K Behrens

Latec

Niversité De B Ourgogne

Jean-Marie Huriot

Christian Michelot

Frédéric Gilli

On the emergence of the urban phenomenon -Part II

Keywords: pre-industrial cities, city-formation, indivisibilities JEL-Classification : RI 1, R12, 018

In this paper, we extend the basic model describing the for mation of urban agglomerations in a pre-industrial setting. As we will show, this setting is flexible enough to allow for the investigation of multiple aspects of the spatial economy. After summarizing the basic model and the numerical examples we will refer to, we successively examine the issues of coordination failure in the emergence of the urban phenomenon, the presence of transactions costs and the possibility of diffusion of manu factured good production through the system of agglomerations. This last aspect will be treated in two different ways : that of increasing agricultural productivity and that of increasing tech nological efficiency. As we will argue, both mechanisms can lead to the formation of an urban system.

Résumé

Dans ce papier nous étendons le modèle de base décrivant la formation d'agglomérations urbaines dans un contexte pré industriel. Comme nous le montrerons, ce cadre est suffisam ment flexible afin de permettre d'explorer de multiples facettes de l'économie spatiale. Après un bref résumé du modèle de base et des exemples numériques auxquels nous allons faire appel, nous examinons successivement les problèmes concernant la co ordination, les coûts de transaction et la diffusion de la production de biens manufacturés au travers du système d'agglomérations. Ce dernier aspect sera traité de deux manières différentes : celle de la productivité agricole croissante et celle d'une diffusion tech nologique. Nous montrerons que les deux mécanismes peuvent mener à la formation d'un système de villes.

Introduction

Despite the significant advances that the theory of urban formation has made these last few decades, especially with the general equilibrium models developed in the "new economic geography", several issues have only been sparsely examined. One of them concerns the formation of urban agglomerations in the context of a pre-industrial econ omy W . Most models are based on hypotheses that do not allow their application to early historical periods and start with characteristics that are already inherently urban; even when there are no agglomerations in the beginning we already suppose that there is a highly developed manufacturing or industrial sector which, through technological and/or pecuniary externalities, generates agglomeration forces. The most ancient form of division of labor, namely between rural-agricultural and urban-manufactured work, so crucial in the explanation of the early phases of urbanization, is hence simply as sumed. We often "[...] find it most natural to think of new cities as emerging, as an economy; that already has an urban structure, grows over time" FUJITA M., Krugm a n P., V enables A., [START_REF] Fujita | The Spatial Economy -Cities, regions and international trade[END_REF]. This leaves us with great models capable of explaining how agglomerations form in "modem" economies but gives no hint at how the urban phenomenon can be initialized when there is no industrial activity and urban struc ture at all in the beginning. We believe that this question is fundamental since there is a large consensus about the fact that the division of labor is intimately related to urbanization and that one can probably not be understood without the other. Even if we will probably never be able to say if the division of labor between agriculture and manufacturing is a source or a consequence of urbanization (a typical hen-and-egg problem), we should jointly consider the emergence of the urban phenomenon and the specialization and diversification of the manufacturing and service activities.

In Part I of this paper, we have developed a spatial general equilibrium model that explains the formation of urban agglomerations in the context of such a pre-industrial and pre-urban setting. As we have emphasized, strict indivisibilities in terms of critical mass of the non-agricultural sector play a major role in the passage from a rural to an urban economy. As we have argued, the economic development can be analyzed in two steps : a local phase, where each agglomeration grows independently and stays self-sufficient and a global phase where the different agglomerations interact in order to exploit jointly the economies of scale inherent in economic activities. A real urban structure, characterized by the existence of agglomerations that can't provide themselves with all the foodstuffs they need, only emerges as the result of the second step. As we have seen, the resulting configuration is highly reminiscent of central place theory where one central place (the urban agglomeration) collects agricultural surplus from the neighboring rural agglomerations and distributes manufactured goods in return.

In this second part of our paper, we will discuss some possible extensions of the basic framework developed in Part I. These extensions will show that this setting is general enough to accomodate various expansions, which should encourage further work in the future. First of all, we will briefly discuss the problem of potential coor dination failure in the emergence of the urban phenomenon. How is the final spatial configuration affected when there are multiple agglomerations that could potentially A major exception is the paper D u r a n to n G., [START_REF] D U R A N T O N | Labor specialization, transport costs and city size , Reg io n a l Scien c e & U r b a n E c o n o m ic s[END_REF].

become urban ? As we will see, several different scenarios, ranging from the complete break-down of the spatial economy to the efficient solution of this problem, are possi ble. This highlights the now well known aspects of non-unicity and path-dependency that are common to a vast class of problems where increasing returns to scale play a major role. Second, we will consider a more realistic setting where there are transaction costs to the organization of the internal structure of each agglomeration. As we will argue, transaction costs directly act as a limit to the size of the economy and are a powerful force foreclosing the emergence of complex urban structures. Although this aspect has been highlighted by several authors (especially authors of the so-called New Institutional Economics, see e.g. North D., [START_REF]Institutions, institutional change and economic performance , T he Po l it ic a l Ec o n o m y o f Institutions a n d D e cisio n s[END_REF]), spatial economic theory has not re ally taken into consideration the important implications transaction costs have on the size and scope of a spatial economy. Since our model works with strict indivisibilities and a critical mass threshhold, the radical implications of transaction costs are clearly visible in this setting.

After analyzing the problems concerning coordination and transaction in our spa tial economy, we turn to the question of the evolution of the urban structure. How can the emergence of an urban agglomeration trigger the emergence of other urban agglomerations in the system ? How can the production of non-agricultural goods spread from the leader to the followers ? In order to examine these two important questions, we adopt two different approaches. In the first one, we consider that the agricultural productivity is endogenous to the system and we analyze how its evolution influences the leader-follower relationships that emerge in the spatial economy. As we will see, endogenous agricultural productivity leads mostly to a stronger polarization of the existing urban-rural structure. In the second setting, we consider that there is a sort of accumulated knowledge and learning effect that allows the critical threshhold of the follower agglomerations to be lower as the leader develops a non-agricultural production. As we will see, this setting is compatible with the diffusion and decen tralization of the production of non-agricultural goods. Although the followers stay self-sufficientf 2> , we no longer have a completely polarized spatial structure.

The remainder of this paper is organized as follows :

In Section 2, we briefly derive the main expressions of the basic model developed in the first part of this paper. All important results are summarized and the principal definitions recalled.

In Section 3 we develop the two numerical examples of Part I and recall the main results obtained in the local and global step. These results will subsequently been used in order to investigate the influence of transaction costs and technological spill-overs.

In sections 4 and 5, the different extensions of the basic model are developed. We will begin by analyzing the problem of potential coordination failure and the pres ence of transaction costs before turning to the question concerning the diffusion of the non-agricultural production in the system. All extensions will first be developed

(2)

We do not develop a model where there are followers that provide other followers with agricultural surplus, even if this setting would be the natural extension of our model. We restrict ourselves to a "simple" leader-follower structure in order to keep the developments as tractable as possible.

theoretically and will then be illustrated numerically with the help of the examples developed in Section 3. Section 6 finally offers some preliminary conclusions.

Summary of the basic model

We consider a linear economy stretching out on the real line and we suppose that land is homogenous. All economic agents are identical and have identical preferences, given by the following Cobb-Douglas function

u(x\,X2) = , 0 < ¡1 < 1 (2.1)
where x\ is the quantity of the agricultural good (A-good) consumed and x<i is the quantity of the manufactured good (M-good) consumed. Let p\ = 1 be the normalized unit cost for the A-good and p be the relative price of manufactured good expressed in A-good. The resolution of the constrained optimization problem with agricultural subsistence constraint

f max u (xi ,# 2) X l , X 2 Xi + px 2 = W X\ > C yields the following demand functions x*(w) = < c c pw if w > - ( (1 -fl)wp~l if w > - c , x*2{w) = < c if w < - (w -c)p~1 if w < - (2.2)
for A-goods and M-goods. Note that these expressions yield the traditional demand functions of the litterature when c = 0, i.e. when there is no subsistence constraint. The Karush-Kuhn-Tucker (K K T ) multiplier associated with the inequality constraint of the corresponding minimisation problem is given by

C ^-l / y j -q \ P a(w) = (fiw -c) (2.
3) P \ P ) which is always positive for pw < c. Indirect utility is therefore given by

{ ^{ i -n y -^p -^-^w if w > - c ( 24
) c^(w -c)1-/ip_(1-/i) if w < - t1
The subsistence revenue is given by w = c, in which case there is no consumption of M-goods possible since the whole revenue is used for the consumption of agricultural goods. Note that revenues below c are not feasible since this would imply that the agent can't even afford the strict minimum for subsistence. The in f M-breakpoint is given by w --= c /p > c which is the point from which on any contraction in revenue leads to a stronger contraction in the demands of M-goods than before. This point captures a phenomenon of structural modification in the demands and will play an important role in the determination of the optimal size of the agglomeration.

Let us turn to the production of the agricultural good now suppose that to produce £ units of A-good per unit of agricultural surface

L : s -^ L(s) > 0
units of labor are needed at a distance s from the agglomeration, where L is supposed to be monotonously increasing beyond a certain value of s. If we note N the total A-worker population and r the agricultural fringe distance

N = 2 f TL(s)ds
(2.5) Jo must hold if there is full employment in the A-sector (we suppose that L is symetric about the origin and that the agglomeration is located in the middle of the agricultural area). Equation (2.5) has a unique solution in r which will be noted r* and which will be called the A-worker exhausting agricultural fringe distance. Let r* = r(N) be the function giving the A-worker exhausting agricultural fringe distance for any given value of N . Total production of A-good is therefore given by Pa = 2 r*£ = 2 r(N)£ while total A-good surplus is given by

S A = P A -Nxl(w) = 2r*£ -Nxl(w).
Of course S a must be non-negative for an agglomeration of size N to be sustainable. We suppose throughout this paper that agriculture is earned out collectively so that the individual revenue in agriculture is given by ....

PA 2£r(N)

W(N) = jjr = -(2 .6 )

Note that since revenue depends on N , the demand functions given by (2.2) can be rewritten as functions of N too. Hence we have

Sa = 2r(N)£ -Nx*(N). (2.7)
Let us turn to the production of the manufactured good. Suppose that this com posite M-good is produced under a constant returns to scale technology using n differentiated intermediate inputs. Production of a final and consumable M-good is given by

Q = ( / q® ' ) ' ' 0<p<1

where q(i) is the quantity of the ith intermediate input used in the production of the consumable final good. Suppose that each intermediate input is produced by a single "firm" with inverse production function

l(i) = f + mq(i).
(2.8)

Thus each intermediate input is sold at price

p(i)* = p* = in quantity q(i)* = q* = f-p (1 -p)m using the amount of labour i(*T = r = 1 -p Therefore we have Q' = n i ^7) < 2 '9 )
as total production of composite M-good. Since p < 1 it is easy to see that Q* has increasing returns in the number of differentiated intermediate inputs used. This can loosely be interpreted as the gains due to diversification of intermediate inputs and labor specialization.

Let L* = n*l* be the total M-worker population. Since M-workers do exclusively produce M-goods, but do also consume A-goods, their maximum number is determined by the agglomeration's agricultural surplus. This is what we will call the A-good exhausting M-worker size. Hence

n*l*x*(N) = S a which implies that n* = = -1- l*x* x* J
must hold if the whole agricultural surplus of the agglomeration is used in order to feed the M-worker population and hence the market for A-goods is cleared. Zero profit condition in the M-industry requires that 7T* = PQ* -Yjp*q* = 0.

i-1

This yields the equilibrium price

P * = _ 2 ^m f l -p \ 1-1/p ( S A( N ) \ 1-1/fir(N) p N P \ f ) \ x i ( N )
J N which is the price which equilibrates demand and supply for M-goods according to Walras' Law. Equality of demand and supply can be checked via the equation

f p = ( N + L*)x$(N) (2.10) m( 1 -p)
which can be written as when n is replaced by its A-good exhausting value. This clearing condition holds for all values of N . A proof is given in Appendix A of Part I of this paper.

Note that the analytical expressions of Q*, P *9 S a , and x \ all depend on the value of the wage w for a given N (all those functions are bi-partite and are not easy to write in a compact analytical way). Their analytical expressions structurally change at the infM-breakpoint.

Note finally that city-size is given by s(N) = N + n*l* = N + as long as the agricultural surplus is positive. Therefore city-size is limited by a value N which corresponds to S a ( N ) = 0.

Let us recall some important definitions we will use in the subsequent developments.

D e fin itio n 2 .1 (URBAN AGGLOMERATION)

An agglomeration will be called urban i f its spatial location is invariant with time, if it has been created on a permanent horizon and i f it is unable to provide itself with all the agricultural goods it needs.

Definition 2.2 (CRITICAL POPULATION SIZE)

We will call N the critical population size. It is the minimum size o f Aworker population needed_ in order for consumable M-good production to take place, that is to say n*(N) > 1. Hence we introduce a strict indivisibility by supposing that

Q*(n*) = [ ° if " * < 1 (r6SP' N < * ) \ Q*(n*) if n* > 1 (resp. N > N ) '
Note that we could choose a different critical value than 1.

Definition 2.3 (POTENTIAL URBAN SITE)

A site will be called a potential urban site i f there exists a value o f N being a critical population size.

A basic numerical example

Our basic model consists of two sequential stages. The "first stage dynamics" consist in selecting one or more rural agglomerations and to check their dynamics by assuming that there is no contact between the agglomerations. Depending on the parameters one gets different results which nevertheless behave qualitatively the same way when one chooses a concave function for r. An analytical investigation of the agricultural surplus has allowed us in the first part of this paper to derive sufficient conditions for the in f M-breakpoint to correspond to the optimal agglomeration size. Once the local dynamics have been checked, the "no-contact assumption" is droped. As has been shown in Part I of the paper, global dynamics consist in the emergence of one or more urban agglomerations, growing at the expense of their neighbors. Global dynamics are optimal responses and adjustment mechanisms, consisting in redistribution of population and production.

Consider the following set of parameters : p, -0.5, p = 0.4, m -0.7, c = 0.7, / = 1 and £ = 1. For this same parameter-set we will investigate two different cases. In the first one we take r(N) = y/N (case 1), in the second one we take r(N) = 0.6 + 0.3N (case 2). This yields the following results :

Let us start with case 1. Note that the critical value for N is given by N = 1.667 in this example. As soon as N > N the agricultural surplus is sufficiently high in order to support the production of M-goods. Therefore N = 1.667 = N is the critical population size. Production of M-goods is sustainable over the interval N e [1.667 ; 4.163]. The point N* = 2.0408 plays a special role. It is the value of N for which we attein the in f M-breakpoint and hence have a structural change in the agents' demand functions. As we have shown in Part I of this paper, N * = 2.0408 corresponds to the optimal size of the agglomeration.

The second case yields different results. In this case we have no interval of sustainable sizes for M-good production, while N* = 1.5 is the in f M-breakpoint (which also yields optimality in this example). Note that the maximal city size, which we will note N , is larger in the second case than in the first one : 8.163 in case one and 12 in case two. The agglomeration of case 2 is not a potential urban site, that is to say no manufacturing activity can be supported by local agricultural surplus and break even (make zero profit while solving its optimization problem). Therefore utility and M-good production is zero, no matter how large the agglomeration gets.

We will use the two numerical situations developped above in order to illustrate how global dynamics work. As we have argued, the agglomeration of case 2 hits its optimal size N* before the agglomeration of case 1 but it is not able to develop any manufacturing activity (it can't generate enough agricultural surplus). Hence popula tion growth continues for both agglomerations with zero utility until agglomeration 1 hits its critical value N = 1.667. Since it is now possible to produce manufactured goods in the first agglomeration, utility becomes strictly positive there and the factor flows from the other agglomerations will push the population size of agglomeration 1 up to its optimal value N* = 2.0408. This corresponds to the first disequilibrating adjustment phase with factor movements. Once the optimal size of agglomeration 1 is reached, adjustment through good flows will take place. This works as follows.

(2) [START_REF] Bairoch | De Jericho à Mexico -Villes et économie dans l'histoire, COLLEC TION A r c a d e s[END_REF] (

(2) Since agglomeration 1 has reached its optimal size, its agricultural surplus is equal to 0. The only possibility to raise this surplus is to import it from the other agglomerations. Suppose that there are k agglomerations of type 2, as depicted in Figure [START_REF] Bairoch | De Jericho à Mexico -Villes et économie dans l'histoire, COLLEC TION A r c a d e s[END_REF], It is clear that the minimal distance between the agglomeration 1 and the other agglomerations must be greater than the extent of the agricultural areas (which must of course not overlap). In this case it is easy to check that must hold. Let us suppose that the agricultural surplus imports for agglomeration 1 are given by since there are k identical agglomerations of type 2 located at the same distance d (see Figure [START_REF] Bairoch | De Jericho à Mexico -Villes et économie dans l'histoire, COLLEC TION A r c a d e s[END_REF]). In order for the "new" M-workers in agglomeration 1 to achieve the same utility level as the "old" workers, they must consume z* units of the imported A-good. Hence additional M-firms can be supported. This leads to a new total quantity of M-goods produced, which is given by

F
dmin(A 1, A 2) = r*(N *) + r£(JVJ) = 2.3282 . Hence d{A 1, A 2) > 2.3282 k A S A = ^S Aie -rd(Al-A<) = k S A2e -Td(A"A^( 3.1) (3 .3 ).
Since p < 1, increasing returns to scale are at work and the increase in the quantity produced will be stronger the smaller p is. Since the "new" workers must achieve the same utility as the "old" ones, they will consume a quantity of this M-good production. Hence the net increase in the local production of M-goods in agglomeration 1 is given by

Aq* = Q * -Q * -q c. (3.4)
So far this is a straightforward reasoning. The two fundamental questions to be an swered are : is Aq* non-negative (in order for this adjustment to be rational) ? is Aq* sufficiently large in order for the agglomerations of type 2 to be compensated ? As we have seen in Part I, those questions are difficult to investigate, even if several interesting results can be highlighted. We have especially shown on the one hand that when economic potential is low (i.e. agricultural surplus is low) and when accessibility is bad (i.e. transport costs are high and/or distances are great), no adjustment through good flows is possible. In this case we have either a) unhealthy growth of the leader agglomeration or b) a constant disequilibrium where a local government restricts entry to the leader agglomeration, a scenario that still applies to most of todays developing countries. On the other hand, when economic potential is high (i.e. agricultural surplus is high due to increases in productivity) and/or when accessibility is good (i.e. transport costs are low and/or distances are small), adjustment through good flows is possible.

In this case we observe the emergence of real urban central places that a) can't provide themselves with the foodstuffs they need and b) collect agricultural surplus and provide their region with manufactured goods.

Extensions of the basic model

In this section we will present four extensions of the basic model summarized in Section 2 and developed more thoroughly in Part I of this paper. We will successively examine the questions concerning the problem of potential coordination failure, the introduction of transactions costs and the presence of diffusion mechanisms.

The problem o f poten tial coordination failure

Let us begin by briefly discussing the problem of potential coordination failure in the emergence of the urban phenomenon. Consider for simplicity the following example which highlights the potential problems when several agglomerations hit the critical population size at the same moment.

Assume that there are k = 5 agglomerations of type 2 in our imaginary setting. Space is supposed to be perfectly homogenous and all 5 agglomerations grow at the same exogenous rate; the set of parameters is the same as in Section 3. As we have shown previously, agglomerations of type 2 are not potential urban sites, i.e. they are not capable of generating enough agricultural surplus in order to support a full-time manufacturing production. Hence several issues to the evolution of the system are possible.

Let us begin with the worst issue : that of a complete break-down of the produc tion of M-goods. This will happen if there is no coordination between the different agglomerations. In that case, no agglomeration can produce M-goods on its own and they will all continue to grow until they hit their malthusian upper bound on population size. This scenario is most likely to be encountered in the context of tribal groups which are relatively closed to the exterior world (as e.g. at the very beginning of the era of agriculture after the neolithic revolution where there is a significant lag between the invention of agriculture and the emergence of the first cities).

The second worst issue is that of an inefficient allocation o f agricultural sur plus. Let us consider a simple numerical example. Consider that our k -5 rural agglomerations of type 2 are located at [START_REF] Puu | Spatial Structures -Advances in Spatial and Network Economics[END_REF][START_REF] Glersch | The Word and the Sword -How techniques of information and violence have shaped the world[END_REF] and A 5 = (0,6) respectively. Consider also that all agglomerations are of surplus-maximizing size (hence N = N* = 1.5) (it is easy to check that the agricultural areas don't overlap since the agglomerations are sufficiently dispersed). For each location s 6 K2 we can calculate the maximal available agricultural surplus as being equal to

A i = (1,3), A 2 = (4,2), A 3 = (0,0), A 4 =
SA(s) = ^5 ^2e~r/l(*-Ai) i= 1
where we use the l\ norm given by

t\ (s -x) = |si -Xi \ + |$2 " X 2\
since one can show that the continuous function S A achieves its maximum at one of the Ai s so that there exists an agglomeration which maximizes global available surplus .

Figure [START_REF] B E C K E R | Intra-industry specialization and urban develop ment (in "The Spatial Economy[END_REF] shows the agricultural surplus surface for our numerical example. The agricultural surplus is maximal for the agglomeration Ai = (1, 3) and is given by S A( Ai ) = 1.535. What is an inefficient allocation o f agricultural surplus ? We will say that the allocation of agricultural surplus is inefficient if an agglomeration Ai develops a manufacturing production such that SA(Ai) < SA(Ai), that is to say the production of M-goods is done in an agglomeration which doesn't maximize the available surplus. Since the available surplus in Ai is less than in A \ , there will be less production of consumable final good and hence the utility level of the economic agents will be less than what it could have been if the final production was done in agglomeration A \ . This is clearly inefficient. Why can the production of M-goods take place in an agglomeration different than A i ? This question is a difficult one and we will just briefly sketch-out a possible answer.

Consider the situation where all agglomerations maximize their respective surplus and still no M-good production is possible. Under reasonable assumptions we will This result holds for the l \ norm, but not necessarily for other i Figure 2 : Agricultural surplus surface with k = 5 consider that there is some exchange of information between the agglomerations so that a complete break-down of the M-good production can be avoided (hence the agglomerations, realizing that they will be better off if they cooperate, escape from the urban underdevelopment trap). The problem of which agglomeration will produce the final good must still be solved. If the agents have sufficient information (and hence know the shape of the agricultural surplus surface given by Figure [START_REF] B E C K E R | Intra-industry specialization and urban develop ment (in "The Spatial Economy[END_REF]), they will (by axiom (2.1)) decide that the production of final good will take place in agglomeration A \. In this case the M-good production increases the agents' utility level from 0 to 0.6626. This scenario is of course efficient since no other agglomeration would allow a similar increase in the agents' utility level. If agents have limited and/or incomplete information there will be the risk of an inefficient allocation, whether by consensus or by force. Consider for example that the agents of agglomeration As = (0, 0) perceive a subjectively modified agricultural surplus surface where the agglomeration As is the maximizing point. In that case, if the agents' bargaining power or violence potential is sufficiently large, they may be able to develop the agglomeration A 3 as the M-good producing location. In that case, the agricultural surplus will be given by SA(As) = 1.2481 and will lead to an increase in utility from 0 to 0.5691 which is clearly less efficient. Several other explanations of allocative inefficiencies are possible, but they are mostly non-economic and we will not develop them here in this paper. The important thing to note is simply that exogenous random perturbations may directly influence the economy's path of development which may lead to persistent inefficiencies. Similar conclusions are obtained in several other works of spatial economics.

Transaction and coordination costs

Suppose now that the coordination of the cooperative production of the A-good requires each agent to pay a part of his revenue (which can be seen as some kind of tax) in order for this cooperation to be possible. In formal terms, the coordination of the production and distribution of the agricultural product absorbs some of the community's resources; hence each agent has the following net revenue

w(N) = w(N) -£(N + L*)
which is his gross revenue minus a transaction cost which depends positively on total agglomeration size N + L*9 i.e.

d N > ° 311(1 dL* > °'
We assume furthermore that each individual agent considers that this coordination cost is a constant which does not interfere with his optimization problem. Hence it is easy to show that the problem

(V){

can simply be written as

max u(xi ,x 2) X\ ,X 2 Xi + px2 = w -Ç(N + L*) X\ > c ' max u{x 1,^2) X\ ,£ 2 ( ? ) 4 X i + px2 = W Where w(N) Xi > c _ 2Çr(N) -NÇ(N + L*) N (4.1)
Although it is known that transaction costs are more or less proportional to the number of transactions and that this number is more or less exponentially increasing with respect to the agglomeration's size, we will, for simplicity, consider that £ is linear with respect to S = N + L*.

Assuming that Ç(N + L *) = v(N -h L *) ^ , where v is a positive parameter, we can show that O ( 2 r ( N ) Z -v N * -N x U N ) ) x U N ) $a(n) = t t m + v N -----------■ (4.2)
Since L* = SA( N)/x*(N), equations (4.1) and (4.2) allow us to derive a new analytical expression for the individual revenue w, which is given by

W[N) = J M W v ' N(x*(N) + N v ) '
(4) Linearity of C is assumed in order to obtain an analytical solution for SA • Or we know x * from equation (2.2). Hence we have

w(N) = < 2£cr(N) N(c + Nv) 2& r ( N ) -v N 2 p N for all N such that w(N) < - Q for all N such that w(N) > - /1
Since r is assumed to be continuous, w is a continuous functions of N . An update of the demand functions yields

f 2r(N)£p -v N 2 . _ _ _ _ ^ c - if AT such that w > - xUN) = N c (1 -n)wp if N such that w < - V c if N such that w > x* 2(N) = » (w -c)p 1 if N such that w < - I1 (4.3) (4.4)
which allows us to establish the A-good exhausting intermediate sector size, given by the following expression

n*(N) = < 2r(AQg -v N 2 -cN 1 -p c + v N f 2r{N)Ç{l -n ) N 1 -p 2r ( N ) & -v N 2 f if N such th at w(N) < - if N such th at w(N) > - 1 1
It is interesting to establish the analytical expression of the elasticity of the size of the non-agricultural sector n* with respect to the level of transaction and coordination costs v. We have

en'/v(N*) = dn* v -2N*Çr(N*)(l -p)v dv n ( c -h vN*)[2r(N*)£ -t;(iV*)2 -vN*]
which is the expression of the elasticity for the optimal population size N*. Some numerical simulations with the example case 1 of section 3 yield the following results: As one can see from the results, city size and the mass of non-agricultural activities that can be supported locally are highly reactive to modifications in the level of trans action and coordination costs. As soon as v > 0.05, the agglomeration is no longer a potential urban site (because n* < 1 stays below its critical mass). The optimal size of the agglomeration decreases with the degree of transaction and coordination costs v, as does the utility level and the quantity of M-good produced. This situation is highly reminiscent of numerous observations anthropologists have made concerning "primitive" tribal groups : when it is costly to transact, transaction and coordination costs are kept low enough by restricting the optimal size of the group but gains from specialization and division of labour are extremely small (see e.g. N o r t h D., [START_REF]Institutions, institutional change and economic performance , T he Po l it ic a l Ec o n o m y o f Institutions a n d D e cisio n s[END_REF]).

V N* n * Q* S U* &n*/v 0.
The higher the actual level of transaction costs, the more reactive the size of the nonagricultural sector to modifications of these costs : for v = 0.05, a 1% increase in the level of transaction costs leads to a 12% decrease in the mass of non-agricultural activities that can be supported locally. In such a context, institutions that efficiently manage and reduce transaction and coordination costs are of the utmost importance. As noted by NORTH D., [START_REF]Institutions, institutional change and economic performance , T he Po l it ic a l Ec o n o m y o f Institutions a n d D e cisio n s[END_REF], "[...] the gains from trade, which economists take to be the bedrock o f economic performance, should make it worthwhile to evolve cooperative solutions among parties to capture jointly those gains. Indeed, under certain circum stances, as I have noted in earlier chapters, these issues are so resolved. Trade does exist, even in stateless societies. Yet, as emphasized earlier, the inability o f societies to develop effective low-cost enforcement o f contracts is the most important source of both historical stagnation and contemporary underdevelopment in the Third World**.

A historically significant example relating urban development and coordination costs is given by the ancient sumerian cities where, in order to increase city-size, an efficient way of large scale irrigation had to be devised. This in turn was synonymous with high coordination costs concerning the collection and redistribution of agricultural supplies.

As D udley L., [START_REF] Glersch | The Word and the Sword -How techniques of information and violence have shaped the world[END_REF] has vividly explained, the growth of the sumerian cities would not have been possible without this generalized form of irrigation. Hence, an efficient way of dealing with those high coordination costs had to be devised in order to make urban development possible. This in turn led to one of the most important "inventions" in the history of humanity : writing, which allowed large scale intertemporal coordination of humans and resources.

Diffusion mechanisms

Until now we have considered a two stage model of urban development. In the first local stage autonomous rural agglomerations develop separately until one of them eventually hits its critical mass and starts developing a non-agricultural sector of production. During the second stage, the different agglomerations interact in order to exploit jointly the scale economies inherent in the non-agricultural sector. While there is a large consensus concerning the direction of causality in these historical developments [START_REF] Glersch | The Word and the Sword -How techniques of information and violence have shaped the world[END_REF] , it is sure that the development of the non-agricultural sector had deep impacts both on the agricultural sector and the diffusion of knowledge, of savoir-Most historians agree upon the fact that the agricultural revolution and the development of a significant agricultural surplus were a necessary condition for the emergence of the first urban structures. Jane Jacobs (1969) on the other hand defends the heterodox theory affirming that it is the process of urbanization which triggered the agricultural revolution.

faire. As MOKYR J., [START_REF]Urbanization, Technological Progress and Economic History in "Ur ban Agglomerations and Economic Growth[END_REF] highlights, BOSERUP already noticed that " [ . . .] urbanization was accompanied by rapid progress in the technology of agriculture. The need to organize the urban economies [...] led to some o f the most important inventions in the history of humanity". So far, our model is too simplistic in that it only emphasizes the role the agricultural sector plays in the emergence of the urban phenomenon but neglects the role the urban phenomenon plays in promoting enhanced technological efficiency in the agricultural sector. Once the leader agglomeration starts developing the non-agricultural production, we obtain an apparently stable configuration in our model. Is this configuration really stable, especially when spill-over phenomena are taken into account ?

In order to investigate this question, we will develop in this section two approaches which are both historically pertinent but which nevertheless yield different results for the spatial structures under examination. The first approach consists in considering that the development of the non-agricultural sector leads to positive developments in the agricultural productivity; the second approach consists in stipulating that the development of the non-agricultural sector leads to a diffusion of knowledge which, through a kind of learning effect, lowers the critical mass of the follower agglomerations (which face a technology already well-known and developed). Formally, these two approaches can be modeled as follows : in the first case, we will consider that the parameter f is partly endogenous (£ = £(n*)) while in the second approach the critical mass s will become partly endogenous (s = s(n*)).

Increasing agricultural productivity

Let us start with the first case and assume, for simplicity, that there are no transaction and coordination costs as developed in the previous section. Consider that £ = f(n*) and that f is of course an increasing function of n* (the agricultural productivity is increasing with the degree of non-agricultural development achieved). We will adopt the following specification

£(n*) = fo + h(n*)

where h is the endogenous agricultural productivity, increasing in n*, and where £0 is the exogenous agricultural productivity which is reasonably assumed to be the same for all agglomerations of the model. Furthermore, we will assume that h(0) = 0 : as long as there is no non-agricultural sector, the new model is striclty identical to the old one. Let us first start by expressing £ as a function of N and the model's parameters. For technical simplicity , we will assume that f is linear in N and given by f(n*) = ib + 0 n *

Using the analytical expression of n* we have

We agree with B airoch P ., [START_REF] Bairoch | De Jericho à Mexico -Villes et économie dans l'histoire, COLLEC TION A r c a d e s[END_REF] that this direction of causality is unlikely, but we also agree upon the fact that the feed-back of the development of non-agricultural activities on the agricultural sector must be accounted for.

Analytical expressions may not be obtained for certain non-linear specifications like for example an exponential function.

ON THE EMERGENCE OF THE URBAN PHENOMENON -PART II

m

= So + e^N)1 9

x i ( N ) f which, by using equation (2.7), yields m = (0 + ) 2r

X1 \ N ) J

Finally, £ is given by

c(N) = x t ( N ) [ ( L o f -O N { l -p ) ] fx*(N) -2r(N)0(l -p)
Two different cases must be considered concerning the analytical expression of x* (which changes at the inf-M breakpoint). Some straightforward calculus, using equa tions (2.2) leads to Figure 3 depicts the evolution of £ for case 1 of Section 3 for 6 = 0.05. As can be seen on this figure, £ is increasing with the size of the agricultural population until the optimal agglomeration size is reached and starts to decrease beyond that size. The positive effect of the non-agricultural sector on the agricultural one has the following implications for our model : no matter which value 9 takes on, the dynamics of the model are not structurally modified and the same Ieader-follower structure (determined essentially by r and 0 will emerge. This result is due to the fact that n. m = L l J i i z £ K

c [ t o f -0 N ( l -p )] f c -2r(N)6{l -p) 6 + if N such th at w(N) < - if N such that w(N) >

H f

does not depend on 0. In this context, the positive dependence of the agricultural productivity on the non-agricultural sector reinforces the urban structure that will emerge by allowing the joint configuration to produce more agricultural surplus.

We conclude that the dependence of the agricultural productivity on the development of the non-agricultural sector is not necessarily a force that will lead to diffusion and a less concentrated urban structure. On the contrary, this dependence can amplify the size and structure of the urban configuration that will emerge : positive feed back leads in this case to more indeterminacy and more clear-cut and large urban structures than before. Once again, sumerian cities yield a striking historical example of these developments : after the generalization of irrigation systems, made possible by developments in the non-agricultural sector, agricultural productivity increased and led to a stronger concentration of population in several cities which, as explained by D u d le y L., [START_REF] Glersch | The Word and the Sword -How techniques of information and violence have shaped the world[END_REF], grew by some orders of magnitude.

Let's take a second look at the simple example we have developed in Section 3 to see how this can work and what the implications are. We will examine the case where 6 takes the same value for all the agglomerations. The numerical values of the parameters are the same as in Section 5 and Appendix A provides some details on how we have run those numerical simulations. We will examine successively the cases 0 = 0, 6 = 0.05 and 6 = 0.1. The last case will be considered twice : once under the implicit assumption of structural stability (stable pattern case) and once under the implicit assumption of structural instability (switch pattern case). We consider that there are k = 4 agglomerations of type 2 and that there is one agglomeration of type 1. The transport cost parameter r is constant and equal to 0.25 (high transport cost case). For simplicity we consider that the spatial distribution of the agglomerations is given as follows (the index 1 refers to the agglomeration of type 1, the other indices to the agglomerations of type 2) : A \ = (2 , 2), A 2 = (0, 0), A 3 = (4,0), A 4 = (0,4) and A$ = [START_REF] B E C K M A N N | Economic growth in a central place system in[END_REF][START_REF] B E C K M A N N | Economic growth in a central place system in[END_REF]. The distance between the agglomerations is measured by the t \ norm and is constant and equal to d -4 between the agglomeration A \ and the followers.

Figure 4 illustrates graphically this sample configuration. In all the following tables, the variables are defined as follows : N* is the optimal agricultural population size of the agglomeration, S is the effective total size of the agglomeration, U* is the utility level corresponding to TV*, Q* is the total non-agricultural output correspond ing to N *, S a is the total agricultural surplus, S"\ is available agricultural surplus, n * is the mass of non-agricultural intermediate inputs produced with an agricultural population size of N* and n ^ax is the maximal local mass of non-agricultural interme diate production possible when all local agricultural surplus is used. When necessary, subscript L refers to the leader agglomeration and subscript F to the follower agglom erations. The results of several numerical examples ^ for the model with "agricultural productivity feed-back" can be summarized as follows : (2)

F ig u r e 4 : S a m p le c o n f i g u r a t i o n w i t h k = 4 f o l l o w e r a g g l o m e r a t i o n s

Let us start by resuming the case where there is no positive feed-back of the nonagricultural sector on the agricultural one (i.e. 0 = 0). In this case, the main results are summerized by Table [ As can be seen, the agricultural imports of the leader, given by equation (3.1), amount to A S a = 1.54509 in this case. This additional A-good supply allows for a larger local non-agricultural sector, the increase of which is given by An* = 1.32436 (refer to equation (3.2); hence we have an increase in the local non-agricultural labor force of A L* = A n* /(1 -p)~l = 2.20727). This increase in the mass of nonagricultural intermediate good production leads, via increasing returns to scale (refer to equation (3.3)), to a new total M-good production of Q* = 9.8781. This leaves us finally with a net production residual (given by equation (3.4)) of A q* = 6.8032 which is available for compensation of the followers by good movements. As one can check, each A-worker in the follower agglomerations needs to consume a quantity x2 = 0.4889 of final M-good in order to achieve the same utility as in the leader agglomeration. Hence a total quantity of 0.4889 x k x S f = 2.9334 is needed in order for the system to be stable (allowing each agent to achieve the same utility level). Since Aq* = 6.8032 > 2.9334 this compensation is possible and an equilibrium can be achieved. The total residual M-good production of6.8032-2.9334 = 3.8698 can be equally redistributed to all agents allowing to keep an equilibrium configuration where everyone (leader and followers) is strictly better off than before. The case where 6 = 0 leads to a city system of total equilibrium population size of 12.2989, with 6.2989 for the leader and 6 for the followers (hence 1.5 for each follower agglomeration). Therefore the leader is approximately 4 times as big as the followers. How is the above configuration modified when there is a positive feedback from the non-agricultural sector on the productivity of the agricultural one ? In order to exa mine this question, suppose that the agricultural productivity £ of each agglomeration is weakly increasing with respect to the maximal mass of non-agricultural production possible in this agglomeration (hence £ is assumed to be increasing with respect to n m ax f°r each agglomeration). In order to account for this week feedback effect, we set 6 = 0.05 and reexamine the resulting configuration. This yields the numerical values given in Table [START_REF] B E C K E R | Intra-industry specialization and urban develop ment (in "The Spatial Economy[END_REF]. As one can see from the table, both types of agglomerations increase in size while the followers still stay below the threshold. Hence local production of non-agricultural final good is still impossible in the agglomerations A 2, A3, A 4 and A$. As one can also see, the agricultural surplus of the followers has increased (despite their increase in population size). Therefore, more agricultural surplus is available for the leader. The new quantity is given by A S a = 1.6827, which allows for an additional mass of An* = 1.4423 of non-agricultural intermediate good production and leads to a new aggregate production of Q* = 12.9963. This corresponds to a net increase of 8.4946 of the production of M-good. As one can check, each A-worker in the follower agglomerations needs to consume a quantity x2 = 0.636 of final M-good in order to achieve the same utility as in the leader agglomeration. Hence a total quantity of 0.636 x k x S F = 4.1559 is needed in order for the system to be stable (allowing each agent to achieve the same utility level). Since Aq* = 8.4946 > 4.1559 this compensation is possible and an equilibrium can be found. Like in the case where 9 = 0, the residual can be equally redistributed to all agents allowing to keep an equilibrium configuration where everyone (leader and followers) is strictly better off. The case where 9 = 0.05 leads to a city system of total equilibrium population size of 13.6120, with 7.0778 for the leader and 6.5342 for the followers (hence 1.6336 for each follower agglomeration). Therefore the leader is still approximately 4 times as large as the followers.

Let us finally take a look at the case where 9 = 0.1. Since this case will allow the follower agglomerations to attein their critical mass at the same time as the leader agglomeration, this scenario is slightly more complex than the previous ones. We will successively take a look at three scenarios ! the stable pattern case where we assume that agglomeration A \ is still the leader, the switch pattern case where we assume that one of the followers becomes the leader and the decentralization pattern case where A t is the leader but where a fraction of non-agricultural production is decentralized to the follower agglomerations. Let us start with the stable pattern case which is characterized by the numerical values of Table [

1.0863

Table [START_REF] Puu | Spatial Structures -Advances in Spatial and Network Economics[END_REF] As one can see, n*lax > 1 for the followers which means that they potentially could develop a local production of final non-agricultural good. Let us suppose that this is not the case in the stable pattern case (refer to Section 6 for a brief discussion on possible reasons). In this case, the additional agricultural surplus imported by the leader is given by A S a = 1.8649 which allows a net increase in the total production of final M-good given by Aq* = 11.0319. An individual compensation amount of 0.8899 is needed for each agent of the followers, so that a quantity of 1.4637 x k x S F = 10.5996 will be absorbed in order to achieve compensation accross the agglomerations. This leaves only a very small quantity of 0.2967 to be distributed among all agents. Yet the configuration is still sustainable as an equilibrium. The resulting city system is of total size 15.4662, with a leader of size 8.2246 and followers of size 1.8104 (hence the leader is approximately 4.5 times as big as the followers). When the strength of the feedback increases further (i.e. if we consider that 8 grows beyond 0.1), the configuration eventually becomes unsustainable with A \ as central leader and the other ^¿ 's as followers. Despite the fact that the leader is growing faster than the followers, a centralized production is no longer feasible since the increasing market size and transport costs are no longer counterbalanced by the corresponding scale economies in the production of the final good.

Since the switch pattern case is relatively similar to the developments in Section 4, we will be brief in explaining the implications. As one can already guess from our example, the switch pattern case is synonymous with coordination failure (although this need not be the case in general of course). Suppose that agglomeration A$ = (4,4) becomes the leader for one reason or another. In this case we have the numerical values given in Table [START_REF] B E C K M A N N | Economic growth in a central place system in[END_REF] below. The total collected agricultural surplus for the leader is of A S a = 18193, which leads to a net increase in production of A q* = 8.3668. After substracting the quantities needed for compensation, we still have a surplus quantity of 5.0992 left, which can be used to increase the utility of all economic agents (leaders and followers). The resulting city system is of total size 12.61, with a leader of size 4.4093 and followers of size 2.7778 (resp. of size 1.8104). Hence the leader is approximately 1.5 times as big as the first follower and approximately 2.5 as big as the second followers. As can be seen, the resulting city system in the switch pattern case is of smaller total size and of more equal size distribution. The joint utility level of the economic agents is lower in this case than in the stable pattern case (refer to Section 4 on coordination failure for a more detailed discussion). Table [START_REF] B E C K M A N N | Economic growth in a central place system in[END_REF] Finally, we have the decentralization pattern case where we assume that the agglom eration Ai is still the leader (as in the stable pattern case) but where a fraction of the M-good production is decentralized to the followers. This case is very similar to the stable pattern case except that we need to take into account the fact that each follower will produce M-goods locally. The fundamental question to be answered is if there is a possible equilibrium in this case. Intuitively, increasing productivity in the A-sector should favor diffusion so that a decentralized production should be possible. That this is, however, not the case can be seen from the numerical results and can be interpreted as follows : the higher agricultural surplus allows each follower to grow larger and to develop a local production of M-goods. However, in this case, there is only few local agricultural surplus left which can be shipped to the leader, so that the size and the production of the leader decreases. Since there is dissipation of scale economies in this case, the local production of the followers does not allow to achieve the same utility level as in the leader agglomeration. Therefore, compensation by goods is necessary, or this is impossible since the leader is no longer able to produce on a sufficiently large scale. As we will see in the next section, the fundamental problem is that the followers must achieve a scale of production that is still too large for an equilibrium to be feasible. The decentralization consumes too much resources for the achieved production efficiency.

Let us summarize some of the results obtained in this section. First of all, the individual sizes (as well as the total size) of the agglomerations naturally increase as the result of an enhanced productivity in the agricultural sector. As long as there is no possibility of a switch pattern case (i.e. the followers are unable to develop any local production) the leader grows relative to the followers and the utility level of every economic agent increases, which shows that joint exploitation of economies of scale are profitable for everyone. As soon as the feedback mechanism gets strong enough (i.e. 6 exceeds a certain value), we can eventually observe a switch pattern scenario where any one of the agglomerations can become leader (this corresponds to a strong structural non-uniqueness of the evolutionary path of the model which is solely determined by strictly exogenous "random" factors). However, once an agglomeration is leader, the resulting path is stable, which corresponds to the historical lock-in effect. The selection of the leader may lead to efficient and inefficient paths, which are both equilibria and stable. As we have seen, as 0 grows beyond a certain value, the joint exploitation of scale economies eventually becomes impossible since the market size grows faster than the output and since compensation by redistribution of M-goods becomes infeasible. In this case, decentralization of the M-good production becomes necessary for an equilibrium configuration to potentially exist. As we have also seen, decentralization may not be feasible if the dissipation of scale economies is too strong with respect to increasing population size. Therefore, the system could potentially get stuck in a configuration which does not allow for an equilibrium W . This result This result is essentially due to the spatial inertia of the cities in our model. Most models, e.g. the models of F u jita , Krugman and M ori, do not allow for spatial immobility of cities, so that the relative position of cities with respect to their agricultural supply is itself variable (some other models like e.g. H en d erson or A bdel-R ahm an do not even care about relative spatial position of cities). In this case, adjustment is (always ?) possible, even if there are so called catastrophic transitions from one spatial structure to another, where cities move, drop out or "jump around". This should not be a surprise since the hypothesis of perfectly malleable urban capital introduces another divisibility which lacks in our model. If we consider in our model that cities are mobile (i.e. that the distance between the agglomerations is a variable), we could also adjust the model and find a way to decentralize the production could shed some light on the persistence of strongly uneven spatial distributions in developing countries : in those countries, all M-good production is traditionally done in primate cities and there is little to no interaction between the technical evolution of the M-sector and the productivity of the A-sector. Hence 6 is very small which doesn't allow to either attein the critical mass in the country side nor to render infeasible the currently existing concentrated equilibrium.

Note finally that the evolution of the parameter 0 yields some interesting results since it is fl-shapedly related to the "degree" of spatial concentration: first, concentra tion of M-good production and total city-system size increases with 6 but as soon as 6 gets too large this system tends to becomes unstable and further increases in 6 can lead to a more decentralized city system. Unfortunately the parameter 6 is exogenous to our system and its evolution is not accounted for .

Increasing technological efficiency

Although there is a large consensus concerning the fact that urbanization and technological progress are intimately related, the exact nature and causal direction of this relationship are much more controversial [START_REF] Huriot | Les extemalités d'information, mythe ou réalité ? in[END_REF] . Without entering this debate, we will consider that there is a causal relationship from urbanization to technological progress (which can give rise to another wave of urbanization, so that we can have a self-reinforcing phenomenon). Two of the "horsemen of economic growth" are clearly related to the phenomenon of urbanization : smithian growth through a market size and labor specialization effect (which is clearly present in most models of the New Economic Geography) and growth through technological evolution (which, as in most models, is singularity absent in our model until now).

In this section, we will examine a second diffusion mechanism : increasing technical efficiency in the production of M-goods through an accumulated knowledge and learning effect. We will consider from now on that the critical mass s (which we have normalized to unity) of the follower agglomerations depends on the accumulated knowledge of the leader. We will use n* of the leader agglomeration as a proxy for the level of accumulated knowledge and aquired savoir-faire. As in the previous section, the subscript L refers to the leader while the subscript F refers to the followers.

Consider once again an initial configuration where there is no non-agricultural activity and production. As we have explained before, increasing agricultural surplus as an equilibrium configuration. We nevertheless believe that, being not only incompatible with our first axiom on spatial immobility, the fiction of moving cities (or cities floating like islands in a spaceless sea) obscurs some really important phenomena like e.g. the functioning of adjustment dynamics and the potential persistence of dysequilibrium configurations.

(9) There is some literature which examines the historical interactions between the agri cultural and the non-agricultural sector. However, relatively little is known on the feedback mechanisms and their economic implications. [START_REF] Huriot | Les extemalités d'information, mythe ou réalité ? in[END_REF] One should also keep in mind that easy generalizations about the positive role of cities in technological progress are not true for all periods of history (but nevertheless for most of them). As MOKYR J., [START_REF]Urbanization, Technological Progress and Economic History in "Ur ban Agglomerations and Economic Growth[END_REF] will eventually allow a rural agglomeration to reach its critical mass and to develop a non-agricultural intermediate sector and final good production. While it seems reason able to assume that the critical mass of knowledge is the same for all agglomerations in the beginning (at least if we assume, as we do, that all agents are identical i.e. that there are no differences in human capital), this hypothesis seems untenable once one of the agglomerations starts developing a non-agricultural production. As is well known, knowledge and technical progress are largely (though not entirely ^ ) irreversible; although technical progress and knowledge are localized and tend to diffuse rather slowly through time and space (see e.g. G u i l l a i n R., H u r i o t J.-M., [START_REF] Huriot | Les extemalités d'information, mythe ou réalité ? in[END_REF]), we need to account for their impact if we want to understand how production can diffuse through a system of cities and how this system can evolve. After all, "[. ..] cities were [...] crucial in the dissemination, exchange and preservation o f information, both by the citizens within a given city and by their contacts with other cities " MOKYR J., [START_REF]Urbanization, Technological Progress and Economic History in "Ur ban Agglomerations and Economic Growth[END_REF].

Assume for simplicity that the endogenous critical mass of the follower agglom erations is given by where 8 > 0 is a parameter accounting for the speed (both temporal and spatial) of diffusion of technical knowledge concerning the production of consumable M-good x 2. Roughly speaking, this can be interpreted as follows: the more technical knowledge and experience is available due to the production of M-goods by the leader, the smaller the feasible scale of production since the mass of non-agricultural resources necessary for the production of the final good decreases (which can be interpreted as a sort of knowledge spillover). The parameter 8 is supposed to be exogenously given in our model and its evolution is not explained.

Before starting to develop our analysis through some numerical examples, we need to discuss a problem which needs some consideration : how does time influence the diffusion process ? While it is clear that both time and space are impedements to the diffusion of knowledge, the temporal aspect needs some more consideration in our sequential model. In fact, we need to determine when the positive effect on sF will take place. At one extreme, we can assume that diffusion is instantaneous and that sf will adjust as soon as N l hits its critical mass. On the other hand we can stipulate that sF is subject to a spillover lag i.e. that it will react to some time after N L has hit its critical mass. In order to cut-short a significant amount of technical complications [START_REF] H Ir R Ia R T -U R R U T Y J.-B | Convex analysis and minimization algorithms -Volume I and n, F un dam en tals o f pure a n d ap p lied m athem at ics NO[END_REF] we will assume, in an admittedly ad hoc way, that the spillover lag coincides exactly with the moment the leader agglomeration hits its optimal size N*. We will An example of reversibility is given by the strange historical path a certain part of Roman knowledge has taken. Forgotten in the Occident during the Dark Ages, some knowledge and techniques have been "rediscovered" by the western civilizations in the Islamic world during the Crusades.

(12) Depending on the lag, there is a continuum of possibilities as to when the spillover effect can take place. There is therefore a continuum of historical paths which could lead to very different equilibrium configurations. We have not investigated this possibility and the generalization of this aspect is left for the future.

nL < SL nL > SL

also assume that the followers develop a non-agricultural sector corresponding exactly to the minimal size given by the critical mass s f (ti* L).

Consider the same spatial configuration as in the previous section (refer to figure 4) where k = 4 and r = 0.25. When will a dispersed equilibrium configuration, where the production is not completely concentrated in the leader agglomeration, be possible (if it is possible at all) ? In order to investigate this question, we will examine different values for the diffusion parameter 0 and the scale economy parameter p.

Let us begin with p = 0.4 and 9 = 0.1 (13^. In this case, since we still assume that the critical mass of the leader s l = 1, the critical mass of the follower agglomerations is given by 0.841985, which leads to followers of size S f = 2.9033 producing a net available agricultural surplus of 0.4606. The total collected agricultural surplus available to the leader is given by A S a = 0.6778, which allows for an increase of A 7i* = 0.5897 (resp. AL* = 0.9683). Hence the new total M-good production of the leader is given by Q* = 4.1714, which corresponds to a net increase of A q* = 1.9355. Since the follower agglomerations are above their critical mass (which is given by 0.841985 due to the spill-over effect), they are able to produce a certain quantity of consumable M-good locally; this is what we will refer to as decentralized production o f M-good. In our numerical example, each follower agglomeration is able to produce a local M-good quantity of 0.6195. In order for its inhabitants to achieve the same utility level as in the leader agglomeration, an additional per capita import of A x 2 = 0.2294 is needed. Therefore a total amount of k x S f x A x 2 = 2.6640 units of M-good needs to be exported from the leader to the followers. This is however impossible since Aq* = 1.9355 < 2.6640 and hence no equilibrium with decentralized production exists in this case.

Consider now that the knowledge spillover effect is stronger and assume that 6 = 0.2. In this case, the new critical mass for the followers is given by 0.68397, which leads to followers of size S f = 2.6399 producing an available agricultural surplus of 0.5712. This surplus allows, after transportation to the leader, for an additional An* = 0.7204 intermediate input firms, leading to an increase of AL* = 1.2008 in the leader's non-agricultural population. The new total quantity of M-good produced by the leader is given by Q* = 5.0245, which corresponds to a net increase of Aq* = 2.6312. The production of decentralized M-good in each follower agglomeration is given by 0.3685 and hence an additional per capita import of A x 2 = 0.3135 is required in order for all agents to achieve the same utility level as in the leader agglomeration. This is however still impossible since the total import requirement of 3.3105 < A q*. There is still no possible decentralized M-good production.

If we consider now that there is a strong spillover effect given by 6 = 0.4, the situation radically changes. In this case the critical mass of the followers, which are of size S f = 2.1132 and produce a net agricultural output of 0.7924, drops to 0.3679. The total A-good surplus collected by the leader is of A S a = 1.1661 which allows for an additional mass of An* = 0.9995 intermediate inputs and A L * = 1.6658 non-agricultural workers. This allows for a new M-good production of Q* = 7.0250 in the leader agglomeration and a decentralized production of 0.07821 in the follower agglomerations. In order to achieve the same utility-level everywhere, the followers need to import a per capita quantity of A x2 = 0.4342, which corresponds to a total quantity of 3.6702. As one can see, this total neccessary import is smaller than the net available export quanitity Aq*, so that a compensation with a partly decentralized production of M-goods becomes sustainable as an equilibrium configuration.

In order to get a better interpretation of the above results, some important numer ical values are summarized in Table [START_REF] Glersch | The Word and the Sword -How techniques of information and violence have shaped the world[END_REF]: Only the last configuration with 6 = 0.4 can be sustained as an equilibrium situation with decentralized M-good production. The reason for this result can be summarized and interpreted as follows : since p is relatively small, scale economies are rather strong in our sample configuration. In case 6 is small (i.e. there is only a slow and local diffusion of knowledge), the critical mass each follower agglomeration needs to achieve stays relatively high; therefore, the scale of production of each follower will need to be large, which significantly spreads production among the different agglomerations and negates the positive effect of the scale economies inherent to the production of the M-good. As can be seen from the above table, if 6 = 0.1, only approximately 4 units of M-good out of 10 will be produced in the leader agglomeration, while the production of the 6 others will be spread between the different follower agglomerations. Hence

0 = 0.
• the division of the production among the different agglomerations does not allow for a sufficient exploitation of scale economies

• much of the agricultural surplus of the followers needs to be consumed locally in order to achieve the technically necessary scale of production. This in turn leaves little agricultural surplus for the leader which cannot exploit sufficiently his economies of scale. When 0 is small, the necessary scale for decentralized production is uneconomically large.

• when the followers are too large compared to the leader (as e.g. 1 : 1.74 in the case where 0 -0.1), the spatial market area the leader must compensate is too large compared with the economies of scale that can be locally realized. Hence the leader must be significantly larger than the followers, which is only possible if there is enough collectible agricultural surplus.

When 6 is large, as is the case in our third configuration, the diffusion and accumulation of knowledge allows for the production of M-goods on a significantly smaller scale of production. Hence a part of the production can be delocalized to smaller follower agglomerations which have still enough agricultural surplus that can be exported to the leader. Hence scale economies can be sufficiently preserved for a compensation to be possible.

What happens now in a situation where scale economies in the production of the non-agricultural good are relatively weak (i.e. p is relatively large) ? Consider the above configuration with 6 = 0.1 and suppose now that p = 0.6. In this case our model yields the following numerical results : although the critical mass of 0.8472 is very similar to the case where p = 0.4, the followers will be of larger size, given by S f = 3.6180, without loosing too much agricultural surplus (which stays at S a f = 0.4569). Hence the leader is able to collect an agricultural surplus of A S a = 0.6724, which allows for an increase of An* = 0.3842 of the mass of non-agricultural intermediate input production. The new total M-good production of the leader is of Q* = 2.9060, which corresponds to a net increase of Aq* = 0.9588 (when comparing with the case where p = 0.4, one sees that scale economies are significantly weaker here). Although the net increase of M-good produced by the leader is significantly smaller in this case, the decentralized production of each follower is significantly higher and amounts to 1.6255. When one determines the per capita amount of M-good that needs to be shipped to the followers, one finds that A x 2 = -0.0631, which shows that there is excess production both in the follower and in the leader agglomeration. This shows that the decentralized production configuration where 6 = 0.1 can be sustained as an equilibrium, provided that scale economies are low enough. Hence, even if the knowledge spillover effect is weak, decentralization of production is possible when the economies of scale are not too important. Table [START_REF] D U R A N T O N | Labor specialization, transport costs and city size , Reg io n a l Scien c e & U r b a n E c o n o m ic s[END_REF] summarizes some interesting values:

As one can see, the city system resulting from a lower degree of scale economies exhibits a more homogenous structure where the leader is not much larger than the followers and where the production is more evenly spread. More important, the exis tence of a decentralized equilibrium configuration becomes more likely the lower the degree of scale economies is (14> : when economies of scale are small, production will Table [START_REF] D U R A N T O N | Labor specialization, transport costs and city size , Reg io n a l Scien c e & U r b a n E c o n o m ic s[END_REF] be spread more evenly throughout the system of cities and sizes will tend to be similar (the differences in sizes being solely due to exogenous first nature factors).

In this section we have tried to take into account the interactions between urban ization and technological change. We have managed to show that it is important to "[...] realize that the level of urbanization determines not the overall level o f tech nology but its rate of change, whereas in the reverse direction, the overall level of technology determines the distribution o f population between city and country-side, and changes in technology to changes in the level o f urban population " M o k y r J., [START_REF]Urbanization, Technological Progress and Economic History in "Ur ban Agglomerations and Economic Growth[END_REF]. As we have shown, changes in technology not only lead to changes in the level of urban population, but can also lead to profound changes in the structure of the urban system. Hence we are able to capture both the quantitative and the qualitative effect that technological evolution can have on systems of cities.

Conclusions

In this paper we have extendend our simple two stage model of the formation of agglomerations in a pre-industrial setting. First, we have discussed the issue concerning potential coordination failure in the emergence of the urban phenomenon. As we have seen, break-down, inefficiency and efficiency are all possible scenarios; which of those scenarios is the most likely is difficult to discuss from a purely economic point of view. The important point of this analysis is to highlight the fundamental role of historical accidents and the subsequent lock-in of the realized urban configurations. Second, we have shown how transaction and coordination costs are a major impedement to urbanization and the size and scope of the economy. This important point has been neglected in spatial economics until now (probably because the link between space and transaction costs is less obvious than the link between space and transportation costs) but needs to be taken into account if we want to better understand both present and past urban development. This point seems to be of particular importance in the explanation of urbanization in the 20th century. Finally, we have analyzed how feed-back effects between urban-and agricultural development can be modeled. Although some of our assumptions seem to be ad hoc, our model accounts for most of the stylized historical and economic facts concerning the development of urban systems. We are especially able to explain a two step concentration-diffusion sequence in a relatively simple trarily small scale the M-goods it needs. In this case there is no transportation at all and the economy is characterized by some sort of "backyard capitalism". This result is of course related to S t a r r e t s spatial impossibility theorem and should not come as a surprise. sequential setting. As we have seen, the two approaches yield relatively different results. While increasing agricultural productivity seems to strengthen the existing polarization of the urban system, the diffusion of technical progress can lead to a dispersion of the production of non-agricultural goods. Clearly more work is called for here.

Despite the variety of results obtained, some major limitations of our model need to be overcome in the future. First, this model is aimed at explaining urban development in a pre-industrial setting. It is clear that any application to more modem periods is not desirable, especially since the role of agriculture has diminished a lot these last 200 years. The absence of an urban land-market could also be criticized, although this hypothesis is often used in order to simplify complex issues in city-system models.

The assumption of collective production, although pertinent in certain circumstances of very early urbanization, needs to be relaxed in the future. The major problem in this case is the need for another logical controlling device concerning the sizes of the agglomerations. We do not believe that the fiction of pure self-organization is very useful in order to understand the evolution of urban structures. Finally, some will argue that our model relies too heavily on numerical examples. While this is true to some extend, we remind the reader that our approach deals with very strong non-convexities, which render analytical investigation extremely difficult. Despite the relatively simple analytical expressions of most of the important functions, our model includes non-differentiabilities, discontinuities, increasing returns to scale, fixed city locations and a constrained consumer problem. Obtaining clear-cut analytical equilibrium results in such a spatial-sequential setting seems to be, if not impossible, at least very complicated.

igu re 1 :

 1 A n exam ple o f s p a tia l c o n fig u r a tio n w ith k = 4

F ig u r e 3 : E v o l u t i o n o f £ in c a s e 1 w i t h 6 =

 36 0.05

  Refer to Appendix A for further explanations on how to run this model numerically.The sample M ath Cad spread-sheet which we have used for running the simulations can be obtained from the author.

( 14 )

 14 When there are no scale economies, each agglomeration produces locally on an arbi-

notwithstanding a priori argum ents, urbanization has been neither a necessary nor a sufficient condition fo r technological progress

  clearly highlights, "[...]

	[...] the

existence of cities does not depend on their role as prom oters of technological progress" .

(13) Refer to Appendix A for more details on the numerical procedures.

n R e f e r e n c e s

Appendix A : Numerical simulation procedure

All numerical simulation results of our model have been computed with Mathsoft's Mathcad 6.0. In this appendix we will briefly outline the main procedures we have used. The sample spread-sheet is given with this appendix and can be implemented as is with Mathcad.

In order to compute the adjustments of Section 8 , proceed as follows :

• set FLAG=1 if you work with the threshold model of Section 8.2, set FLAG=0 if you work without threshold as in Section 8.1.

• set the values of v (coordination), £ (agricultural productivity eifect) and e (threshhold parameter)

• if you work with threshold (case FLAG=1), set f( x ) := y/x, goto "Setting the followers' critical mass", copy the value of SEUIL and paste it on the redefine line below

• set / (x) := 0.6 + 0.3x; goto "Setting the followers' critical mass", copy the value of SA and paste it on the redefine line below, copy the value of POP and paste it on the redefine line below

• copy the value of SA and paste it at the beginning on the line "Follower surplus"

• re-set f( x ) y/x\ goto "Additional M-firms possible" to see the results that are obtained in the following lines below