Jean Cavailhès

Carl Gaigné
email: gaigne@rennes.inra.fr

Takatoshi Tabuchi

Jacques-François Thisse

K Behrens

V Henderson

T Mayer

A Venables

Trade and the structure of cities

Keywords: F12, F22, R12, R14 city structure, secondary business centers, commuting costs, trade costs, communication costs

Our purpose is to investigate how the interplay between trade, commuting and communication costs shapes the economy at both the interregional and intra-urban levels. Speci…cally, we study how economic integration a¤ects the internal structure of cities and show how decentralizing the production and consumption of goods in secondary employment centers allows …rms located in a large city to maintain their predominance. Several new results in both economic geography and urban economics are established, which all agree with empirical evidence.

Introduction

Cities are major actors in the process of trade. It is, therefore, fundamental to understand (i) how the intensity of trade is in ‡uenced by their size and structure and, conversely, (ii) how economic integration a¤ects the internal structure of cities. This is what we undertake in this paper by modelling the interplay between trade costs, commuting costs and communication costs. Our approach, which combines basic ingredients from urban economics and new economic geography, explains how decentralizing the production of goods in secondary employment centers may allow large cities to retain a large share of …rms and jobs in an integrating world.

Our starting point is that …rms'performances are a¤ected by the level of housing and commuting costs, which we call "urban costs". This occurs through the land rent they pay to occupy central urban locations, and through the higher wages they have to pay to their workers to compensate them for their longer commutes and/or higher land rents. Hence, high urban costs render …rms less competitive on local and foreign markets alike. As a result, despite scale economies arising from urban agglomeration [START_REF] Duranton | Micro-foundations of urban increasing returns: theory[END_REF], increasing urban costs could shift employment from large monocentric cities either to their suburbs or to distant and smaller cities, where these costs are lower, at least once trade costs have su¢ ciently declined to permit large-scale exports to distant markets. In other words, economic integration could well challenge the supremacy of large cities in favor of small cities. The main point we wish to stress in this paper is that the emergence of subcenters within cities is a powerful strategy for large cities to maintain their attractiveness.

Despite the many advantages provided by the inner city through a good access to highly specialized services [START_REF] Porter | Competitive advantage of the inner city[END_REF], …rms or developers may choose to form secondary employment centers, enterprise zones, or edge cities [START_REF] Henderson | New urban landscape: developers and edge cities[END_REF]. In this way, …rms are able to pay lower wages and land rents while retaining most of the bene…ts generated by large urban agglomerations. And, indeed, [START_REF] Timothy | Intra-urban wage variation, employment location and commuting times[END_REF] report large variations in wages according to intra-urban location (15% higher in central Boston than in outlying work zones, 18% between central Minneapolis and the fringe counties). As they enjoy living on larger plots and/or move along with …rms, workers may also want to live in suburbia [START_REF] Glaeser | Sprawl and urban growth[END_REF]. Consequently, the creation of subcenters within a city, i.e. the formation of a polycentric city, appears to be a natural way to alleviate the burden of urban costs. It is, therefore, no surprise that Anas et al. (1998) observe that "polycentricity is an increasingly prominent feature of the landscape". 1Thus, the escalation of urban costs in large cities seems to prompt a redeployment of activities in a polycentric pattern, while smaller cities retain their monocentric shape. However, for this to happen, …rms set up in the secondary centers must maintain a very good access to the main urban center, which requires low communication costs. Indeed, as pointed out by [START_REF] Schwartz | Subservient suburbia[END_REF], about half of the business services consumed by US …rms located in suburbia are supplied in city centers. As a result, by focusing on urban and communication costs, we recognize that both agglomeration and dispersion may take two quite separate forms because they are now compounded by centralization or decentralization of activities within the same city. Such a distinction is crucial for understanding the interaction between cities and trade.

To achieve our goal, we develop a two-region model where regions have a spatial extension that imposes commuting and communication costs whereas interregional shipments of commodities imply trade costs. Unlike [START_REF] Helpman | The size of regions[END_REF], [START_REF] Tabuchi | Urban agglomeration and dispersion: a synthesis of Alonso and Krugman[END_REF] and others, our framework allows cities to be polycentric. We will see how this seemingly minor change sheds light on several important concrete issues that have been pretty much overlooked until now. More precisely, we organize our main conclusions around two main ideas: (i) local factors may change the global organization of the economy, whereas (ii) global forces may a¤ ect the local organization of cities. This interaction arises because the global organization of production and employment may take di¤erent forms as either a single polycentric city or two monocentric cities may emerge, thus yielding very contrasted economic landscapes and trade patterns. Regarding the …rst idea, we focus on communication and commuting costs. When these costs are high, the economic landscape is likely to be formed by several small cities trading di¤erentiated varieties. By contrast, when commuting and/or communication costs reach low values in each region, a large city emerges. In particular, by facilitating the formation of secondary centers, the development of New Information and Communication Technologies (in short NICTs) may prevent the re-dispersion of activities between regions that a deep economic integration is expected to trigger [START_REF] Ottaviano | Agglomeration and economic geography[END_REF].

Concerning the second idea, our thought experiment is about trade costs. When trade and commuting costs are large, agglomeration within a polycentric city is the market outcome. This is because the decentralization of jobs leads to lower land rents for workers and allows …rms to pay them lower wages. This result agrees with the formation of megalopolises in which employment is decentralized in several centers that all belong to the same metropolitan area [START_REF] Macmillen | The number of subcenters in large urban areas[END_REF]. When trade costs fall below some threshold, the agglomeration becomes partial in that it loses jobs to the bene…t of smaller cities because inter-city trade is cheaper. Yet, even for fairly low trade costs, the large city is able to maintain its prevalence because its polycentricity allows it to preserve its competitive position through lower wages and land rent.

In the sections that follow, we …rst describe our modeling strategy (section 2). The intra-urban equilibrium is characterized in section 3, whereas the subsequent section analyzes the urban system when cities have given structures. In section 5, we study the impact of trade and communication costs on the size and structure of cities. Section 6 concludes.

The model

The spatial economy

Consider an economy with two regions, labelled r = 1; 2, separated by a given physical distance, one sector and two primary goods, labor and land. Each region can be urbanized by accommodating …rms and workers within a city, and is formally described by a one-dimensional space X. Whenever a city exists, it has a central business district (in short CBD) located at the origin 0 2 X. One would expect us to explain why this CBD exists as well as why …rms leaving the CBD want to be together and form a secondary business district (in short SBD). Doing that would require the introduction of local spatial externalities and local public goods that would render the analysis much more involved from the technical point of view, without adding much to our results. Indeed, our model has nothing new to add to what is known in this domain. By contrast, we determine the sizes of CBD and SBDs, thus the structure of each city, in the presence of inter-city trade and factor mobility. 2**Firms are free to locate in the CBD or to set up in the suburbs of the metro where they form a SBD. Both the CBD and SBDs are assumed to be dimensionless. 3 In what follows, the superscript C is used to describe variables related to the CBD, whereas S describes the variables associated with a SBD. Without loss of generality, we focus on the right-hand side of the city, the left-hand side being perfectly symmetrical. Distances and locations are expressed by the same variable x measured from the CBD located at x = 0 in city r = 1; 2 whereas the SBD, if any, is established at x S r > 0, which is endogenous.** **Even though …rms consume services supplied in each SBD, the higher-order functions (speci…c local public goods and non-tradeable business-to-business services such as marketing, banking, insurance) are still located in CBD [START_REF] Schwartz | Subservient suburbia[END_REF]. Hence, for using such services, …rms set up in a SBD must incur a communication cost, which is given by

K(x S r ) = K + kx S r (1)
where K and k are two positive constants. Indeed, communicating requires the acquisition of speci…c facilities, thus explaining why communication costs have a …xed component. However, relationships between the CBD and a SBD also involves face-to-face communication. Therefore, some workers must go to the CBD, thus making communication costs dependent on the distance x S r between the CBD and the SBD. For simplicity, we assume that this cost is linear in distance, but this does not a¤ect the nature of our results.** Both the CBD and the SBD are surrounded by residential areas occupied by workers. Furthermore, as the distance between the CBD and SBD is small compared to the intercity distance, we disregard the intra-urban transport cost of goods. Finally, for analytical convenience we restrict ourselves to the case of two SBDs. It should be clear, however, that our analysis can be extended to several subcenters as communication costs get smaller.

Under those various assumptions, the location and size of the SBDs as well as the size of the CBD are endogenously determined. In other words, apart from the Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 assumed existence of the CBD, the internal structure of each city is endogenous.

Workers

The economy is endowed with L mobile workers. The welfare of a worker depends on her consumption of the following three goods. The …rst good is unproduced and homogenous. 4 It is assumed to be costlessly tradeable and chosen as the numéraire. The second good is produced as a continuum n of varieties of a horizontally di¤erentiated good under monopolistic competition and increasing returns, using labor as the only input. Any variety of this good can be shipped from one city to the other at a unit cost of > 0 units of the numéraire. The third good is land; without loss of generality, we set the opportunity cost of land to zero.

Each worker living in city r consumes a residential plot of …xed size chosen as the unit of area. 5 The worker also chooses a quantity q(i) of variety i 2 [0; n], and a quantity q 0 of the numéraire. She is endowed with one unit of labor and q 0 > 0 units of the numéraire. The initial endowment q 0 is supposed to be large enough for her consumption of the numéraire to be strictly positive at the market outcome. Each worker commutes to her employment center -without cross-commuting -and bears a unit commuting cost given by t > 0, so that for the worker located at x the commuting cost is either tx or t x x S r according to the employment center. The budget constraint of an individual residing at x 2 X in city r and working in the corresponding CBD can then be written as follows:

Z n 0 p r (i)q(i)di + q 0 + R C r (x) + tx = w C r + q 0 ( 2
)
where R C r (x) is the land rent prevailing at a distance x from the CBD. The budget constraint of an individual working in the SBD is obtained by replacing tx by t x x S r , R C r (x) by R S r (x), and w C r by w S r . Thus, as in [START_REF] Ogawa | Equilibrium land use patterns in a nonmonocentric city[END_REF], commuting costs and wages are endogenously determined by the global distribution of …rms and workers within the city.

Preferences over the di¤erentiated product and the numéraire are identical across workers and represented by a quasi-linear utility encapsulating a quadratic subutility:

U (q 0 ; q(i); i 2 [0; n]) = Z n 0 q(i)di 2 Z n 0 [q(i)] 2 di 2 Z n 0 q(i)di 2 + q 0 (3)
where > 0 and > > 0. The condition > implies that workers have a preference for variety.

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002

Firms

Technology in manufacturing is such that producing q(i) units of variety i requires a given number of labor units. 6 There are no scope economies so that, due to increasing returns to scale, there is a one-to-one relationship between …rms and varieties. Thus, the total number of …rms is given by n = L= . Labor market clearing implies that the number of …rms located (or varieties produced) in city r is such that n r = r n, where r stands for the share of workers residing in r.

Denote by C r (resp., S r ) the pro…t of a …rm set up in the CBD (resp., the SBD) of city r. Let r be the share of …rms located in the CBD of city r and, therefore, by (1 r )=2 the share of …rms in its right-hand side SBD. When the …rm producing variety i is located in the CBD of city r, its pro…t function is given by:

C r (i) = I r (i) w C r ( 4
)
where I r (i) stands for the …rm's revenue earned from local sales and from exports (see ( 14) below). When the …rm sets up in the SBD of the same city, its pro…t function becomes:

S r (i) = I r (i) w S r K(x S r ) (5)
where the …rm's revenue is the same as in the CBD because shipping varieties within the city is costless so that prices and outputs do not depend on …rm's location in the city. Those two expressions encapsulate the trade-o¤ faced by …rms located in city r: by locating at the SBD, …rms are able to pay a lower wage to workers, but must incur the communication cost K(x S r ).

Market structure

Solving the budget constraint for the numéraire consumption, plugging the corresponding expression into (3) and taking the …rst order condition with respect to q(i) yields ( )q(i)

Z n 0 q(j)dj = p(i) i 2 [0; n]:
The demands for a variety i produced in city r by a worker living in city r and a worker living in city s can then be written, respectively, as follows:

q rr (i) = a (b + cn) p rr (i) + cP r (6)
q rs (i) = a (b + cn) p rs (i) + cP s (7)
where p rr (i) (resp., p rs (i)) denotes the price a variety-i …rm located in city r charges to consumers living in city r (resp., city s 6 = r) and P r the average price (up to n) of all varieties in city r:

P r Z nr 0 p rr (i)di + Z ns 0 p sr (i)di s 6 = r: (8)
Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi Parameter a expresses the desirability of the di¤erentiated product with respect to the numéraire and may, therefore, be viewed as a measure of the size of this market; b gives the link between individual and industry demands: when b rises, consumers become more sensitive to price di¤erences. Finally, parameter c is an inverse measure of the degree of product di¤erentiation between varieties; when c ! 1, varieties are perfect substitutes, whereas they are independent for c = 0.

Firm i located in city r faces a downward sloping demand in city r and city

s 6 = r: Q rr (i) = r Lq rr (i) Q rs (i) = s Lq rs (i)
where q rr (i) and q rs (i) are given by ( 6) and ( 7), respectively.

As empirical evidence suggests that …rms practice some form of spatial price discrimination [START_REF] Greenhut | Spatial pricing in the United States, West Germany and Japan[END_REF][START_REF] Engel | How wide is the border?[END_REF][START_REF] Haskel | The law of one price -A case study[END_REF], we assume that markets are spatially segmented, which means that each …rm chooses a delivered price speci…c to the city in which its variety is sold. As the price of a variety does not vary within a city, the total revenue of …rm i located in city r is given by

I r (i) = p rr (i)Q rr (i) + [p rs (i) ]Q rs (i):
Because there is a continuum of …rms, each …rm has a negligible impact on the market outcome in the sense that it may accurately ignore its in ‡uence on, and hence reactions from, other …rms. However, aggregate market conditions of some kind (here the price index P r ) a¤ect any single …rm. This de…nes a setting in which individual …rms are not competitive (in the classic economic sense of having in…nite demand elasticity) but, at the same time, they have no strategic interactions with one another. Because varieties are symmetric, all …rms located in the same city charge the same price. As shown by [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF], the equilibrium prices are as follows:

p rr = 1 2 2a + c (1 r )n 2b + cn (9)
p rs = p ss + 2 s 6 = r: (10)
It thus appears that the equilibrium price prevailing in a city decreases with the number of …rms located there, but increases with the level of trade costs. Finally, even though factor prices do not enter ( 9)-( 10) because they have the nature of a …xed cost, they have a negative impact on the number of …rms set up in city r, whence an indirect positive impact on equilibrium prices. Substituting [START_REF] Glaeser | Sprawl and urban growth[END_REF] into the demands ( 6)-( 7) and using [START_REF] Giuliano | Subcenters in the Los Angeles region[END_REF], the equilibrium consumption levels can be expressed as follows:

q rr = a bp rr + cn s =2 (11)
q rs = q ss (b + cn) =2: (12)
Not surprisingly, high trade costs raise the local demand for each locally produced variety at the expense of varieties produced in the other city. This substitution e¤ect decreases when varieties becomes more di¤erentiated.

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. ( 2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002

Hence, evaluated at the equilibrium prices ( 9)-( 10), the consumer surplus is given by:

S r = a 2 n 2b a(n r p rr +n s p sr )+ b + cn 2 n r (p rr ) 2 + n s (p sr ) 2 c 2 (n r p rr +n s p sr ) 2 (13)
while the equilibrium revenue of a …rm located in r is may be expressed as follows:

I r = (b + cn) h r Lp 2 rr + s L (p rs ) 2 i : (14)
Both ( 13) and ( 14) depend on the distribution of workers between the two cities. It remains to determine the conditions to be imposed on for trade to occur between cities at the equilibrium prices regardless of the interregional distribution of workers. This is so if and only if the equilibrium demand q rs is positive for any distribution of workers. It is readily veri…ed that this condition is equivalent to:

< trade 2a 2b + cL
which is assumed to hold throughout the paper. This condition also guarantees that it is always pro…table for a …rm to export to the other city (p rs > 0). To sum-up, we consider a full- ‡edged general equilibrium model involving labor, land as well as a di¤erentiated product and a homogeneous good. At the global level, increasing returns at the plant level are the agglomeration force whereas urban costs are the dispersion force. At the city level, communication costs act as the agglomeration force and commuting costs as the dispersion force. In the next section, we study the city equilibrium within one city before considering the case of an urban system in the subsequent section.

Decentralization within a city

A city equilibrium is such that each individual maximizes her utility subject to her budget constraint, each …rm maximizes its pro…ts and markets clear. Individuals choose their workplace (CBD or SBD) and their residential location with respect to given wages and land rents. In each workplace (CBD or SBD), the equilibrium wages are determined by a bidding process in which …rms compete for workers by o¤ering them higher wages until no …rm can pro…tably enter the market. Given such equilibrium wages and the location of workers, …rms choose to locate either in the CBD or in the SBD. At the city equilibrium, no …rm has an incentive to change place within the city, and no worker wants to change her working place and/or her residence. In this section, we analyze such an equilibrium, taking as …xed the number of workers. To ease the burden of notation, we drop the subscript r.

Land rents, wages and workplaces

Within each city, a worker chooses her location so as to maximize her utility (3) under the budget constraint (2). Let C (x) and S (x) be the bid rent at x 2 X of an individual working respectively in the CBD and the SBD. Land is allocated to Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 the highest bidder. 7 Because there is only one type of labor, at the city equilibrium it must be that R(x) = max C (x); S (x); 0 :

Denote by y the right endpoint of the area formed by residents working in the CBD. Let z 1 be the endpoint of the residential area on the left-hand side of the SBD, and z 2 the symmetrical residential endpoint, which is also the outer limit of the city.

Because communication costs to the CBD increase with distance, the two residential areas are adjacent when the city is polycentric, which implies y = z 1 . Therefore, the critical points are as follows:

y = l 2 x S = 1 + 4 l z 2 = l 2 ( 15
)
where l is the city size and the share of …rms located in the CBD. Note that the bid rents at y and z 2 are equal to zero because the lot size is …xed and the opportunity cost of land is zero. An illustration of the land rent pro…le is provided in Figure 1.

Figure 1 Because of the …xed lot size assumption, at the city equilibrium the value of the equilibrium consumption of the nonspatial goods

E = Z n 0 p(i)q(i)di + q 0
is the same regardless of the worker's location. Then, the budget constraint of an individual residing at x and working in the CBD implies that w C + q 0 R(x) tx = E, whereas the budget constraint of an individual working in the SBD is w S + q 0 R(x) t x x S = E: At the city equilibrium, the worker living at the right-endpoint y of the CBD residential area (or at the left-endpoint z 1 of the SBD residential area) is indi¤erent between working in the CBD or in the SBD, which implies w C R(y) ty = w S R(z 1 ) t(x S z 1 ):

Because y = z 1 and R(y) = R(z 1 ) = 0, this equation becomes

w C w S = t(2y x S ) = t 3 1 4 l ( 16
)
where we have used the expressions of x S and z 1 given in [START_REF] Henderson | New urban landscape: developers and edge cities[END_REF]. Thus, the di¤ erence in the wages paid in the CBD and in the SBD compensates exactly the worker for the di¤ erence in the corresponding commuting costs. The wage wedge w C w S is positive as long as > 1=3, thus implying that the size of the CBD exceeds the size of each SBD (recall that another SBD exists on the left-hand side of the CBD).

Observe that a rise in the population size increases the wage wedge: as the average commuting cost rises, …rms located in the CBD must pay a higher wage to their workers.

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002

Equilibrium wages and the city structure

Regarding the labor markets, the equilibrium wages of workers are determined by the zero-pro…t condition. In other words, operating pro…ts are completely absorbed by the wage bill. Hence, the equilibrium wage rates in the CBD and in the SBD must satisfy the conditions C (w C ) = 0 and S (w S ) = 0, respectively. Thus, setting (4) (resp., ( 5)) equal to zero, solving for w C (resp., w S ), we get:

w C = I w S = I K(x S ) . ( 17
)
Hence w C w S = K(x S )= , which means that the equilibrium wage wedge is proportional to the level of the communication cost that prevails at the SBD.

Substituting (17) **and (1)** into ( 16) and solving with respect to yields:

= 4K + (t + k)l (3t k)l
which is positive and exceeds 1=3 if and only if k=3 < t, which means that commuting costs are large relative to distance-sensitive communication costs. When this condition is not satis…ed, the city is monocentric regardless of the value of commuting costs. 8 Assuming from now on that k < 3t holds, we have

= min 1; 4K + (t + k)l (3t k)l : (18)
Observe …rst that = 1=3 when K = k = 0 because the city is formed by three identical employment centers. Furthermore, when < 1, increasing the population size leads to a decrease in the relative size of the CBD, though its absolute size rises, whereas both the relative and absolute sizes of the SBD rises. Indeed, increasing the size of the labor force leads to a more than proportionate increase in the wage rate prevailing in the CBD. This is because of the corresponding rise in the average commuting cost. The number of …rms being …xed, this in turn implies that more …rms choose to set up in the SBD at the expense of the CBD. Last, as long as < 1, the higher the communication cost (either K or k), the larger the CBD. In the same way, the lower the commuting cost, the larger the CBD size.

It is readily veri…ed that the city is monocentric if and only if

t 2K + kl l : (19)
Hence, a polycentric city is more likely to occur when commuting costs are large, communication costs are low, and the population size is large. This agrees with Anas et al. (1998) who observe that by the end of the 19th century telephones have made it possible for US …rms to decentralize, whereas NICTs play nowadays a comparable role. By contrast, a high degree of increasing returns favors the centralization of production.

We may summarize the main results of that analysis in the following proposition.

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 Proposition 1 A city is monocentric if and only if t (2K + kl)= l. Otherwise, the city is polycentric.

Finally, it is worth noting that the equilibrium land rents are given by

R(x) = C (x) = t l 2 x for x < y (20)
where we have used the expression of y and the condition C (y) = 0 and by

R(x) = S (x) = t 1 4 l + x S x for x > x S : (21)
Workers'bid rents around the SBD are maximized at x S whereas S (z 1 ) = 0. The gap C (x) S (jx x S j) > 0 at any given x rises as the relative size of the CBD increases. Note also that the households'bid rents functions C and S in the CBD and the SBD are identical once the employment centers have the same size, that is, = 1=3.

Urban system and inter-city trade

Consider now our two-city setting in which workers are free to choose the city in which they want to live. Let be the endogenous share of workers residing in city 1. A global equilibrium arises at 0 < < 1 when the utility di¤erential

V ( ) V 1 ( ) V 2 ( ) = 0, or at = 1 when V (1)
0. An interior equilibrium is stable if and only if the slope of the indirect utility di¤erential V is strictly negative in a neighborhood of the equilibrium, i.e., d V ( )=d < 0 at , whereas an agglomerated equilibrium is stable whenever it exists. What makes our analysis richer, but more complex, is the fact that the utility di¤ erential V is not uniquely de…ned in that it varies with the internal structure of cities. Consequently, when we deal with a con…guration involving only monocentric (resp., polycentric) cities, we must consider the possibility of a deviation toward a polycentric (resp., monocentric) city when studying the existence and stability of an equilibrium.

The indirect utility of an individual working in the CBD is given by

V C ( ) = S + w C C C + q 0 ( 22
)
where S is the consumer surplus given by ( 13) and C C the urban costs borne by this individual. Using [START_REF] Ottaviano | Agglomeration and economic geography[END_REF], it is readily veri…ed that

C C R C + tx = t L 2 : (23)
If she works in the SBD (if any), her indirect utility becomes

V S ( ) = S + w S C S + q 0
where C S now denotes the urban costs the individual bears. Using [START_REF] Porter | Competitive advantage of the inner city[END_REF], we have

C S R S + tjx x S j = t (1 ) L 4 : (24)
Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 Two comments are in order. First, the equilibrium allocation of workers within each city depends on the global distribution of workers between cities through the value of . In particular, workers are distributed at the city equilibrium in a way such that V C ( ) = V S ( ):

Likewise, when n …rms are established in city 1, …rms are distributed at the city equilibrium such that C ( ) = S ( ) = 0 when SBDs exist. Second, when deciding whether or not to move from one city to the other, workers know whether the cities of origin and destination are monocentric and/or polycentric; they also know the land rent that prevails in each one of them.

In order to determine the stable con…gurations, we de…ne two critical values of t by replacing l by L and ( 1)L, respectively, in [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF]:

t 1 2K L + k t 2 2K (1 )L + k : (25)
Without loss of generality, we may assume that 1=2, so that t 1 t 2 . Using Proposition 1, it is easily seen that the following three global patterns may emerge: (i) when t < t 1 , both cities are monocentric, (ii) when t 1 < t < t 2 , city 1 is polycentric and city 2 is monocentric, and (iii) when t 2 < t, both cities are polycentric. Hence, under dispersion ( = 1=2), we have t 1 = t 2 = T D where T D 4K= L + k= so that the two cities are monocentric if t < T D and polycentric if t > T D . Similarly, under agglomeration ( = 1), t 1 = T A where T A 2K= L + k= < T D and t 2 ! 1; thus, agglomeration arises in a monocentric city when t < T A or in a polycentric city when t > T A .

In order to determine what a global equilibrium is, we must consider the utility di¤erential corresponding to each of the three foregoing patterns. In what follows, we study the cases in which (i) no city is polycentric, (ii) one city is polycentric and the other monocentric, and (iii) no city is monocentric. Because one region may be empty in the …rst and third cases, our analysis will allow us to identify all stable equilibria.

The monocentric case

Assume that t < t 1 so that no city is polycentric:

= 1 for all 2 [1=2; 1]. As mentioned above, the equilibrium wages are given by a bidding process in which …rms compete for workers by o¤ering them higher wages until no …rm can earn positive pro…ts, given by (4), in the CBD of either city. Using [START_REF] Helpman | The size of regions[END_REF] and the equilibrium wages given in Appendix, it is readily veri…ed that the utility di¤erential with two monocentric cities (with subscript mm) is as follows (up to a positive and constant factor):

mm V ( ) mm ( 1=2) (26)
Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 where mm " 1 2 + " 2 " 3 t

with

" 1 (b + cL)(6b 2 2 + 6b cL + c 2 L 2 ) > 0 " 2 4a (b + cL)(3b + 2cL) > 0 " 3 2(2b + cL) 2 2 > 0:
When t < T D , Proposition 1 implies that = 1=2 with two monocentric cities is feasible. Note that the condition t < T D also prevents a marginal deviation to a polycentric city to occur because, in the vicinity of = 1=2, the city in region 2 must be monocentric. The dispersed con…guration with two monocentric cities is, therefore, a global equilibrium when t < T D . It is stable if mm < 0 or, equivalently, t > t m with

t m ( " 1 + " 2 ) " 3
which is positive for all admissible value of because trade < " 2 =" 1 . Thus, = 1=2 is stable when t > t m and t < T D .

When t < t m , dispersion between two identical monocentric cities is no longer stable. If t < T A , = 1 with a monocentric city is feasible. Because mm V (1) > 0 when t < t m whereas the city in region 2 is always monocentric in the vicinity of = 1, = 1 is a stable equilibrium when t < t m and t < T A . The discussion above may be summarized as follows.

Proposition 2 If t < t m and t < T A , there exists a stable global equilibrium in which the industry is agglomerated into a single monocentric city. If t m < t < T D , there exists a stable global equilibrium in which the industry is dispersed between two monocentric cities of equal size.

Note, …rst, that communication costs have to be su¢ ciently large to ful…ll the condition t < T A . If communication costs are very low (formally K = k = 0), the global economy never involves monocentric cities. When they are large, agglomeration in a monocentric city may occur provided that commuting costs are su¢ ciently low (t < t m ). Once commuting costs get larger (t > t m ), the industry is dispersed between two monocentric cities. Hence, a pattern involving two symmetric and monocentric cities is more likely to emerge when both commuting and communication costs are high.

However, for the dispersed pattern with two monocentric cities to arise, it must be that t m < T D . As " 2 increases with a whereas " 1 , " 3 and T D are independent of a, this condition is satis…ed when the parameter a does not exceed the unique solution a m to the equation

" 2 = " 1 2 + " 3 T D :
In other words, the size of the di¤erentiated product market cannot be too large, a < a m , for two monocentric cities to be a global equilibrium.

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. ( 2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002

Otherwise, when a > a m -hence t m > T D -the industry is agglomerated in a single monocentric city as long as t < T A : there is both agglomeration and centralization. This form of extreme agglomeration arises because the intensi…cation of price competition that such a global structure brings about is itself compensated by a su¢ ciently large market size e¤ect (a > a m ).

The mixed case

We now come to the case in which cities have di¤erent internal structures. Without loss of generality, we may assume that city 1 is polycentric whereas city 2 is monocentric: 1 < 1 and 2 = 1 or, equivalently, t 1 < t < t 2 where both t 1 and t 2 are unknown. Note that, when this condition holds, it implies > 1=2 so that the industry is split between the two regions in a way such that the polycentric city hosts the majority of workers.

Using the equilibrium wages given in Appendix, it is readily veri…ed that the equation of motion -with subscript pm -is now given (up to a positive and constant factor) by

pm V ( ) 1 + 2 = 1 2 1 with 1 " 1 2 + " 2 " 3 2t 2 3t k 2 1 2 " 1 2 " 2 + " 3 t 1 4K=L 3t k :
Since pm V ( ) is linear, the intermediate value theorem implies that the interval (1=2; 1) contains a unique equilibrium, given by = 2 = 1 2 (1=2; 1), if and only if pm V (1=2) and pm V (1) have opposite signs. Furthermore, this equilibrium is stable when the two inequalities pm V (1=2) > 0 and pm V (1) < 0 hold because pm V ( ) is then monotone decreasing. This in turn implies that 1 < 0. The …rst condition ( pm V (1=2) > 0) is equivalent to t > T D , whereas the second ( pm V (1) < 0) amounts to

" 1 2 + " 2 " 3 t (t + k + 4K=L) 3t k < 0 or K > K L 4 " " 1 2 + " 2 (3t k) " 3 t t k # :
Note that city 1 is polycentric and city 2 monocentric because the two conditions pm V (1=2) < 0 < pm V (1) are met.

To summarize, Proposition 3 If t > T D and K > K, there exists a stable global equilibrium in which the industry is split between a large polycentric city and a small monocentric city.

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. ( 2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002

The polycentric case

It remains to consider the case where t > t 2 so that no city is monocentric (with subscript pp):

< 1 for all 2 [1=2; 1]. Using the equilibrium wages given in Appendix, the equation of motion becomes

pp V ( ) = pp ( 1=2) (27)
where

pp " 1 2 + " 2 " 3 t (t + k) 3t k = 4" 3 tK (3t k) L : (28)
As mm and pp di¤er only through the last term, the argument is similar to the one developed in subsection 4.1.

When t > T D , Proposition 1 implies = 1=2 with two polycentric cities is feasible. Note that t > T D also prevents a marginal deviation to a monocentric city to occur because the city in region 2 must be polycentric in the vicinity of = 1=2. Therefore, the dispersed con…guration with two polycentric cities is a global equilibrium when t > T D . It is stable if pp < 0 or, equivalently, t > t p , which is the larger solution of pp = 0. This is because the smaller solution violates the condition k < 3t . Note that t p is positive for all admissible value of because trade < " 2 =" 1 . Thus, = 1=2 is stable if t > t p and t > T D . When t < t p , dispersion between two identical polycentric cities ceases to be stable. If t > T A , agglomeration within a polycentric city is feasible. Because pp V (1) > 0 when t > t p , = 1 is a stable equilibrium as long as pm V (1) > 0, which is itself equivalent to K < K.

To summarize, Proposition 4 If t > T A and K < K, there exists a stable global equilibrium in which the industry is agglomerated into a single polycentric city. If t > T D and t > t p , there exists a stable global equilibrium in which the industry is dispersed between two polycentric cities of equal size.

Hence, high distance-sensitive communication costs can prevent the emergence of a single polycentric city because K < 0, even when commuting costs are large (t > T A ). In addition, for agglomeration in a polycentric city to occur, it must be that t > T A and K < K. It is easy to show that the latter condition holds if and only if a > a p , where a p is the unique solution to the equation

" 2 = " 1 2 + " 3 t (t + k + 4K=L) 3t k :
Consequently, when the market size e¤ect is strong (a > a p ), agglomeration within a polycentric city takes place. Because a p is smaller than a m , agglomeration is sustained for weaker market size e¤ects in one polycentric city than in one monocentric city. This shows once more that dispersion is a less likely outcome when the size of the di¤erentiated product market is large. When commuting costs are large (t > t p ) and communication costs low (T D is small), workers alleviate the burden of urban costs by having two polycentric cities:

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 there is both dispersion and decentralization. In other words, the global organization of the production is associated with the lowest level of urban costs in the economy. Observe that a strong reduction in trade costs leads to low values of t p , thus fostering the dispersion of the industry, which now takes the form of two polycentric cities. Because t p exceeds t m , agglomeration is sustainable over a larger set of t-values in the polycentric case than in the monocentric case.

Note, …nally, that the con…guration involving one polycentric city and one monocentric city is sustainable over a larger set of parameter values than the dispersed con…guration with two polycentric cities. The reason is as follows. When a dispersed con…guration with two polycentric cities is a stable equilibrium, the conditions t > T D and t > t p must hold. We know that t > t p is equivalent to pp < 0. If pp < 0, then K < 0 also holds by (28). This implies the condition K > K is always met, so that Proposition 3 holds. In other words, partial agglomeration is also a stable outcome.

Summary results

Propositions 2 to 4 reveal the existence of …ve stable con…gurations: (A) a single monocentric city ( = 1 and = 1), denoted (m; 0); (B) two identical monocentric cities ( = 1=2 and = 1), denoted (m; m); (C) a single polycentric city ( = 1 and < 1), denoted (p; 0); (D) two identical polycentric cities ( = 1=2 and < 1), denoted (p; p); and (E) one large polycentric city and one small monocentric city ( 2 (1=2; 1), 1 < 1 and 2 = 1), denoted by (p; m). Those …ve equilibria are illustrated in the (K; t)-and ( ; t)-planes in Figures 2 and3, respectively. It is readily veri…ed that, in both …gures, the positive orthant is fully covered by the set of conditions identi…ed through Proposition 2 to 4. However, this covering does not de…ne a partition because of the multiplicity of stable equilibria.

Figure 2 Figure 3 Interestingly, a partial agglomeration may emerge as an equilibrium outcome once it is recognized that SBDs can exist, while such a con…guration never arises in standard NEG-models [START_REF] Ottaviano | Agglomeration and economic geography[END_REF]. Another interesting feature of the partial agglomeration pattern is that it involves intra-industry and asymmetric trade in the di¤erentiated product together with intersectoral trade.

Interaction between local and global forces

In this section, we study the impact of changes in commuting costs, communication costs and trade costs on the location of …rms within and between cities. Our results are organized around the following two ideas: (i) local factors may well change the global organization of the economy, whereas (ii) global forces may a¤ect the local/urban organization of production and employment. From the historical point Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 of view, we …nd it interesting to consider two cases. We …rst study how falling communication and commuting costs a¤ects the space-economy. Next, we will consider the usual thought experiment of NEG, namely the impact of falling trade costs on the global distribution of …rms and workers. The di¤erence is that the structure of cities now depends on the evolution of the interregional distribution of activities, which a¤ects itself the way cities are organized.

How the local a¤ects the global

The impact of communication costs

Consider the case of communication costs decreasing from high to low values. For conciseness, we focus on the sole …xed component K. As shown by Figure 2, which is drawn for the case where t m > k= , the evolution of the space-economy changes according to the value of commuting costs. Three cases appear to be relevant.

Assume, …rst, that t exceeds t p . When K is large, the economy involves two monocentric cities because commuting costs are too large to a¤ord the agglomeration of activities within a single city. Once K decreases and crosses the line t = T D , the economy may follow two very di¤erent paths. Along the …rst one, dispersion still prevails but the two monocentric cities are now polycentric. This is because communication costs have su¢ ciently decreased to permit the decentralization of some activities within each city. However, SBDs remain small when K has not decreased by a su¢ ciently large amount. By contrast, when K is very small, each city is formed by three areas having almost the same size. Between these two polar cases, as K decreases the CBD of each city shrinks whereas its SBDs grow, the whole process being smooth (see Proposition 1). Along the second path, there is a bifurcation at the crossing point as revealed by the emergence of a polycentric city whose additional population comes from the other city, which then remains monocentric. The economic space becomes asymmetric and, as communication costs decrease, the large city grows at the expense of the small city. In this case, falling communication costs increase the size of the polycentric city (d =dK < 0). This is because su¢ ciently low values of K trigger the decentralization of production. This in turn increases net wages

d(w S r C s r ) dK = d(w C r C C r ) dK = 2 3 < 0
and makes the agglomeration forces stronger. The entire process may then be summarized as follows:

(m; m) ! (p; p) (case 1:1) (p; m) (case 1:2)

In the second case, we have t m < t < t p . Again, the economic space involves two monocentric cities when K is large. Ignoring for the moment the shaded area in which two stable equilibria exist, we see from Figure 2 that, when K falls, city 1 expands gradually under the form of a polycentric structure, whereas the monocentric city 2 shrinks. Eventually, when K becomes smaller than K, the large city accommodates all workers and city 2 vanishes. There is agglomeration within a single polycentric city. Again, as shown by Proposition 1, the size of city 1 increases gradually from L=2 to L. We thus have:

(m; m) ! (p; m) ! (p; 0) (case 2)
It remains to consider the third case in which k= < t < t m . For large values of K, the economy has a single monocentric city, which may correspond to an isolated "city-state". When the line t = T A is crossed, the city-state ceases to be monocentric and two small SBDs are created. As K keeps decreasing, the city still retains the whole population of workers, but its CBD loose more and more activities at the bene…t of its SBDs. Here also, Proposition 1 implies that the transition process is smooth. This is summarized as follows:

(m; 0) ! (p; 0) (case 3)
To sum up, when communication costs decrease, two contrasted pictures emerge. In the former (cases 1.1 and 3), the global organization of the economy remains the same ( = 1=2 or = 1). The decrease in communications cost a¤ects only the city structure, which becomes more decentralized. In the latter (cases 1.2 and 2), the reduction of communication costs leads to an imbalanced distribution of activities as well as to asymmetric trade (1=2 < < 1). In other words, the global economy, the structure of cities and the nature of trade are all a¤ected. It should then be clear that any simple prediction made about the impact the development NICTs could have on the spatial organization of the economy is likely to be inaccurate.

Note also that in all, except one, cases, the spatial evolution of the economy is smooth, a result that vastly di¤ers from what is known in standard NEG-models [START_REF] Ottaviano | Agglomeration and economic geography[END_REF]. The only case that shows a strong discontinuity arises when the evolution path goes through the shaded area. Urban inertia suggests that the economy retains its previous (m; m) structure when K and t belong to this area. 9 When the path gets out of the shaded area, the economy shifted abruptly from dispersion with two monocentric cities to one polycentric city, that is, the most drastic transformation the spatial economy may experience.

The evolution of the urban landscape described above is to be contrasted to what happens when cities are, by assumption, monocentric. In this case, it follows from Figure 2 that the global equilibrium is always given by (m; 0) as long as t is smaller than t m , whereas this equilibrium is given by (m; m) when t exceeds t m . This shows that the (partial) agglomeration of activities may be sustained over a larger domain of parameters when cities can be polycentric.

The impact of commuting costs

Regarding commuting costs, the following remarks are worth making. First, whatever the value of K, agglomeration is always the single global equilibrium when Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 commuting costs are low (t < t m ), but never when t is high (t > t p ). This may be explained as follows. When cities are monocentric, it is readily veri…ed that

dC C dt =1 > 0
where C C is given by [START_REF] Tabuchi | Urban agglomeration and dispersion: a synthesis of Alonso and Krugman[END_REF]. When cities are polycentric, some tedious, but standard, calculations show that

dC j dt 2[1=3;1) > 0 j = C; S
where C j is given by [START_REF] Tabuchi | Regional specialization, urban hierarchy, and commuting costs[END_REF]. Hence, regardless of the city structure, urban costs borne by workers decrease as commuting costs fall. Consequently, net wages increase regardless of workers'residential location. This implies that more workers are willing to choose to set up in a single city. This larger concentration of workers then makes the agglomeration forces stronger, which in turn increases workers' utility. Eventually, all workers end up living in the same city when commuting costs are su¢ ciently low. Second, increasing commuting costs from low values of t does not necessarily induce the decentralization of production within cities. When communication costs are low, agglomeration still prevails and the city becomes polycentric instead of being monocentric. However, this ceases to be true when communication costs are large. In this case, workers get dispersed between two monocentric cities. Likewise, as shown by Figure 2, a polycentric city may emerge even when commuting costs are low (t < t m ) provided that communication costs are low too. This is because low commuting costs foster agglomeration, thus raising, all else equal, land rents, and inducing, in …ne, the decentralization of production within the same city.

Third, when communication costs are fairly low, a progressive decrease in commuting costs yields an interesting picture. Again, Figure 2 will be our backbone. It shows that the economy follows the path described below:

(p; p) ! (p; m) ! (p; 0) ! (m; 0)
When commuting costs are large but communication costs low, the market forces reduce the burden of urban costs by dispersing workers across cities as well as employment in each city. Below the threshold t p , commuting costs can sustain a larger polycentric city, thus implying that city 2 becomes monocentric because its population is too low for this city to retain its polycentric structure. As t decreases further, city 1 keeps growing at the expense of the small one, which vanishes when t is su¢ ciently low. Indeed, both communication and commuting costs take low enough values to permit the agglomeration of activities within a single polycentric city. Finally, when commuting costs take extremely low values, city 1 becomes monocentric. This is because such a spatial arrangement allows …rms and workers to save on communication costs as commuting is almost inexpensive.

Given the discussion above, we may safely conclude that allowing for structural changes within cities a¤ ects both the global economy and the nature of trade.

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 In what follows, we consider the case of high commuting costs (t > T D ). When trade costs are high, we see that the economy involves a single polycentric city. The fact that commuting and trade costs are high prevents both the existence of monocentric cities and the emergence of a dispersed pattern. Once trade costs cross the line K = K, there is partial dispersion with the creation of a small city that allows workers to reduce the burden of urban costs in the large city. However, trade costs remain too large for that city to capture a large share of activities. If trade costs keep decreasing and cross the line t = t p , two stable equilibria exist.

How the global a¤ects the local

In the …rst one, the small city remains monocentric but attracts a growing share of the labor force at the expense of the large city:

d( 2 = 1 ) d > 0 for all < trade :
In other words, as the global economy gets more integrated, less workers settle in the large city, the partial re-dispersion of the economic activity being gradual. Yet, the large city keeps hosting a large share of mobile activities. Stated di¤erently, as trade costs keep decreasing, the regional gap shrinks but does not disappear because converges to 3 4

T D t 2 1 2
; 3 4 when ! 0. It is worth stressing the fact that this process, which is summarized below, is smooth in that it involves the progressive growth of the small city together with the gradual decline of the large one: In the second equilibrium, full dispersion with two polycentric cities prevails. Because trade costs are low enough, the dispersion of activities does not hamper inter-city trade, whereas high commuting costs leads to the decentralization of activities. Although there is multiplicity of equilibria here, urban inertia is likely to select the …rst equilibrium. In this case, the economy shifts from agglomeration (and no trade) to partial agglomeration (and asymmetric trade) when trade costs steadily decline. Such an outcome is reminiscent of the bell-shaped curve of spatial development -the sequence dispersion/agglomeration/re-dispersion -obtained in several NEG-models with urban costs [START_REF] Ottaviano | Agglomeration and economic geography[END_REF].

Again, it is worth comparing the …rst scheme of evolution to what happens when cities are assumed to be monocentric (t < T A ). In this case, Figure 3 shows that agglomeration prevails below the line t = t m ; otherwise there is always dispersion. Therefore, when cities are allowed to become polycentric, the larger city is able to maintain its primacy over a larger set of structural parameters, thus preventing the complete re-dispersion of activities that would arise in the case of monocentric cities. Thus, there is a need to extend the analysis of urban systems to cope with polycentric cities, instead of focusing almost exclusively on monocentric cities as existing theories do (Abdel-Rahman and [START_REF] Abdel-Rahman | Theories of systems of cities[END_REF].

Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002

Concluding remarks

We have presented a simple model that uncovers how the interplay between di¤erent types of spatial friction a¤ects the location of economic activities between and within cities. Historical evidence shows that both trade and commuting costs have been decreasing since the beginning of the Industrial Revolution. Ever since the end of the 19th century, the development of the new communication technologies has allowed …rms to alleviate the burden of urban costs in large metropolitan areas, through the emergence of secondary employment centers. We have shown how these various technological changes have impacted on the way …rms and workers locate. Our results agree with empirical evidence. Thus, we may safely conclude that what matters for the organization of the space-economy is the relative evolution of three types of costs: the commuting of workers, the transfer of information and the transport of commodities.

When cities are open to trade, the organization of the space-economy varies with the ability of cities to accommodate a small or a large population. In particular, our results show that, once agglomeration within a polycentric city has been achieved, the core maintains its primacy over a large range of trade cost values, thus con…rming the idea that the polycentric structure fosters agglomeration. Our analysis also highlights the importance of local factors in the emergence of regional inequalities. For example, agglomeration is more likely to occur when commuting costs take very low values, but never arises when these costs are high.

The multiplicity of stable equilibria observed here has also an important implication that has been very much overlooked in the literature: di¤ erent types of spatial patterns may coexist under identical technological and economic conditions. It should be no surprise, therefore, to observe a variety of urban systems in the real world.

Finally, our analysis also has an important policy implication. It is well known that quite a few American and European cities have lost employment and population for a few decades. This state of a¤air has led city managers to seek local policies that would prevent the decentralization of industrial activities toward small and/or remote places. However, most of them did fail. This paper suggests that urban decision-makers would have been better inspired to foster the development of SBDs, endowed with high-quality business-to-business services and consumption amenities that attract both …rms and workers.

  , T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002 Furthermore, we have a b, b 1=[ +(n 1) ] and c [ =( )]b.

  Comment citer ce document : Cavailhes, J., Gaigné, C., Tabuchi, T., Thisse, J.-F. (2007). Trade and the structure of cities. Journal of Urban Economics, 62 (3), 383-404. , DOI : 10.1016/j.jue.2006.12.002

Figure 3

 3 Figure 3 displays our main conditions in the ( ; t)-space.In what follows, we consider the case of high commuting costs (t > T D ). When trade costs are high, we see that the economy involves a single polycentric city. The fact that commuting and trade costs are high prevents both the existence of monocentric cities and the emergence of a dispersed pattern. Once trade costs cross the line K = K, there is partial dispersion with the creation of a small city that allows workers to reduce the burden of urban costs in the large city. However, trade costs remain too large for that city to capture a large share of activities. If trade costs keep decreasing and cross the line t = t p , two stable equilibria exist.In the …rst one, the small city remains monocentric but attracts a growing share of the labor force at the expense of the large city:

  (p; 0) ! (p; m) ! (p; m) (p; p)

To illustrate, Giuliano and Small (1991) identify

job centers in Los Angeles, McMillen and McDonald (1998) …nd 15 in Chicago, and Creveso and Wu (1997) count 22 for the San Francisco Bay Area.

Thus, we di¤er from[START_REF] Fujita | On the evolution of hierarchical urban systems[END_REF] because cities have a spatial extension and an endogenous structure. Unlike them, however, the inter-city distance is given. We also di¤er from[START_REF] Henderson | The sizes and types of cities[END_REF] who considers monocentric cities and zero transport costs between cities.

[START_REF] Cavailhès | Trade costs versus urban costs[END_REF] In Cavailhes et al. (2004), we suppose that …rms consume land, thus implying that clusters have a spatial size. This makes the analytical treatment of the model more cumbersome without changing the nature of our results.

The model can easily be extended by introducing a second sector producing the homogenous good under constant returns and perfect competition, using an immobile factor.

Allowing for a variable lot size makes the analysis much more involved without a¤ecting the nature of our results. See Tabuchi (1998) for a study of the monocentric-city case.

When a second sector is considered, we may assume that the production of q(i) units of variety i requires mq(i) units of the immobile factor. Without loss of generality, we may then set m = 0[START_REF] Ottaviano | Agglomeration and trade revisited[END_REF].

Utilities being quasi-linear, the structure of land ownership across indivuals is immaterial for our analysis provided that the distribution is atomless.

In this case, the analysis provided in[START_REF] Tabuchi | Regional specialization, urban hierarchy, and commuting costs[END_REF] in which all workers are mobile applies.

Urban inertia is strengthened by the durability of the housing stock, a variable not taken into account here[START_REF] Glaeser | Urban decline and durable housing[END_REF].

Appendix

The corresponding equilibrium wages (w C r and w S r for r = 1; 2) are such that all …rms, located either in the CBD or in the SBD of each city, earn zero pro…ts (given, respectively, by (4) and ( 5)).

When city 1 is polycentric whereas city 2 is monocentric, the equilibrium wages (w C 1 , w S 1 and w C 2 ) are such that no …rm established in city 1 and located either in the CBD or in the SBD of this city, or established in the CBD of city 2 is able to make positive pro…ts. More precisely, the equilibrium wages in city 1 are given by

where

and the equilibrium wage in city 2 by

where

When both cities are polycentric, wages w C r ( ) and w S r ( ) are given by (A) for r = 1; 2.


Sanket S Kalamkar
email: sanket.kalamkar@inria.fr

Reliability and Local Delay in Wireless Networks: Does Bandwidth Partitioning Help?

This paper studies the effect of bandwidth partitioning (BWP) on the reliability and delay performance in infrastructureless wireless networks. The reliability performance is characterized by the density of concurrent transmissions that satisfy a certain reliability (outage) constraint and the delay performance by so-called local delay, defined as the average number of time slots required to successfully transmit a packet. We concentrate on the ultrareliable regime where the target outage probability is close to 0. BWP has two conflicting effects: while the interference is reduced as the concurrent transmissions are divided over multiple frequency bands, the signal-to-interference ratio (SIR) requirement is increased due to smaller allocated bandwidth if the data rate is to be kept constant. Instead, if the SIR requirement is to be kept the same, BWP reduces the data rate and in turn increases the local delay. For these two approaches with adaptive and fixed SIR requirements, we derive closed-form expressions of the local delay and the maximum density of reliable transmissions in the ultrareliable regime. Our analysis shows that, in the ultrareliable regime, BWP leads to the reliability-delay tradeoff.

I. INTRODUCTION

A. Motivation

The performance in a wireless network depends critically on the allocated bandwidth. For instance, in an interferencelimited wireless network, the data rate R over a link depends on bandwidth W as R = W log 2 (1 + SIR), where SIR is the signal-to-interference ratio for the link under consideration. Often, an operator divides available bandwidth into smaller frequency bands, and the users select one or more subbands for their communication. Such a bandwidth partitioning (BWP) affects the data rate. In particular, suppose the entire bandwidth W is split into N sub-bands and each transmitter selects one sub-band for transmission. In this case, the supported data rate is W N log 2 (1 + SIR). Increasing the number of sub-bands N has two conflicting effects. On the one hand, it reduces the interference since the concurrent transmissions are divided into N bands. On the other hand, to maintain the target data rate at R, the SIR requirement, which is 2 N R/W -1, grows with N . This exponential rise in the SIR requirement could potentially hurt the SIR-based performance in wireless networks. For example, commonly in wireless communication, a packet transmission is considered successful if the receiver achieves the minimum SIR. Hence, any increase in the SIR requirement reduces the probability of a successful packet transmission, which often quantifies the reliability in wireless networks. But, if the SIR requirement is to be kept the same, splitting the bandwidth into N sub-bands results in N -fold reduction in data rate. This means that the time required to send a fixed size packet increases N -fold, leading to a higher transmission delay. Taking these two conflicting aspects into account, this paper attempts to answer the following question: Does bandwidth partitioning help improve the reliability and the delay performance in infrastructureless wireless networks?

B. Reliability and Delay

We focus on outage scenarios where a successful transmission needs the received SIR to be larger than a threshold. That is, the reliability is determined by the events of SIR exceeding some threshold. In this context, subject to an outage (reliability) constraint, the maximum number of successful transmissions per unit area of the network may be thought as an indicator of the number of users that can reliably be accommodated in the network. This density of reliable transmissions is an extremely useful performance metric in modern wireless networks because it sheds light on the questions of network densification under strict reliability constraints.

The delay is characterized by the amount of time required to successfully transmit a message. Specifically, we quantify the delay as the average number of time slots needed for a transmitter to successfully transmit a packet to its next-hop receiver, which is called the local delay [START_REF] Baccelli | A new phase transitions for local delays in MANETs[END_REF], [START_REF] Haenggi | The local delay in Poisson networks[END_REF]. 1

C. Contributions

1) For infrastructureless wireless networks, depending on whether the SIR requirement is changed or not with the number of sub-bands N , we analyze and compare two approaches using stochastic geometry to study the effect of BWP on the density of reliable transmissions and the local delay. 2) We obtain simple closed-form expressions of the local delay and the maximum density of reliable transmissions in the ultrareliable regime. 3) We show that, in the ultrareliable regime, optimizing the number of sub-bands with the sole aim of maximizing the density of reliable transmissions leads to infinite local delay for both the adaptive and fixed SIR requirement approaches.

D. Related Work

From the stochastic geometry approach, the works in [START_REF] Jindal | Bandwidth partitioning in decentralized wireless networks[END_REF] and [START_REF] Zhong | Managing interference correlation through random medium access[END_REF] are perhaps the most relevant. The goal of [START_REF] Jindal | Bandwidth partitioning in decentralized wireless networks[END_REF] coincides with one of our goals-the use of BWP to maximize the density of transmissions that meet an outage constraint in ad hoc networks. But, [START_REF] Jindal | Bandwidth partitioning in decentralized wireless networks[END_REF] applies the outage constraint at the typical link which represents the average performance of all links in a snapshot of the network. Thus the outage constraint is not applied at each individual link. In contrast, our framework applies the outage constraint at each individual link and hence actually calculates the density of transmissions that meet an outage constraint. Also, [START_REF] Jindal | Bandwidth partitioning in decentralized wireless networks[END_REF] does not study the effect of BWP on delay, while our work does. Finally, [START_REF] Jindal | Bandwidth partitioning in decentralized wireless networks[END_REF] considers only the approach where the SIR requirement is changed with N to keep the data rate the same. The work in [START_REF] Zhong | Managing interference correlation through random medium access[END_REF] focuses on the local delay analysis with BWP for the fixed SIR requirement approach and does not consider the reliability aspect.

In [START_REF] Mehanna | Sharing vs. splitting spectrum in OFDMA femtocell networks[END_REF], for two heterogeneous networks coexisting together, a comparison between spectrum sharing and spectrum splitting is studied in terms of the average spectral efficiency. A recent work [START_REF] Lu | Spatial transmitter density allocation for frequencyselective wireless ad hoc networks[END_REF] focuses on BWP for frequency-selective channels with an aim to maximize the total density of concurrent transmissions subject to the outage constraint at the typical link. In [START_REF] Kalamkar | The spatial outage capacity of wireless networks[END_REF], the density of reliable transmissions is maximized for ALOHA-based uncoordinated wireless networks without BWP. There are also several works on BWP for coordinated random wireless networks such as cellular networks (see e.g., [START_REF] Stefanatos | Access point density and bandwidth partitioning in ultra dense wireless networks[END_REF], [START_REF] Saha | Bandwidth partitioning and downlink analysis in millimeter wave integrated access and backhaul for 5G[END_REF]) and wireless networks with deterministic node locations (see e.g., [START_REF] Yeung | Channel management in microcell/macrocell cellular radio systems[END_REF]).

II. SYSTEM MODEL A. Network Model

We consider a network model where the transmitter locations follow a homogeneous Poisson point process (PPP) Φ ⊂ R2 of intensity λ. Each transmitter has an associated receiver at unit distance in a random direction. This model is known as the Poisson bipolar network and is widely used to study infrastructureless networks such as device-to-device (D2D) and machine-to-machine (M2M) networks [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF]. Due to the stationarity of the PPP, it is sufficient to focus on the reference link between the receiver located at the origin and its associated transmitter. In other words, we condition on the transmitter located at x 0 ∈ Φ whose receiver is located at unit distance at the origin, i.e., x 0 = 1. The link between the transmitter at x 0 and its receiver is called the typical link after averaging over the PPP in the sense that this link has the same statistical properties as all other links in the network.

The network operates in a time-slotted manner where the time is divided into slots of equal duration. Each packet is of a fixed size, and it takes exactly one slot for a packet transmission if the entire bandwidth W is used. We focus on the interference-limited regime. 2 A transmission is considered successful if the received SIR exceeds the predefined threshold θ. If successful, a transmitter can send information at spectral efficiency log 2 (1 + θ) bits/s/Hz. Let R denote the data rate of a transmitter when it uses the entire bandwidth W for its transmission, i.e., R = W log 2 (1 + θ), which corresponds to the SIR threshold θ = 2 R/W -1.

All transmitters transmit at unit power and always have packets to transmit. A transmission over distance r is subject to the path loss as ℓ(r) = r -α , where α > 2 is the pathloss exponent. We assume Rayleigh fading, where the channel power gain is exponentially distributed with mean 1. Let h k,x denote the channel power gain between the transmitter x ∈ Φ and the typical receiver at the origin in kth time slot.

B. Performance Metrics

We focus on the performance metrics that are based on the SIR threshold model discussed in the previous subsection.

1) Density of Reliable Transmissions: For a stationary and ergodic point process Φ, the density of reliable transmissions, i.e., the density of concurrent transmissions that meet the outage constraint, is defined as

λ s λ F (θ, ε), θ > 0, ε ∈ [0, 1], (1)
where F (θ, ε) is the fraction of links that achieve an SIR of θ in each realization of the point process Φ with probability at least 1 -ε. Alternatively, F (θ, ε) represents the fraction of reliable links in each realization of Φ. Note that ε corresponds to the target reliability or the target outage probability. The intensity λ of Φ affects λ s positively as well as negatively: an increase in λ increases the density of links but reduces the fraction of links that meet the outage constraint due to the increased interference. Hence there exists an optimal λ that maximizes the density of reliable transmissions, which we shall calculate later in Theorem 3.

For an ergodic point process-based model, the fraction F (θ, ε) is the SIR meta distribution (MD) [START_REF] Haenggi | The meta distribution of the SIR in Poisson bipolar and cellular networks[END_REF], which is a fine-grained performance metric that allows one to calculate per-link reliability in each realization of the network. The SIR MD is defined as the distribution of the conditional success probability P s given a realization of Φ. Mathematically,

F (θ, ε) P ! o (P s (θ) > 1 -ε), where P ! o (•)
denotes the reduced Palm probability conditioned on the typical receiver at the origin o and its associated transmitter being active. The link success probability P s (alternatively, the link reliability) is given by

P s (θ, Φ) P(SIR > θ | Φ),
where the averaging is done over the fading and the channel access scheme.

Letting ε → 0, one can analyze the network performance in the ultrareliable regime, where the target outage probability is close to 0. 32) Local Delay: The local delay is defined as the mean time (in terms of the number of time slots) until a packet is successfully delivered. Conditioned on the point process Φ, the transmissions in different time slots are independent and succeed with probability P s . Hence the time until a successful packet delivery is a random variable with the geometric distribution and conditional mean 1/P s (after averaging over the fading and the channel access scheme). Consequently, the local delay D is given by

D E ! o (1/P s )
, where the expectation E ! o (•) corresponds to the probability P ! o (•) and is taken over the point process. Notice that the local delay is the -1st moment of the SIR MD.

In summary, the density of reliable transmissions corresponds to reliable communication and the local delay corresponds to the latency as it yields the transmission delay.

C. Bandwidth Partitioning

The total bandwidth W is partitioned into N orthogonal sub-bands of equal bandwidth. Thus the bandwidth of each sub-band is W/N . Each transmitter randomly selects a subband for transmission. Let s k (x) ∈ {1, 2, . . . , N } denote the index of the sub-band that the transmitter at the location x ∈ Φ selects in kth time slot. Then the SIR at the typical receiver in kth time slot is given as

SIR k = h k,x0 x∈Φ\{x0} h k,x x -α 1(s k (x 0 ) = s k (x))
, where 1(•) is the indicator function and x is the distance of the transmitter x from the origin. The data rate R N depends on the allocated bandwidth as

R N = W N log 2 (1 + θ).
With BWP, there are two ways to send a packet of fixed size depending on whether one wishes to keep the data rate the same or not:

1) Adapt the SIR threshold: In this approach, the data rate is maintained despite the reduction in the allocated bandwidth due to BWP, i.e., we wish to have R N = R. 4To maintain the data rate, we have to adapt the SIR threshold accordingly. The required SIR threshold θ(N ) can be obtained by inverting the rate expression as

R N = R = W N log 2 (1 + θ(N )) ⇒ θ(N ) = 2 N R W -1.
A packet transmission still requires exactly one time slot. 2) Adapt the packet transmission time: In this approach, the SIR threshold θ is kept unchanged irrespective of the value of N . Hence the data rate R N = W N log 2 (1 + θ) is reduced by N compared to the case with no BWP, i.e.,

R N = R/N ⇒ θ = 2 R/W -1.
This increases the time required to transmit a packet. Since a packet transmission requires one time slot when the allocated bandwidth is W , it needs N time slots when the bandwidth W is split into N sub-bands.

III. DENSITY OF RELIABLE TRANSMISSIONS

A. Adaptive SIR Threshold Approach

As defined in (1), the SIR MD framework can directly be used to calculate the density of reliable transmissions. But the direct calculation of the SIR MD is infeasible. Hence we take a detour where we first calculate the bth moment of the conditional success probability P s .

Theorem 1 (bth moment of P s ). For the adaptive SIR threshold approach with N sub-bands, the b-th moment M b of P s is

M b (N ) = exp -bλC 2 F 1 (1 -b, 1 -δ; 2; 1/N ) θ(N ) δ N , (2)
where

δ 2/α, C = πΓ(1 -δ)Γ(1 + δ), 2 F 1 (•, •; •; •) is the Gauss hypergeometric function, and θ(N ) = 2 N R W -1.
Proof: See Appendix A. The exact SIR MD FPs (x) = P(P s (θ) > x) can be calculated using the Gil-Pelaez theorem [START_REF] Gil-Pelaez | Note on the inversion theorem[END_REF] as

FPs (x) = 1 2 + 1 π ∞ 0 ℑ(e -jt log x M jt (N )) t dt, (3)
where ℑ(ν) is the imaginary part of ν ∈ C. Although the expression in (3) calculates the SIR MD exactly, no useful insights can be obtained due to its complexity. Surprisingly, it is possible to get a very simple closed-form expression of the SIR MD in a practically important regime, which is the ultrareliable regime where the target reliability is close to 1.

In our notation, the ultrareliable regime corresponds to the one where ε → 0, i.e., the target outage probability goes to 0. To obtain the SIR MD in a closed-form in the ultrareliable regime, we use the following two lemmas. The first lemma identifies an asymptotic property of the Gauss hypergeometric function.

Lemma 1. For b ∈ R, 2 F 1 (1 -b, 1 -δ; 2; 1/N ) ∼ b N δ-1 Γ(1 + δ) , b → ∞.
The second lemma is borrowed from [14, Theorem 1], which basically states the de Bruijn's Tauberian theorem.

Lemma 2. For a non-negative random variable X, the Laplace transform E[exp(-tX)] ∼ exp(pt u ) for t → ∞ is equivalent to the cumulative distribution function P(X ≤ ε) ∼ exp(q/ε v ) for ε → 0, when 1/u = 1/v + 1 (for u ∈ (0, 1) and v > 0), and the constants p and q are related as

|up| 1/u = |vq| 1/v .
The following theorem obtains the SIR MD in a closedform as ε → 0.

Theorem 2. As ε → 0, the SIR MD is

F (θ(N ), ε) ∼ exp -C δ θ(N ) N ε δ 1-δ λ 1 1-δ , ( 4
)
where C δ

(πδΓ(1-δ)) 1/(1-δ) δ/(1-δ)
.

Proof: See Appendix B.

Consequently, in the ultrareliable regime, the density of reliable transmissions λ s can be calculated as

λ s ∼ λ exp -C δ θ(N ) N ε δ 1-δ λ 1 1-δ , ε → 0,
which reveals that the intensity λ has two competing effects on λ s . To maximize λ s , we need to obtain the optimal pair (λ, N ). The following theorem gives the maximum density of reliable transmissions in the ultrareliable regime.

Theorem 3. As ε → 0, the maximum density of reliable transmissions S is

S sup λ,N λ s ∼ ε θ 1 δ 1 πe 1-δ δ δ Γ(1 -δ) . ( 5
)
It is achieved at λ =

ε δ πδ δ Γ(1-δ)θ δ 1 and N = 1, where θ 1 = 2 R/W -1.
Proof: We relax N > 0 to be continuous and take the nearest integer as the optimal value. The objective at hand is:

f (λ, N ) = λ exp -B θ(N ) N δ 1-δ λ 1 1-δ , where B = C δ ε δ/(1-δ) and θ(N ) = 2 N R/W -1. We obtain the critical point λ 0 (N ) by letting ∂f (λ,N ) ∂λ = 0, i.e., exp -B θ(N ) N δ 1-δ λ 1 1-δ >0 1 - B 1 -δ θ(N ) N δ 1-δ λ 1 1-δ = 0. It follows that λ 0 (N ) = θ(N ) N -δ 1 -δ B 1-δ .
Since the objective function is quasiconcave for any given N > 0, we have

S ∼ sup N f (λ 0 (N ), N ), ε → 0 = 1 -δ eB 1-δ sup N θ(N ) N -δ
.

From the Taylor series of 2 N R/W , it can be shown that θ(N )/N = (2 N R/W -1)/N increases strictly monotonically with N . Hence N = 1 maximizes the density of reliable transmissions. Consequently, we have S as in [START_REF] Mehanna | Sharing vs. splitting spectrum in OFDMA femtocell networks[END_REF], which is achieved at λ =

ε δ πδ δ Γ(1-δ)θ δ 1 .
Remark 1. For the adaptive SIR threshold approach, the density of reliable transmissions is maximized when there is no bandwidth partitioning. This is because the exponential rise in the SIR threshold with N to maintain the data rate (which reduces the reliability) dominates the advantage of smaller interference due to bandwidth partitioning.

B. Adaptive Transmission Time Approach

In this approach, the SIR threshold does not depend on the number of sub-bands N . Hence the bth moment of P s and the SIR MD in the ultrareliable regime can be calculated using ( 2) and ( 4), respectively, by simply replacing θ(N

) = 2 N R/W -1 with θ = 2 R/W -1.
Following the steps discussed for the adaptive SIR threshold approach (Section III-A), in the ultrareliable regime, the maximum density of reliable transmissions behaves as S → ∞, ε → 0, which is achieved as N → ∞ and λ → ∞. We observe this behavior because the fraction of reliable links, i.e., the SIR MD, is maximized as N → ∞ since the SIR threshold does not depend on N unlike the adaptive SIR threshold approach and the advantage of smaller interference due to BWP helps increase the reliability. This behavior is in contrast with that in the adaptive SIR threshold approach where N = 1 (the other extreme end of BWP) maximizes λ s .

IV. THE LOCAL DELAY

A. Adaptive SIR Threshold Approach

As discussed in Section II-B2, the local delay is the -1st moment of the SIR MD. Hence the local delay D can be obtained by directly substituting b = -1 in (2) as

D(N ) = M -1 (N ) = exp λC θ(N ) N δ (N -1) -(1-δ) , (6)
where

C = πΓ(1 -δ)Γ(1 + δ).
Remark 2. For N = 1 which maximizes the density of reliable transmissions, the local delay is infinite, i.e., a successful packet transmission requires infinite time slots. This is because all transmitters share the same frequency band and there are interferers close enough to the typical receiver to result in unsuccessful packet transmissions. Hence, bandwidth partitioning helps achieve a finite local delay.

From [START_REF] Lu | Spatial transmitter density allocation for frequencyselective wireless ad hoc networks[END_REF], we also observe that the local delay is infinite as N → ∞. Hence there exists a finite optimum N that minimizes the local delay, which we obtain in the following theorem.

Theorem 4. Let N 0 > 1 be the unique solution of (aδN (N -1) log(2) -N + δ)2 aN + N -δ = 0, [START_REF] Kalamkar | The spatial outage capacity of wireless networks[END_REF] where a = R/W . Also, let Round(x) denote the nearest integer if 2 < x < ∞ and equals 2 when 1 < x ≤ 2. Then N * = Round(N 0 ) is the optimum number of sub-bands that minimizes the local delay.

Proof: To obtain the optimum number of sub-bands N * , we relax N to be continuous. Then N * is its nearest integer.

From ( 6), we can see that N = 1 leads to the infinite local delay. Hence N * is at least 2. For N > 1, we take the derivative of D(N ) given in (6) w.r.t. N , which is

D ′ (N ) = g(N ) 2 aN -1 N δ (N -1) δ-1 , where g(N ) = (aδN (N -1) log(2)-N +δ)2 aN +N -δ N (N -1)(2 aN -1)
. It can be observed that g(N ) strictly monotonically increases with N , which implies that there exists only one optimal value N * of N that satisfies D ′ (N * ) = 0. Consequently, N * is obtained from solving g(N ) = 0, i.e., (aδN (N -1) log(2) -N + δ)2 aN + N -δ = 0. Since N is a positive integer, we take the nearest integer greater than 1 as the optimum N .

Although it is not possible to obtain a closed-form solution to [START_REF] Kalamkar | The spatial outage capacity of wireless networks[END_REF], it can easily be obtained numerically.

B. Adaptive Transmission Time Approach

For the local delay, this approach has been studied in [START_REF] Zhong | Managing interference correlation through random medium access[END_REF]. We briefly discuss it here for the sake of completeness. In this approach, the expression of the local delay is obtained by replacing θ(N ) in ( 6) by θ = 2 R/W -1 and multiplying the complete expression by N since a packet transmission (irrespective of whether it is successful or not) requires N time slots. Thus the local delay can be expressed as

D(N ) = N exp λC (θ/N ) δ (N -1) -(1-δ) ,
which is the same as (5) of [START_REF] Zhong | Managing interference correlation through random medium access[END_REF].

Remark 3. For this approach as well, N = 1 and N → ∞ make the local delay infinite. Hence there exists a finite N that minimizes the local delay.

As shown in [START_REF] Zhong | Managing interference correlation through random medium access[END_REF]Theorem 5], the optimum number of subbands N * is the unique solution of the following fixed point equation 1+θ) against the number of subbands (N ). For the adaptive SIR threshold approach, θ = 2 NR/W -1. For the adaptive transmission time approach, θ = 2 R/W -1. R = 0.25, W = 1, α = 3, and λ = 1.

1 N Cθ δ N N -1 1-δ N -δ N -1 = 1. (8)
It has also been shown in [START_REF] Zhong | Managing interference correlation through random medium access[END_REF] that N * is tightly bounded as against the number of sub-bands N . 5 It shows that there exists a finite N for which the delay is minimized. Hence, BWP helps reduce the local delay. Also, we observe that when N is small to moderate, the adaptive SIR threshold approach experiences a smaller delay. But as N increases, the adaptive transmission time approach has a better delay performance since the exponential rise in the SIR requirement for the adaptive SIR threshold approach reduces the success probability (reliability), which causes frequent failed transmissions and thus requires more number of time slots to successfully transmit a packet. This negative effect in the adaptive SIR threshold approach dominates the negative aspect of the adaptive transmission time approach that each packet transmission (successful or not) requires N time slots.

N * ∈ [⌊λCθ δ ⌋, ⌈λCθ δ ⌉ + 2].
Table I summarizes the optimum number of sub-bands N * corresponding to the maximum density of reliable transmissions in the ultrareliable regime and the minimum local delay.

V. CONCLUSIONS

The answer to the question "Does BWP help improve the reliability and delay performance in wireless networks?" is both yes and no. For the adaptive SIR threshold approach, BWP is not helpful if one wishes to maximize the density of reliable transmissions in the ultrareliable regime. On the other hand, "extreme" BWP, i.e., N → ∞, maximizes the density of reliable transmissions for the adaptive transmission time approach. In fact, both cases of N = 1 and N → ∞ make

Fig. 1 .

 1 Fig. 1. The 3D plot of the density of reliable transmissions against λ and N for the adaptive SIR threshold approach. The maximum density of reliable transmissions is achieved at λ = 0.0584 and N = 1 and is equal to 0.0354. R = 0.1, W = 1, α = 4, and ε = 0.01.

Fig. 1

 1 Fig.1shows the behavior of the density of reliable transmissions λ s against λ and N .

Fig. 2 .

 2 Fig. 2. The normalized local delay D(N)log 2 (1+θ) against the number of subbands (N ). For the adaptive SIR threshold approach, θ = 2 NR/W -1. For the adaptive transmission time approach, θ = 2 R/W -1. R = 0.25, W = 1, α = 3, and λ = 1.

Fig. 2

 2 Fig. 2 compares the delay performance for the adaptive SIR threshold and the adaptive time approaches. Specifically, Fig. 2 plots the normalized local delay given by D(N ) log 2 (1+θ)

In this paper, we focus on fully backlogged transmitters that always have packets to transmit. Hence, the local delay is the transmission delay and not the queuing delay.

The results in this paper can easily be extended to scenarios with both interference and noise taken into account.

This regime is of high interest in wireless networks. The ultrareliable communication is a key requirement in modern wireless networks. For example, in vehicular networks, an extremely high reliability is mandatory.

Note that we have denoted R to be the data rate when there is no BWP, i.e., R = W log 2 (1 + θ).

Note that the local delay is measured in number of time slots, and the time-slot duration is proportional to 1 log 2 (1+θ) since the packet size is fixed and the spectral efficiency is proportional to log 2 (1 + θ). Hence, to compare actual delays for different SIR thresholds θ, we need to normalize the local delay D(N ) by log 2 (1 + θ).

the local delay infinite for both the approaches. The optimum N that minimizes the local delay takes none of the extreme values of N (i.e., N = 1 or N → ∞) and lies somewhere in between depending on the system parameters such as the path-loss exponent and the intensity of the underlying point process. Hence, one needs to select N appropriately to maximize the density of reliable transmissions in the ultrareliable regime while keeping the local delay below a threshold. From a broader perspective, since the reliability and the delay are the components of the ultrareliable low-latency communication (URLLC), the results in this paper may help understand the effect of BWP on the URLLC dynamics.

ACKNOWLEDGMENT The author would like to thank Martin Haenggi and Franc ¸ois Baccelli for the initial discussion on this problem.

This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme grant agreement number 788851.

APPENDIX A PROOF OF THM. 1

In the kth time slot, the success probability P s conditioned on Φ is

where

. By averaging over the fading h k,x0 on the desired link, we have

Averaging over the random sub-band selection and the fading on interfering links, it follows that

The bth moment M b = E(P b s ) of P s is given by

where (a) follows from the probability generating functional (PGFL) of the PPP [START_REF] Haenggi | Stochastic Geometry for Wireless Networks[END_REF]Chapter 4].

APPENDIX B PROOF. OF THM. 2

The Laplace transform of X = -log(P s ) is E(exp(-tX)) = E(P t s ) = M t , which is the tth moment of P s . From (2), Lemma 1, and Lemma 2, it follows that

Also, P(X ≤ ε) = P(P s ≥ exp(-ε))

(a)

∼ P(P s ≥ 1 -ǫ), ε → 0

, where (a) is due to exp(-ε) ∼ 1 -ε as ε → 0 and (b) is due to Lemma 2.


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