuming that workers are attracted (resp., repelled) by regions having a utility higher (resp., lower) than the average utility (Fujita et al., 1999b). It is also assumed that the power of attraction of such a region increases with its size because this makes it more visible. Formally, we study stability through the replicator dynamics defined as follows: λ r = λ r [V r (λ r ) λ r V r (λ r ) (1 λ r ) V s (λ r )] = λ r (1 λ r ) V (λ r ) A spatial equilibrium is asymptotically stable if, for any marginal deviation of the population distribution from the equilibrium, the equation of motion above brings the distribution of skilled workers back to the original one. When some skilled workers move from one region to the other, we assume that local labor markets adjust instantaneously. Wages are then adjusted for each firm to earn zero profits. For simplicity, we assume that individual consumption of the numéraire is positive during the adjustment process. This assumption is made to capture the idea that individuals need both types of goods. It is then readily verified that V (λ r )=K (1/2 λ r ) (2) 11

12 where K b(6b2 6bc + c 2 )τ 2 4ab (3b c) τ +(2b c) 2 φθ 2(2b c) 2 φ is a constant. It follows immediately from (2) that λ r =1/2isalways a spatial equilibrium. Furthermore, the symmetric equilibrium is stable if K > 0. Otherwise, the industry is agglomerated into a single region, with λ r =0, 1 according to the initial distribution. It is straightforward to compute the corresponding threshold at which the spatial structure changes: θ = bτ [4a (3b c) (6b2 6bc + c 2 )τ] (2b c) 2 φ It is readily verified that θ > 0 as long as τ does not exceed τ trade. The following result thus holds: Proposition 1 If θ > θ, then the symmetric configuration is the only stable spatial equilibrium; if θ < θ, there are two stable spatial equilibria corresponding to the agglomerated configurations; if θ = θ,thenanyconfiguration is a spatial equilibrium. This result exhibits the evolutionary process from dispersion to agglomeration as θ decreases. We have here a setting that bears some resemblance with the core-periphery model developed by Krugman (1991) and by Ottaviano et al. (2002) in that improvements in intraregional (instead of interregional) transport technologies induce a catastrophic transition from dispersion to agglomeration. Even though our model does not allow for an income effect on commuting time (because the labor supply is fixed), the observed increase value of time associated with higher individual incomes amounts here to a larger value of θ. Proposition 1 thus provides a rationale for the relative deconcentration of activities observed in highly industrialized countries (Vining and Kontuly, 1978; Elliott, 1997). Indeed, higher commuting costs are likely to yield a dispersed configuration for the industry. 12

13 3 The two-industry economy 3.1 The model We now come to the case of I > 1 differentiated goods. Assume that all varieties of each good are consumed. The utility (1) may then be extended as follows: U(q 0 ; q i (v),v [0,N i ],i=1,...,i)= (β γ)n i 2 P I j=1 N j Z Ni 0 [q i (v)] 2 dv IX α i=1 γ 2 P I j=1 N j Z Ni 0 µz Ni 0 q i (v)dv 2 q i (v)dv + q 0 (3) where q i (v) is the quantity of variety v of good i =1,...,I.IntheRHSof(3), the second term in each bracketed term is weighted by the relative size of its sector N i / P j N j to capture the idea that, when they are several industries, a good with a small range of varieties has less impact on the consumer wellbeing than a good with a large array of varieties. This formulation allows us to account for some interaction between the I goods,whichisinaccordance withtheconceptofloveforvariety.letusmakethispointmoreprecise. Since β > γ, (3) encapsulates both a preference for diversity between the I goodsaswellasapreference for variety across varieties of the same good. Assume, first, that an individual consumes a given mass of Q i units of good i =1,...,I and that consumption of good i is uniform and equal to Q i /x i on [0,x i ] and zero on (x i,n i ]. Evaluating (3) at this consumption pattern yields U = = " IX α i=1 Z xi 0 Q i dv (β γ)x i x i 2 P I γ 2 P I j=1 x j " IX αq i i=1 µz xi 0 βq2 i 2 P I j=1 x j j=1 x j Z xi 0 µ Qi x i 2 dv # 2 Q i dv + q 0 x i # + q 0 (4) which is strictly increasing in x j for each j. Hence, x j = N j must hold for each good j, that is, each consumer prefers to consume all the varieties 13

14 of each of the N available goods. Assume now that the total consumption Q = P i Q i is fixed. Then, maximizing (4) with respect to Q i subject to Q = P i Q i yields Q i = Q/I (i =1,...,I). In words, each good is equally consumed and, hence, each variety of any good is equally consumed. From now on, we assume that there are two sectors (I = 2). There are several ways to make the two industries asymmetric: γ 1 6= γ 2 (different degrees of product differentiation), φ 1 6= φ 2 (different levels of fixed cost), and τ 1 6= τ 2 (different transport costs). Since we focus on spatial parameters and variables, it is worth investigating the impact of different transport costs with τ 1 < τ 2. Let λ ir be the share of the labor force established in region r and working for sector i. Because the sum of the λ ir is one, there are three endogenous variables to be considered: λ 1r, λ 2r and λ 1s. Since workers can change both places and jobs, the labor market clearing conditions imply that N i = λ ir + λ is φ i =1, 2 so that the number of varieties produced in each industry is now endogenous. An individual working in sector i = 1, 2 and residing in region r = H, F earns awageequaltow ir and maximizes (3) under her budget constraint: Z N1 0 p 1r (v)q 1r (v)dv + Z N2 0 p 2r (v)q 2r (v)dv + θ 4 (λ 1r + λ 2r )+q 0 = w ir + q 0 where the individual demand for the variety v of good i =1, 2 by a worker located in region r = H, F is: q ir (v) = 1 a bp ir (v)+c P ir (5) φn i N i the price index P ir being defined as follows: P ir Z Ni 0 p ir (y)dy It follows from (5) that the demand q ir (v) is affected by P ir,whichconsistsof prices of varieties belonging to industry i in the two regions. As a result, the cross-elasticity of demand between any two varieties belonging to different industries is zero, but is positive when they belong to the same industry. 14

15 This is consistent with the classical definition of an industry given by Triffin (1940). The demand for each variety of good i is negatively affected by the share of the corresponding industry because consumers distribute their consumption over a larger range of varieties. As seen in the foregoing, one of the main differences with the one-sector model is that the number of varieties N i produced in industry i is endogenous, being determined by the sectoral mobility of workers. Equilibrium prices and wages are now given by p irr = 2a + τ icλ is 2(2b c) w ir = p irs = p iss + τ i 2 with s 6= r (6) bn (p λ ir + λ irr ) 2 (λ 1r + λ 2r )+(p irs τ i ) 2 (λ 1s + λ 2s ) is which are independent of the number of firms operating in each sector. However, equilibrium prices and wages, hence the indirect utility, depend on the distribution of the labor force across sectors as well as on the interregional distribution of workers within each industry. This suffices to show that the analysis is likely to be much harder than in the one-sector model considered above. The welfare of an individual working in sector i in region r is given by her indirect utility V ir evaluated at the foregoing equilibrium prices and wages. A spatial-sectoral equilibrium thus arises when no worker has an incentive to change place and/or to switch job. Formally, this means that V exists such that V ir = V if λ ir > 0 V ir V if λ ir =0 The value of w ir reveals that the expression for V ir is highly nonlinear so that the set of equilibria is very hard to characterize. Yet, because of sectoral mobility, equilibrium wages must be equal across sectors in each region so that we may write w 1r = w 2r = w r (r = H, F). Note also that the expression of w ir shows that goods 1 and 2 are to be supplied in equilibrium because λ ir + λ is must be positive for i =1, 2. However, we do not know whether or not λ ir is uniquely determined in equilibrium. As in the one-sector model, we propose to study stability through the replicator dynamics, thus treating the choice of a job or of a location in a 15

16 symmetric way: Ã λ ir = λ ir V ir X X i=1,2 r=h,f λ ir V ir! f ir (7) This means that workers out-migrate (resp., in-migrate) from sector i in region r when her utility V ir is lower (resp., higher) than the intersectional and interregional average utility. Among other things, this implies that intersectional mobility and interregional mobility are determined simultaneously by allowing them to interact in a symmetric manner. In addition, a change in the population of industry i-workers in one region is no longer accompanied by a corresponding change in the number of firms in that industry. Finally, all spatial-sectoral equilibria are steady-states of (7). It is, indeed, easy to show that f ir =0when V ir is equal to the average utility for i =1, 2 and r = H, F, whereasλ ir =0when working in sector i in region r yields a utility level lower than the average utility. By solving V 1r = V 1s = V 2r = V 2s simultaneously, we can show that there exists at least one equilibrium given by where bµ 1 µ λ 1r λ 2r λ 1s λ 2s = µ µ1 µ µ 1 1 µ a 2 16a(b c)τ 1 +(8b 2 12bc +5c 2 )τ 2 1 [1/2, 1) 32a 2 16a(b c)(τ 1 + τ 2 )+(8b 2 12bc +5c 2 )(τ τ 2 2) This equilibrium is interior. However, there are also equilibria involving boundary and interior values, the set of which is much richer than what we get in the one-industry case. 3.2 Agglomerated and dispersed equilibria As seen above, there exists a fully dispersed equilibrium (8) for all τ 1 < τ 2, which involves a symmetric distribution of firms between the two regions. However, the labor force is not evenly distributed between the two industries within each region: the low transport cost industry attracts a larger share. As prices are lower in this sector, consumers are induced to consume more varieties, thus requiring more workers (when τ 1 = τ 2,wehavebµ 1 =1/2). 16 (8)

17 This equilibrium becomes unstable at the symmetry breaking point, which may be obtained by computing the Jacobian of (7) evaluated at (8). More precisely, the characteristic polynomial of the Jacobian of (7) may be written as follows: ϕ (x) =x 3 + C 1 (θ) x 2 + C 2 (θ) x + C 3 (θ) where x stands for an eigenvalue. We know from Routh s theorem that the system is asymptotically stable if the following conditions hold (Samuelson, 1947, Appendix B): C 1 (θ) > 0 C 3 (θ) > 0 C 1 (θ) C 2 (θ) >C 3 (θ) (9) This yields (at most) three lower bounds on θ. Clearly, the largest bound θ 2 is the symmetry breaking point. Indeed, the symmetric equilibrium ceases to be stable when θ < θ 2. Here too, agglomeration may be an equilibrium µ λ 1r λ 2r λ 1s λ 2s µ 1 0 = (10) When both industries are agglomerated within the same region, the work force is equally split between the two industries because λ 1r = λ 2r is the unique solution of the equation V 1r = V 2r in which we have plugged λ 1s = λ 2s =0. This is because the differences in transport costs no longer matter once the two industries are together. This is an equilibrium if and only if the following three conditions V 1r V 1s V 2r V 2s V 1r = V 2r (11) hold for r 6= s. The stability of this equilibrium is guaranteed because we also have (V 1r V 2r ) λ1r +λ 2r =1 λ 1r (λ1r,λ 2r )=(1/2,1/2) = 8a2 b (2b c) 2 φ < 0 Let θ 1 be the sustain point associated with full agglomeration, which is the unique solution of V 1r = V 1s evaluated at (10). At θ = θ 1, the three conditions (11) are satisfied because V 1r = V 1s and V 1r = V 2r hold when evaluated at (10), whereas V 2r >V 2s is readily shown to hold. Hence, full agglomeration is a stable equilibrium when commuting costs are sufficiently low (θ < θ 1). The foregoing discussion may be summarized in the following proposition. 17

18 Proposition 2 If θ > θ 2, then full dispersion is a stable equilibrium. 0 < θ θ 1, then full agglomeration is a stable equilibrium. If This is identical to Proposition 1 when θ 1 = θ 2. In the case of two industries, however, the interval (θ 1, θ 2) is typically nonempty and we expect some form of partial agglomeration to arise as an equilibrium outcome when commuting costs take intermediate values. Unfortunately, due to the strong nonlinearity of the equilibrium conditions, it is beyond our reach to determine the whole set of stable equilibria against decreasing commuting costs. In particular, some preliminary analysis reveals that equilibria vanish whereas others emerge as these costs decrease. This makes the use of numerical analysis fairly problematic. Thus, extending our model (and, since ours is very simple, any economic geography model) to two sectors appears to be a very difficult task. This is why we choose to study two special, but meaningful, cases in which good 1 can be carried at zero cost, whereas good 2 either is shipped at a positive cost lower that does not exceed τ trade or is nontradeable. 3.3 Regional specialization It is possible to show the existence of a unique stable equilibrium when one good is perfectly tradeable. Formally, we assume from now on that good 1 can be shipped at zero cost from one region to the other: τ 1 =0. By setting τ 1 =0in (6), it is readily verified that the price of good 1 is the same and equal to p a 2b c regardless of the region where the varieties are sold. Furthermore, profits beingzeroineachregion,τ 1 =0implies that w1r = w1s as long as λ 1r > 0, r = H, F. The mobility of workers across sectors then ensures that factor price equalization w w 1r = w 1s = w 2r = w 2s = b(p ) 2 φ (λ 1r + λ 1s) holds when λ ir > 0, i =1, 2 and r = H, F. Consequently, at any interior distribution of workers, there is no wage differential between regions at the labor market equilibrium once we allow for the existence of a footloose industry. Note, however, that w is not constant because λ 1r + λ 1s changes with θ. 18

19 We are now able to provide a full characterization of the set of equilibria. In the absence of good 2, the market equilibrium always involves dispersion because workers seek to minimize urban costs. We will show that the presence of good 2 leads to a richer set of outcome for intermediate values of commuting costs. When τ 1 =0, the sustain point and the symmetry breaking point are respectively: θ 1 = 2b2 [2a (b c) τ 2 ] τ 2 (2b c) 2 φ θ 2 = 4b2 [2a (b c) τ 2 ] 2 c (2b c) 2 φ where θ 1 < θ 2 holds. The symmetry breaking point θ 2 is obtained as follows. The stability conditions of the symmetric configuration (9) become θ θ A > 0 θ θ 2 > 0 (θ θ A )(θ θ B ) > (θ θ 2) where the parameters θ A and θ B are defined in Appendix 1. Since θ 2 > θ A and θ 2 > θ B can be shown to hold, θ 2 is the symmetry breaking point. Furthermore, it is readily verified that the conditions V1r = V1s = V2r = V2s for an interior equilibrium yield a unique solution given by (8). Thus, when θ < θ 2,itmustbethatsomeλ ir =0because (8) is unstable. In what follows, we determine the stable equilibrium for θ (θ 1, θ 2). In Appendix 2, we show that λ 1r + λ 1s > 1/2 > λ 2r > λ 2s =0 is the unique stable equilibrium when θ (θ 1, θ 2) (see Appendix 2 for the values of λ ir). In words, the production of good 2 is fully concentrated in region r whereas good 1 is produced in the two regions. This result may be explained as follows. As shipping good 1 is costless, sector 1-firms are a priori indifferent about their location. Therefore, sector 2-firms, which seek to minimize their transport costs, will do so by being located all together in the larger market, say region r. However, the agglomeration of sector 2-firms in region r implies that λ 1s > 0, for otherwise we would have full agglomeration, acasethatisruledoutbecauseθ exceeds θ 1. Finally, λ 1r must also be strictly positive. If not, all sector 1-workers would reside in region s and would then have to pay for shipping good 2 from region r. In addition, because more than half of the labor force works for sector 1 (λ 1r + λ 1s > 1/2), sector 1-workers would also incur higher urban costs if they were agglomerated in region s. Consequently, the configuration λ 1r = λ 2s =0cannot be an equilibrium, so 19

20 that both λ 1r and λ 1s must be positive. Stated differently, full specialization à la Henderson (1974) is never an equilibrium. Among other things, the foregoing equilibrium implies that trade in good 2 is unilateral, while trade in good 1 is bilateral. By solving V 1r = V 1s and V 1r = V 2r for θ (θ 1, θ 2), it is readily verified that λ 1r and λ 2r increase when θ decreases and reaches the value 1/2 at θ = θ 1. Similarly, λ 1s decreases when θ decreases and reaches the value 0 when θ = θ 1. Hence, region r accommodates an increasing number of workers as commuting costs decrease. This is because the agglomeration force generated by the transport costs of good 2 is unaffected whereas the dispersion force associated with the urban costs weakens. For a certain value of θ, the majority of sector 1-workers live in region r, which becomes a dominant city because we have λ 1r > λ 1s and λ 2r > λ 2s. Simultaneously, the number of workers in sector 2 increases because the two industries tend to congregate. These results are summarized in the following proposition. Proposition 3 Assume that good 1 can be shipped as zero costs and good 2 at positive costs τ 2.Ifθ (θ 1, θ 2), thenindustry2 is agglomerated. As θ decreases, the share of industry 1 collocated with industry 2 rises, whereas the share of the labor force working in sector 1 decreases. The uneven distribution of workers implies the existence of an urban cost differential. In equilibrium, this one is just compensated by the variety price differential because wages are equalized across regions. In other words, some workers choose to live in the larger city in which they bear higher urban costs because they enjoy there the whole range of varieties at lower prices (Glaeser et al., 2001). This is consistent with the empirical fact that, in cities of different sizes, the price index differential and/or the nominal wage differential is lower than the differential in housing rent (Tabuchi, 2001). Furthermore, the equilibrium distribution of industry 1 now varies continuously with the parameter θ, unlike what we observe in the one-sector model. This is sufficient to show that the existence of a catastrophic change in the economic landscape is an artefact of the one-sector model. The foregoing proposition is important because it provides us with an explanation for differences in the spatial patterns of various industries. It is, therefore, worth investigating what can be the impact of differences in other parameters. In the next proposition, we focus on another relevant case 20

21 in which two industries supply products characterized by different degrees of substitutability, but identical transport costs (τ 1 = τ 2 ). In order to keep things simple, we consider the special case where γ 1 =0< γ 2 = β/2: the varieties of industry 1 are independent, whereas those supplied by industry 2 is mildly differentiated. 11 The proof being lengthy and cumbersome, we have chosen not to include it, but it may be obtained from the authors upon request. Proposition 4 Assume that γ 1 =0and γ 2 = β/2. Ifθ (θ 1, θ 2), then industry 2 is agglomerated. As θ decreases, the share of industry 1 collocated with industry 2 rises, while the share of the labor force working in sector 1 remains constant. This proposition suggests that whereas an industry producing more differentiated varieties is likely to be more dispersed, an industry producing close substitutes tends to be agglomerated. This is because keen price competition lowers prices, which attract consumers. 12 We might think of industry 1 as being the electric appliances industry, which tends to be dispersed among several medium-sized cities (Henderson, 1988). On the other hand, industry 2, which supplies an array of close substitutes, would be concentrated in large cities, which agree with Christaller s (1933) central places. 3.4 Urban hierarchy We now deal with another limiting case in which good 1 can be traded at zero costs, whereas good 2 is nontradeable (τ 2 >a/(b c) so that even one-way trade is precluded) think of a business-to-consumer service industry. We will see that the existence of such a nontradeable consumption good (other than land) is sufficient to generate an urban hierarchy. Let the total number of varieties available in region r be N r N 1 +λ 2r N. Since good 2 cannot be shipped, its price in each region is equal to p a/(2b c), asshownbymaximizingp rr q r (p rr )(λ 1r + λ 2r ). As the indirect utility of an individual working in industry i and residing in region r is given 11 In this case, factor price equalization no longer holds, thus making the problem difficult to handle. Setting γ 2 = β/2 vastly simplifies the analysis. 12 This is a sharp contrast to the case with immobile demand (d Aspremont, Gabszewicz and Thisse, 1979). 21

22 by V ir = [a (b c) p ] 2 Nr + w θ b c 4 (λ 1r + λ 2r) i =1, 2, r = H, F (12) an uneven distribution of workers implies the existence of an urban cost differential (the third term in (12)). In equilibrium, this one is just compensated by the differential in the number of varieties available in each region (the first term in (12)). In other words, workers now choose to live in a larger city in which they bear higher urban costs because they enjoy there a larger variety of differentiated services. Solving V 1r = V 1s = V 2r = V 2s simultaneously, we obtain the following two interior equilibria: 13 µ λ 1r where µ λ 1r λ 2r λ 1s λ 2s λ 2r à = 1 4 T λ 1s λ 2s µ 1 = (4 T θ) + 3 T θ 4T θ T θ 4 4 (b c)φ b 2 (p ) θ, 3 θ (4 T θ) 3 T θ 4T θ 1 3 T θ 4 4! (13) (14) The first equilibrium (13) involves full dispersion. This is always an equilibrium, whatever the value of the commuting cost θ. However, (14) is an equilibrium if and only if θ [2/T, 3/T ] becauseeachvariablemustbereal and must take its values in [0, 1]. This equilibrium can be shown to be stable by applying Routh s theorem to the characteristic polynomial of the Jacobian of (7) evaluated at (14). Note that the fact that λ ir + λ is =1/2 holds for i =1, 2 eases the stability analysis when compared to the case with finite and positive values for τ 2. Furthermore, for all θ (2/T, 3/T ), 1/4 < λ 1r < λ 2r < 1/2 holds, thus implying that the industry producing the nontradeable good is more agglomerated than the other. The two equilibria coincide with full dispersion when θ =3/T. On the other hand, (14) involves full agglomeration (10) at θ =2/T.Thus,θ =3/T is the symmetry breaking point and θ =2/T is the agglomeration sustain point. Since all possible equilibria have been accounted for, we may conclude that the labor force is equally split between the two industries despite the fact 13 The mirror image of the asymmetric equilibrium is disregarded. 22

23 that workers can shift jobs. Indeed, when compared to firms producing a costlessly tradeable good, firms producing a nontradeable good have fewer customers, but each individual demand is higher. Because these two effects just cancel out, the labor share is never higher in one sector than in the other. The foregoing argument may then be summarized as follows: Proposition 5 Assume that good 1 canbetradedatzerocostswhereasgood 2 is nontradeable. As θ decreases, there is only one stable equilibrium path: (i) if θ 3/T, then each industry is fully dispersed; (ii) if 2/T < θ < 3/T, then both industries are partially agglomerated within the same region and the industry producing the nontradeable good is more agglomerated than the industry producing the costlessly tradeable good; (iii) if θ 2/T, then both industries agglomerate in one region. The following remarks are in order. First, although varieties of good 1 can be shipped at zero cost, the existence of a nontradeable good allows for a well-defined distribution of industry 1 between regions. More precisely, except for fairly high commuting costs (θ 3/T ), the nontradeable sector acts as a centripetal force that yields partial or full agglomeration of the other industry. Stateddifferently, cities are more attractive when they offer more local consumption goods. This agrees with Glaeser et al. (2001) for whom the capacity of a city to attract industries depends on its ability to supply a wide array of consumption goods and services, many of which are nontradeable. When commuting costs are sufficiently low (θ 2/T ), the two industries are located into a single region while the other region is empty. As discussed above, the availability of more differentiated services compensates workers for the higher urban costs they bear within the agglomeration. Second, the locational pattern chosen by each of the two industries is in general different. Thisisbecausethefootlooseindustrydoesnotcarea priori about its location, whereas the service industry cares about the spatial distribution of its demand. As a result, the former industry tends to be more dispersed than the latter industry. This agrees with empirical facts: within modern cities, the share of the manufacturing industries tends to be smaller than the share of the service sectors. 14 However, one region accommodates 14 For example, the Tokyo Metropolitan Area consists of Tokyo, Kanagawa, Chiba and Saitama prefectures, whose population share is 26% in Its employment share is 24% in manufacturing industries, which is slightly less than population. However, the service sectors are more concentrated than population: the employment share is 30% in transport 23