Prof Louis De Mesnard
email: louisae-mesnard@u-bourgogne.fr

Infected persons by the fact of a public

Keywords: AIDS, hepatitis, infection, compensation, graph. JJEL Çl^ific^0P.I18

organism: is a fair compensation by the community realistic? A graph-theory approach

Introduction

A general tendency inside developed countries is the following: the national community have to compensate financially the disease generated by an infection contracted by the direct fact of a public organism \ as in the case of AIDS or hepatitis C contracted during a blood transfusion. This is fair: the community must support the consequences of its negative actions, even if they are involuntary 2 3 and even if the number of persons that are concerned by this 1

The paper focuses on the case of public health organisms, even if in some countries, as France, both public and private health organisms can be sued, as private individuals, as hospital directors, doctors, etc. However, this is not the aim of this paper, even if some conclusions can be adopted. A first difference is that the number of victims of a same private individual is lower, even if the possibility of compensation that can be paid by a private agent is lower. A second difference is that insurance systems can be suitable more easily to pay the compensation of the injury.

Note that, the same principle can be discussed for other diseases contracted by the fact of a private organisms as cigarette producers, because it can be argued that the infected persons were knowing the risk: it is not the aim of this paper. In the paper, only infectious diseases will be considered.

2

The question of a faulty behavior of the infecting organism is left out of this analysis; in the French law, the health organism must have done a fault to be condemned to pay a compensation but the victim does have to prove the fault of the health organism: this one has to prove its no-fault behavior; for [START_REF] Brazier M | Can no-fault analysis ease the problems of medical injury litigation?[END_REF] and [START_REF] Harris John | The injustice of compensation for victims of medical accidents[END_REF], compensation must be automatic because the distinction between two categories of victims infection can be large (even if data, are difficult to know) what is problematic for a compensation of their injury (perhaps this is why the French compensation law for post-transfusion hepatitis C infection that were planned in 1994 has never be achieved).

However, the case of indirectly infected persons can be examined. I call indirectly infected persons, the persons that have contracted an infectious disease as AIDS or hepatitis C from another person that have itself directly contracted it by the fact of a public organism, or that have contracted it from a person that have itself indirectly contracted it, etc. Usually, these persons are excluded from compensation but the fairness of such a decision can be discussed: do indirectly infected persons have to be compensated by the community? Generally, the programs of compensation by the community do not envisage to indemnify the individuals that are not directly infected by the public agent but that are only indirectly. Yet, the responsibility of the public agent is engaged, because if the initial infection could have been removed, the indirect infection could have be avoided itself: the initial cause is the behavior of the public organism 4 .

(victims after a faulty behavior of the health organism and other victims) is unfair.

3

The question of how much has to be paid for a fair compensation does not concern this paper. About the theoretical aspects of the question, see [START_REF] Johansson Per-Olov | Evaluating health risks[END_REF]. About the measure of how high is the disability introduced by such a disease, see for example (Bowling, 1995, pp. 271-273). Note that a potential disease can be compensated even if the infection simply generates a fear (Dyer, 1998b).

4

Note that the responsibility of a public health organism cannot be reached for infections that are easy to contract in the ordinary life, even for nosocomial infections; for So it is unfair not to indemnify the disease of indirectly infected persons when the responsibility of the public agent is indirectly engaged. Justice imposes a yes as answer to the question of the compensation of indirectly infected private individuals by the national community when infection has been caused by a public organism, even indirectly 5 . However, the determination of who have been infected is difficult. In this paper, graph theory will be used to define precisely what means "indirectly infected", then to explore the consequences and the difficulties generated by the question of compensation of indirectly infected persons. example, B hepatitis can be transmitted involuntarily by a simple oral contact when hepatitis C or AIDS can be transmitted only by blood transfusion or sexual contact. Also, some public health organisms -as in France -have an obligation of result, not of mean: they have to guarantee the patients against all infectious agents, known or unknown. This idea is not similar to the idea of compensation of the victim by ricochet. In the case of the neonatal infection both concepts, indirect infection and infection by ricochet, can be confused: the child is a victim by ricochet if its parents decease or cannot have a job.

Another example: for the Creutzfeldt-Jakob disease (remember that this disease cannot be compared with AIDS or hepatitis C: it is not proved that it can be transmitted to another human), some families have obtain compensation for the infection and/or the death of their children [START_REF] Dyer | British families win compensation over CJD deaths[END_REF]1998a). About these juridical problems, see [START_REF] Gromb | Les problèmes médicaux-légaux de la transfusion sanguine[END_REF].

II. Indirect infection by the public health organism

A. The idea of indirect infection

Denote i and j the indices for private individuals that have infected or that have been infected.

A binary relation is set up: considering a private individual j that was not infected before, i is in relation with j if i have directly infected j\ what is denoted: iRj <=> hy = 1 and hy = 0 else: the term hy is a boolean variable that indicates that j have been infected by / 6 . Naturally ha = 0. Re-infection is excluded at this step, so there are no circuits; for example, if hy = 1, then hjt = 0. All terms hy are regrouped in a matrix H. Matrix H has only one 1 by column (even if it is possible to find more than one 1 in each row). For reasons that will be explained now, matrix H has one additional row and column.

H = 0 . • h u . • h\" 0 ha • . 0 . • h in 0 h"i . • h ni . . 0 0 0 . 0 . . 0 0
A supplementary agent has to be introduced: it is any public organism, of which index is denoted s (for simplicity, it is assumed that the number of public organism is 1). Also, some private individuals can have initially contracted the disease without the help of the public agent and without the contact of any other persons infected by the help of a public organism. I call them "naturally infected agents". The vector e denotes the initial infection. One has e s = 1; the fact that / is a naturally infected agent is denoted e 7 = 1. There can be many zeros in vector e At this step, the relation is assumed to be deterministic. because all individuals, naturally infected or not, are in e: for individuals / that are not naturally infected, the term / in e is zero.

The public organism and naturally infected persons can be seen as exogenous entries of the system made up of private individuals. One have only two types of initially infected individuals: those that are initially infected by the public organism s --compensation by the community is fair for them -and those that are initially infected by an agent k -compensation by the community does not seem fair (even if these persons are relevant of an health care system to pay the cost of their disease, as for everybody) --. The direct infection by the public organism or by initially infected agents is given by the vector e (1) = H 7 e.

Example.

H =

0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 00000000 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 00000000 00000000 1 0 0 0 0 0 0 0 12 3 4 5 6 7 So, hf = h\/ * h/j, where * is the boolean product, obtained by multiplying H with itself by the boolean product, denoted: (HO 2 = (HO * (HO because e (2) = H' e (1) = (HO 2 e. This can be generalized for a /-arcs chain: hf is the term of (H0 / , where t is an index of iteration. As the length of the chain is ly = t, the higher t is, the more indirect is the infection between individuals. 1 SJ indicates the length of the infection chain from the public organism to the person j: l sj = 1 + ly if i was directly infected by s and if j was infected by / indirectly. In the above example, s has infected 1, that has infected 2, that has infected 3, etc. This corresponds to a chain in a tree (see the figure 1 that corresponds to the above example):

s -> 1 -» 2 -» 3.
Note that a same individual has only one backward chain, beginning either by s, either by a naturally infected person, the individual 5 in the figure. indicates who was infected by all chains of direct or indirect infection from the public organism s or from a "naturally infected agent": in W can be found the "successors" of the public health organism or of naturally infected agents. This tends to (I -HO" 1 e.

Example (following^:

f 1 ] fo f o > r i) 0 1 0 0 i 0 0 1 0 i 1
, e< 2 > = 0 , e< 3 > = 0 , e< 4 > = 0 , etc., and w = i 0 0 0 0 , etc., and

w = i 0 1 0 0 i 0 1 0 0 i 1° J lo J lo J lo J li J
As only infected individuals are taken into account, the vector w is full of zeros. •

To know who is infected by the public agent, it is simply sufficient to replace the vector e by the vector e p such that (e 7 ')' = ( 0 ... 0 1 ). The direct infection caused by the public agent is given by e P(1) =e /> H'. A vector w p is computed as Example (following).

w /> =
( 1 ] f° 1 fo ) fo f 1 1

0 1 0 0 i 0 0 1 0 i 0 0 0 0 , etc.
, and w p = 0 0 0 0 0 , etc., and

w p = 0 0 1 0 0 1 0 0 0 0 0 lo J lo J lo J lo J ll J
Here, agents 1, 2, 3 and 6 are directly or indirectly infected under the direct or indirect responsibility of the public agent. •

Compensation for indirect infections

One has two types of individuals.

• The individuals, j\ that are not directly or indirectly infected by a public organism, along a chain of any length. For those individuals, there exists no chain of any length exists beginning by s. Thus, all chains, that can be set up to j, do not begin by s: wf = 0 for all n.

For these individuals, clearly a compensation by the community has not to be set.

• The other individuals, for whom a chain -of any length ~ can be set from s to them: wf = 1. Many of the individuals are not directly infected by s, but only indirectly.

Unfortunately, there are more people with wf = 1 than people with ef (1) = 1, so the cost of compensation is higher when indirect infection is considered. This intuitive idea is easy to prove. First, the number of individuals directly or indirectly infected by the public organism can be globally estimated by y~, where N is the number of persons initially infected by the public agent and p is the overall probability of transmission of any agent / to any other agent j (remember that it is the overall probability and not the probability at each period): you have N agents directly infected, Np indirectly infected at the first iteration, Np 2 at the second, etc.;

this is decreasing but the total number Y~ ls higher than n and can be much higher if p is strong. Second, in the boolean product (HO 2 , you have some "1" that are not in the same place than in H'; so, in the boolean sum H' + (HO 2 , you have generally more " 1" than in H';

a similar thing happens for (HO 3 and H' + (HO 2 , etc.

Compared to the number of persons indirectly infected by a naturally infected person, the number of persons indirectly infected by the public organism can be large or low: it depends on the data 7 , i.e., on the structure of H and e. The computation has to be done, so an information problem occurs. Only a precise study of the direct causes of contamination of all infected individuals of the society (even if their infection is not yet declared), followed by the computation of indirect infection chains, can help to answer to this question 8 . This is not realistic, or this can be realized at a very high cost.

What is the information cost? There are two ways to evaluate who is infected. First, starting from an infected person, the question is: who has been infected by this person from the moment of its infection; it is the forward search. Potentially, there are n persons, minus the persons that are known to be yet infected. Second, starting from an infected person, the question is: by whom this person has been infected; this is the backward search. There is, by definition, only one person, and the complexity of the search is similar to the forward search but this will be not the case in an improved version of the model where some indecision may occur.

7

That is on the behavior of the people directly infected and on the disease: probably, these people have a less risky behavior than naturally infected persons regarding to the problem of AIDS infection, but for hepatitis C, the possibility of transmission does not depend on a risky behavior.

Do not confuse this question with the question of the estimation of HIV incidence over time from surveillance data, that is to say the estimation of the number of people that develop and AIDS from the number of persons simply infected by the HIV virus (Lessner, 1998a and b). I leave out the distinction between the fact to be infected -by example by the HIV virusand the fact to be effectively ill -by example to have effectively develop and AIDS syndrome.

Moreover, matrix H and vector e have a very large number of rows and column, even when the study is restricted to actually infected persons 9 , so the cost of search is very high because it is proportional to the number of arcs. Fairness, that implies here perfect information, is impossible to reach in a democratic society unless to establish the inefficient Orwell's "big-brother" society.

C. Decreasing responsibility and probabilistic transmission

1.

Decreasing responsibility

The hypothesis of a decreasing indirect responsibility of the public organism can be also introduced: in this case, the responsibility of the public organism decreases when the length of the infection chain increases. So the jcan be an indicator of the decreasing responsibility of the public organism on the indirect infection of the person j. A less artificial view consists into considering that in the chain s -» / -»y ->f ->/' ->it is not acceptable that j could obtain a higher compensation than / has obtained itself, only by the fact that j has the possibility to sue both / and s (even / is a private individual) when i is able to sue only s. So, the responsibility of the infection of j must be shared between s and i: s is fully responsible of the infection of /, s and / become equally responsible of the infection ofy, then s , / and j become equally responsible of the infection of /', etc. So, naturally, the initial responsibility of the public organism becomes decreasing. The choice between one principle or the other depends on legal principles what goes outside the subject of this paper.

But I have said that an individual can be infected not initially but only later: potentially, all the population of the country could be considered.

2.

Probabilistic transmission

It has been assumed that only one person can be infected by only one person but, when is considered the possibility of multiple partners and multiple acts per partner, some indecision remains about who have been really infected by whom: often, it is only possible to know with whom a person has had a potentially infectant contact (as a sexual relation) but not if this contact was really infectant (for example, Xhas been infected; he has had a contact with Fand Z, who are declared infected after a test, but Fand Z declare that they have had also a contact with other persons 10 ).

For these reasons, the transmission relation h becomes probabilistic; see: [START_REF] Pinkerton Steven | Cost effectiveness of post-exposure prophylaxis following sexual exposure to HIV[END_REF]Bloom, 1998, p. 1070). So, the precision of the search of infection causes can be considered as rapidly decreasing. Consider a probability py of truth for the information:"/ has infected /'. Naturally, considering the complementary event, "j has been infected by another cause than /" and considering its probability p-j, it follows that Py+Py = 1. Along a chain s -> / ->j ->/ -» ...j (n ~l ) -»j (n) , these probabilities of truth are multiplicative:

Psj-PsiPyPjj'

...pfli-Djb).

As all probabilities are between 0 and 1, the precision of the determination of infecting causes decreases rapidly when the length of chains increases: the probability psj tends toward zero when n tends to infinite. When a probability near zero is assumed to be equal to zero, the responsibility of a given infectious organism or person is 10 An additional difficulty occurs when a person is infected but the disease is not declared (or is hidden by the medical treatment (Zhang et alii, 1998): this person can be infectant without itself knowing). In this case, investigation can be difficult, or even impossible, because uncertainty is maximal.

limited forward: in practical terms, chains with only two up to four steps have to be considered instead of infinite chains what is more practicable.

On the other hand, when the transmission relation is probabilistic, if an indecision remains, one can admit that there are equally placed in the infection problem: one person can have been infected by multiple partners (the person can have had multiple potentially infectant contacts with many persons that are all infected, without the possibility to say that this is one or the other that has infected him). In this case, there can be more than one "1" in a same column in matrices H. As a consequence, forward and backward searches have to be combined, the complexity of search is larger, the cost also.

III. Re-infection

1.

Complications induced by the idea of re-infection

Depending on the considered disease, a re-infection may have no additional effect, or may have a serious effect. When re-infection does not add to the disease of the person, this re-infection can be neglected without consequences and the resulting graph remains a tree again (trees are more simple to study).

In the second case, if re-infection adds to the disease of the re-infected person, one can think about the case of the re-infection of any private individual /: a same individual has now many backward chains. As a general principle, a fair attitude could consist into compensate only the increasing of the disease introduced by the additional infection, when it can be calculated what is no so easy: again detailed investigations are necessary but they can appear as incompatible with respect of of private life and democracy.

However, the most important case is those of naturally infected persons / that are re-infected by a person j that have itself the public agent s in its chain: even partially, these persons i leave the category of naturally infected persons and are able to receive compensation, as the other persons that will be infected by them later.

Loops in the chain are even possible but they are not true loops because time flows here: in a static ordinary graph, a loop is run through an infinity of times, so re-infection is repeated again and again, what is not realistic. A dynamic version of the model has to be introduced by taking time into account: loops become pseudo-loops and they will be run through only one time.

Remark. Short pseudo-loops, as -> i are normally poorly re-infectious (the infectious agent is exactly the same and has a low additional effect). If they are long, these pseudo-loops can have an additional infectious effect because the infectious agent can have been modified between the first and the second infection (example: mutations of the virus)

and even because a simple re-infection with the same infectious agent can reactivate the immunological process 11 . •

In dynamics, things are more complicated than in statics. It can be assumed that there is an initial period, denoted t = 0, where the infection by the public organism (and by the naturally infected agents) occurs, and there are following periods where private individuals that have 11

The case of multiple infections is left aside: it is known that an infection with a some virus as hepatitis C or AIDS can worsen another infection as rheumatoid polyarthritis.

been infected at preceding periods can infect other private individuals. Denoting by H, the matrix H for period t then the product (HO 2 has to be replaced by H' (2) = H' i * H' 2 , then (HO 3 by H /(3) = H' 1 *H / 2*H / 3 , etc., up to H' (/) = H' 1 * H' 2 *... *H that replaces (HO'. Finally: w = e + e (1) + e (2) +... + e ( ' } +... = e + H' (0) e + H' (1) e (1) +H' (2) e (2) +H' (3) e (3) + ... + H' ( ' > e (/) + ... where H' (0) e corresponds to the initial infection by s and by the naturally infected agents, without indirect infections, at the beginning of the epidemic study (f = 0 stands for the beginning date of the study).

Remark. With this dynamic model, even the decease of some persons can be taken into account. For the persons i that are deceased at a date /, the row i of H (/) becomes entirely full of zeros. • Example.

0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 6 H<» = 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 1 0 0 0 0 0 0 _ s 0 0 0 0 1 0 0 0 0 0 s 1 2 3 4 5 6 7 8 9 s 1 2 3 4 5 6 7 8 9 s 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 0 6 H< 3 > = 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 1 0 7 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 _ 5 _ 0 0 0 0 0 0 0 0 0 0 _ s 1 2 3 4 5 6 7 8 9 s 1 2 3 4 5 6 7 8 9 5 0 0 0 0 0 0 0 0 0 0 " 1 f 11 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 6 and e = 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 s 1J 1 2 3 4 5 6 7 8 9 s In this scholarship example, where agent 1 is a naturally infected individual, you can find two pseudo-loops, 1 -• > 2 -»7-»9->l and 1 ->3 -»4 ->6 -> 1 (see figures 2 and 3). In the first case, there is a loop but it is run through one time only. In the second case, the same problem occurs but, in addition, the temporal causality is not respected: this pseudo-loop is not run through even one time and it is only an artefact: the naturally infected agent 1 has been indirectly re-infected by s through 4 and 6 but 1 has no responsibility in the infection of 6 because the re-infection of 4 (through 3) is later. The same thing happens for 1 -»8-»5-»2-^7-^9->1; for 8 5 -> 2 and 5 -» 2 -» 7, temporal causality is not respected: even if 2 has been indirectly re-infected by s through 5, s has no responsibility in the infection of 7 and 9. •

The idea of period itself can be discussed: is it the year, the month, the week, the day, or less?

It is not only a question of time basis: the responsibility of agents depends on it. Inside a same period, all infections are assumed to be simultaneous. For example, if the organism s infects j, and at the following period, j infects j\ so s has infected j indirectly, when both periods are confused into only one, either s appears to have infected s and ? directly, either s appears to have infected only j and for / the origin of infection appears to be unknown. The second possibility is more plausible. Again, the complexity of the problem, and its contestability, appears clearly.

2.

Responsibility and compensation

For the individuals that are infected by a naturally infected individual / that have been indirectly re-infected by the public organism, before the re-infection, the public organism is not responsible, when after it is (again the static model does not work and the model must be dynamic). Moreover, when time passes, by the mechanism of re-infection of naturally infected persons, the ratio of the number of individuals that have s in their chains under the number of persons that fall under the first category will increase. So, if it is considered that re-infection has bad additional consequences, then when time passes the infected population, eligible for a compensation by the community, will increase even if the compensation concerns only the additional consequences. So, the cost of compensation will higher than in the case without re-infection, unless compensation per capita is reduced.

IV. Conclusion

It is not fair that the community compensates only individuals that are directly infected by the public organism. Persons that are indirectly infected by the public organism have to be compensated by the community. With the help of graph theory, it is demonstrated that this implies two consequences. First, the number of persons to be indemnified is larger than for direct infection, so the cost of compensation is higher, unless if compensation per capita is lower. Second, the quantity of information that is required is so large that it is unrealistic and contradictory with democracy. Third, the precision of the determination of the infecting causes, as well as the responsibility, decreases rapidly when the length of chains increases. In remember that loops are only pseudo-loops)

  a naturally infected individual. • Chains of indirect infection can appear. One say that an agent j is indirectly infected by an agent i if there is a chain between / and j. For an indirect infection implying only one intermediary individual (the chain is two arcs long): Yif = 1 <=> there exists a person / such that {/*,/ = 1 and hfj = 1}

Figure 1

 1 Figure 1 about here

  e /> [i + H' + (HO 2 + ... + (H0' + ...]: it indicates all individuals to be compensated for their infection by the public agent.

Figure 2 about here Figure 3

 here3 Figure 2 about here

Figure 2 .

 2 Figure 2. Example: graph with re-infections (arcs in bold: re-infections; time flows from up to down)


Carl Gaigné

Pamina Koenig-Soubeyran

Pierre-Philippe Combes

Jacques Thisse

Jean-Marie Huriot

Christian Mi

Keywords: trade liberalization, trade costs, agglomeration JE L Classification : D ll, F12, L13, R12

Trade liberalization and trade policy obviously affect the internal distribu tion of economic activities within a liberalizing country. We show that the impact of decreasing international trade costs on the regional distribution of economic activities crucially depends on the value of transport costs in ternal to the country. Trade liberalization in developing countries with poor internal infrastructures and small volumes of interregional trade is likely to increase regional disparities, while developed countries with good internal \ infrastructures and large volumes of interregional trade are likely to experi ence redispersion. We argue that the way transport and trade costs are modeled might have a crucial impact on the results obtained. Résumé La libéralisation du commerce et la politique commerciale affectent mani festement la répartition interne de l'activité économique d'un pays. Nous montrons que l'impact d'une baisse des coûts d'échange sur la répartition régionale de l'activité économique dépend de manière cruciale de la valeur des coûts de transport à l'intérieur du pays. Tandis qu'une libéralisation dans les pays en voie de développement, caractérisés par de mauvaises infrastructures et de faibles volumes d'échanges interrégionaux, est susceptible d'accroître les inégalités régionales, cette même libéralisation dans les pays développés est susceptible de mener à une redispersion de l'activité. Nous montrons que la manière dont nous modélisons les coûts de transport et d'échange pourrait significativement influencer les résultats obtenus.

Intro d uction

It is obvious that world economic activity is unevenly distributed around the globe. At a large spatial scale, the most striking manifestation is to be found in the North-South dualism opposing the developed and the developing world (Fujita and Thisse [START_REF] Fujita | Economics o f Agglomeration: Cities, Industrial Location and Regional Growth[END_REF]). At a much smaller spatial scale, the internal distribution of economic activity within countries is also highly uneven. Most countries are made up of core regions, characterized by high "economic density" and a large share in both GDP and employment, as well as of peripheral regions which contribute only marginally to both GDP and employment. This phenomenon seems to be universal in the sense that it affects more or less all countries independently of their level of technological, social and economic development. The Sao Paulo region in southeast Brazil is as much an economic core of the developing world as are the Ile-de-France and Tokyo MA for the developed world. The really striking difference between the developed and the developing world is that the overall level of regional inequalities seems to be even larger in developing countries than in industrialized ones. The most visible manifestation of those regional imbalances are urban giants like Mexico City and Bangkok, that dot the developing world since several decades. Although it is extremely difficult to weight the economic costs and benefits of those unequal regional patterns, we need to understand how they form in the first place and how changes in economic key parameters as well as in policy possibly affect those imbalances.

Since the seminal paper by Krugman [START_REF] Krugman | Increasing returns and economic geography[END_REF] and its subsequent refinements (refer to Fujita and Krugman [START_REF] Fujita | When is the economy monocentric: Von Thünen and Chamberlin unified[END_REF], Puga [START_REF] Puga | The rise and fall of regional inequalities[END_REF] and Fujita et al. [START_REF] Fujita | The Spatial Economy -Cities, regions and international trade[END_REF]), it is well known that decreas ing transport and trade costs within a country are susceptible to increase the spatial concentration of economic activity. [START_REF] Bairoch | De Jericho à Mexico -Villes et économie dans l 'histoire[END_REF] While this result fits well the trend observed in developed countries since the Industrial Revolution, it fits less well that of developing [START_REF] Bairoch | De Jericho à Mexico -Villes et économie dans l 'histoire[END_REF] As recently shown by Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF], those conclusions remain valid in an alterna tive modeling framework with different specifications for consumer preferences and transport technology. Therefore, the main result of the core-periphery model, namely that economic agglomeration arises for sufficiently low values of trade costs, seems to be robust with respect to alternative modeling choices, at least in a two region setting.

V 3 countries in which, despite usually high values of interregional trade costs, huge urban agglomerations and regional concentrations of economic activity have emerged. How come? As recently shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], interregional trade is not necessary for an uneven distribution of economic activities to arise. Despite high levels of internal trade costs, a process of agglomeration can trigger in a country as the ratio of mobile to immobile factors of production increases. This finding concurs with Bairoch [START_REF] Bairoch | De Jericho à Mexico -Villes et économie dans l 'histoire[END_REF], who argues that population growth and increasing pressure on limited amounts of arable land have released large numbers of workers from the yoke of immobility and have redirected them towards the growing urban areas. While these internal causes might partly explain the core-periphery structure of several developing countries, external causes must also be accounted for. Yet there is, to the best of our knowledge, no clear consensus on which forces are susceptible to play an important role nor on how they work for or against regional imbalances. A major exception, that has attraced a lot of attention from both development economists and international trade theorists, is the potential role of international trade policy.

That international trade and trade policies have a significant impact on the in ternational distribution of economic activity is hardly debatable. Those aspects are emphasized in new trade theory, in which it is argued that factor mobility and self reinforcing forward and backward linkages can lead to regional specialization as well as to significant and persistent differences in economic development and real wages (see Krugman and Venables [START_REF] Krugman | Globalization and the inequality of nations[END_REF], Venables [START_REF] Venables | Equilibrium locations of vertically linked industries[END_REF] and Puga and Venables [START_REF] Puga | Agglomeration and economic development: Import substitution vs. trade liberalisation[END_REF]). While factor mobility (essentially in the form of capital) increased between developed countries and from developed to certain developing countries, factor mobility between developing countries and from developing to developed countries usually remains quite low. This is most clearly seen from the figures provided by Wong [START_REF] Wong | International trade in goods and factor mobility[END_REF] (p. 18), which show that developing countries' share in world FDI never exceeded 7% during the period 1971-1990. We argue in this paper that, even in the absence of international factor mo bility, international trade and trade policies have a significant impact on the internal geography of a country, as soon as production factors are at least mobile within that country. There is a huge literature on whether international trade liberalization leads to economic growth and long-run income convergence between countries (refer, e.g., to

McCulloch et al. [START_REF] Mcculloch | Trade liberalization and poverty: a handbook[END_REF]). There is nearly no literature, and especially no consensus, on whether international trade liberalization leads to stronger agglomeration of economic activities within a given country or whether we can observe a gradual dispersion as the country progressively opens up to trade. Five major exceptions are the papers by Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF], Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF], Crozet and Koenig-Soubeyran [START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF],

Paluzie [START_REF] Paluzie | Trade policy and internal geography[END_REF] and Montfort and van Ypersele [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF]. Those papers develop general equi librium models that analyze how trade liberalization (given by a progressive decrease in international trade costs) possibly affect the spatial structure within a country. While Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF] "predict" internal dispersion of economic activities due to a progressive trade liberalization, the other papers "predict" an increase in regional dis parities within countries. 2 This divergence in results, obtained within otherwise quite similar Dixit-Stiglitz-Iceberg modeling frameworks, is probably due to a difference in the modeling of dispersion forces. While Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF] focus on urban rents/commuting costs, the other papers retain the more traditional assumption of a partially immobile population. [START_REF] Baldwin | The core-periphery model and endogenous growth[END_REF] We believe it is important to mention right away that, no matter the dispersion forces used, the results established in the current literature do not depend on the relative values of international to interregional trade costs. As we argue in this paper, the conclusions derived in a Dixit-Stiglitz-Iceberg framework might crucially hinge on the fact that, under such a specification, decreasing international trade costs do not modify the share of interregional trade costs in consumer prices. Yet, decreasing international trade costs exacerbate price competition within countries, which can significantly increase the share of interregional trade costs in final consumer prices.

2 Both Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF] and Montfort and van Ypersele [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF] develop models with two countries and four regions. They show that international trade liberalization affects the internal geography within countries and that those internal geographies are interdependent.

Hence, there is spatial correlation in the sense that countries' internal structures influence each other mutually. Whether this correlation is positive or negative seems to depend on the chosen modeling framework. O See also Fujita et al. [START_REF] Fujita | The Spatial Economy -Cities, regions and international trade[END_REF] (Chapter 18) for a simpler approach in terms of a pure congestion externality.

Hence, it becomes relatively more expensive to ship goods within the country, which can actually lead to a reversal of the conclusions established in the aforementionned papers once international trade costs are sufficiently low.

We develop in this paper a linear three region general equilibrium model based on Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF] and Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF]. We show that the relative level of international to interregional trade costs, the share of the domestic industry in the world industry and the composition of trade are all crucial in explaining the impact of trade liberalization on the internal geography of the liberalizing country. Trade liberalization in countries characterized by high values of interregional trade costs and small volumes of inter regional trade usually leads to regional divergence, while that same liberalization in countries with low values of interregional trade costs and large volumes of interregional trade usually leads to regional convergence. Those results could partly help clarifying the debate on whether trade liberalization leads to regional convergence or not. They could also explain why trade liberalization does not seem to lead to regional conver gence in developing countries, while several developed countries like Great Britain seem to experience a gradual redispersion of economic activities.

The remainder of this paper is organized as follows. In Section 2, we lay out the basic three region model. In Section 3, we examine the case of a developing country with high values of interregional transport costs and no interregional trade. In Section 4, we turn to the polar case of a developed country with low values of interregional transport costs and bilateral interregional trade. In Section 5, we tie together the seem ingly conflicting results of the literature and offer a unifying interpretation. Finally, Section 6 offers some conclusions and points towards future research directions.

The model

Let us briefly lay out the formal framework we use in the rest of this paper. As sume there are three "regions" labeled H , F and R respectively. Variables associated with each region will be subscripted accordingly. Regions H and F belong to the same country C, while region R corresponds to the "rest of the world" (henceforth ROW).

There are two production factors in the economy: geographically mobile manufac turing workers, which produce a differentiated good under monopolistic competition and increasing returns to scale, and immobile agricultural workers, which produce a homogeneous good under constant returns to scale and perfect competition. Denote by L (resp. by L r ) the mass of mobile and by A (resp. by A r ) the mass of immobile factor in country C (resp. in the ROW). The immobile factor A is evenly split between the two regions H and F , each of which accommodates a mass A /2 of it. In accord with empirical observations, we assume that manufacturing workers are characterized by a low international mobility. In order to keep things as simple as possible and to remain in the spirit of international trade theory, we make the assumption that labor is internationally immobile (see Venables [START_REF] Venables | Equilibrium locations of vertically linked industries[END_REF] and Fujita et al. [START_REF] Fujita | The Spatial Economy -Cities, regions and international trade[END_REF]). Hence, both L and L r are constant throughout the analysis. Nevertheless, workers in the manufacturing sector are mobile within country C. We denote by A G [0,1] the share of mobile factor L located in region H and we assume, without loss of generality, that there is no immobile agricultural sector in the rest of the world (i.e. A r = 0). Denote finally by N = nH + n F + n R the mass of firms in the world economy, where n r denotes the mass of firms located in region r = H ,F , R.

All agents in the economy have the same quadratic utility and solve the consump

tion problem r* 8 -1 f N 7 ( f N \2 maxXOiX a ^ x ( i)d i----J ^ x(i)2di --i x (i)d i) + x0 N p (i)x (i)di + Xq = W -b XQ
where the price of the homogenous good xo is taken as the numéraire, x0 > 0 is the (sufficiently large) initial endowment in good xq and a > 0, / ? > 7 > 0 are exogenous parameters. Let e := /? -7 for notational convenience. The parameter e is an inverse indicator of the elasticity of substitution between any two varieties. As shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], the extended demand functions of ( V q ) are given by

x*(i) = [a -(b + cN)p(i) + cP(t)]\ (1)
where

and where

P (i) = [ p (j)d j Jo (3 )
is an aggregate price index of the differentiated industry.

Each firm in the differentiated industry produces a single variety under increasing returns to scale using labor only. Hence, there is a one-to-one correspondence between firms and varieties, so that nr denotes both the mass of firms and of varieties located in region r. Without loss of generality, we assume the fixed cost 0 is incurred in mobile labor while marginal costs are normalized to zero (see Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF]). Under those assumptions, labor market clearing implies that

We assume the differentiated good can be transported at no cost in the interior of each region r, while there is a unit transport cost of r > 0 (resp. a unit trade cost of t r > 0)

units of the numéraire in order to ship one unit of manufactured good between the two regions of country C (resp. between a region of country C and the ROW). We refer to r as internal transport costs and to t r as external trade costs. Note that we assume that both regions can access world markets at equal costs. Hence, international trade involves no gate effects. Denote by wr the wage rate in region r and by pr8 the price a firm located in region r charges in region s. In accord with empirical evidence (refer to Greenhut [START_REF] Greenhut | Spatial pricing in the USA, West Germany and Japan[END_REF], Wolf [START_REF] Wolf | Patterns ofintra-and inter-state trade NBER Working Paper no 5939[END_REF] and Head and Mayer [START_REF] Head | Non-Europe: The Magnitude and causes of market fragmentation in the EU[END_REF]), we assume that markets are segmented and that each firm sets a particular price in each market it sells its output in. Further, firms bear initially all transport and trade costs, even if they subsequently pass a fraction of them on to the final consumer. The profit of a firm located in region r = H , F is hence given by riH --r, tif = < P XL [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF] + <fm ^jx*r r p * r + { ^ + ^s ^K s l P r s -T]

+ 4 > n R X * R \p*R -t r \ -(j)W r , s ^r , ( 5
)
while that of a firm located in the ROW is given by nR

= ( f + ^n ) X* RH^*RN ~ + ( f + < ^riF) Xrf^rf ~ T fi] + <jm Rxk RBp*RR -<fmR. ( 6
)
As shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], the profit-maximizing prices are given by Using ( 4), ( 7) and ( 8), the aggregate price index (3) in region r is given by

Pr -Tlrprr(Pr) -f Tl8p8r (Pr) ■+■ TltPtriPr)> S ^ V ^ t . ( 9
)
Note that all profit-maximizing prices depend on the price aggregate Pr of the region the firm wants to sell in, which depends itself on individual prices set by rival firms.

Equilibrium prices are determined as in Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF]. Each firm sets its optimal price, taking aggregate market conditions as given, but taken together these aggregate market conditions must be consistent with firms' optimal individual prices. Hence, the equilibrium price index Pr is a fixed-point of [START_REF] Fujita | The Spatial Economy -Cities, regions and international trade[END_REF]. Because, as shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF],

the expressions of the price indices (9) depend on the structure of regional and international trade, we have to distinguish several cases.

First, we investigate the case of a developing country characterized by high values of internal trade costs and low volumes of interregional trade. For analytical simplicity, we assume that internal trade costs are so large that no interregional trade occurs at all.

Although this is clearly a strong assumption, we believe it captures the essence of the argument we develop later on. Second, we investigate the case of a developed country characterized by low values of internal trade costs and large volumes of interregional trade. We assume that internal trade costs are so low that interregional trade in such a country is always bilateral. [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF] Let us begin with the case of high values of internal trade costs.

Trade liberalization in developing countries

Development economists and international trade theorists have thoroughly in vestigated how trade policies and trade liberalization possibly affect the economic performance of developing countries. Yet, as argued by Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF], the question on how those policies affect the internal geography of such countries has been largely neglected. This neglect might turn out to be mistake, because it is known that the spatial distribution of economic activities within a country can have a profound im pact on growth and convergence (see Baldwin and Forslid [START_REF] Baldwin | The core-periphery model and endogenous growth[END_REF], Baldwin [START_REF] Baldwin | Agglomeration and endogenous capital[END_REF] and Martin and Ottaviano [START_REF] Ottaviano | Growing locations: industry location in a model of endogenous growth[END_REF]). In this section, we examine how international trade liberalization possibly affects the equilibrium distribution of firms in a country characterized by poor internal infrastructures and small volumes of interregional trade.

We make three assumptions on the structure of trade. First, we consider that internal transport costs r in country C are so high that there is no interregional trade no matter the spatial distribution of firms between regions H and F . Second, we assume that external trade costs t r are sufficiently low such that the ROW sells to both regions of country C no matter the internal distribution of firms. Third, we assume, without loss of generality, that neither firms in region H nor firms in region F export 4 Note that this assumption, though seemingly less strong than that of regional au tarky, is not susceptible to hold for all industries in developed countries (see Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF] and Behrens [START_REF] Behrens | Asymmetric trade and agglomeration[END_REF]).

their varieties to the ROW (refer to Appendix A for a justification). Under those three assumptions, expressions ( 7), ( 8) and ( 9) yield the equilibrium price indices

F . = « ( * + y + n«(l' + f ) m sj!r ( 10
) 2 b + c(ur -b Till)
Substituting the equilibrium price index ( 10) into ( 7) and ( 8) yields the equilibrium prices

2 a + c n R T R Prr 2[2 b + c(nr + nR)}' { li)
Note that equilibrium prices [START_REF] Fujita | Economics o f Agglomeration: Cities, Industrial Location and Regional Growth[END_REF] are increasing with respect to external trade costs tr, which shows that trade liberalization (i.e. a decrease in tr) erodes the monopoly profits of country C firms through fiercer price competition. Note further that the impact of a modification in the mass Ur of firms in the ROW is ambiguous, because a, b and c depend on the total mass N of firms in the world economy. [START_REF] Behrens | Asymmetric trade and agglomeration[END_REF] Substituting expressions [START_REF] Fujita | Agglomeration and market interaction[END_REF] and ( 11) into (1), the individual equilibrium demands in region r are given by

_ (b + cN)[2a + cnRrR} _ 2[2b + c(nr + nR)] _ ( + )Pr'" (12)
Next, we derive the import prices of goods produced in the ROW. Using ( 8) and [START_REF] Fujita | Agglomeration and market interaction[END_REF], the price of imports to region r = if , F is given by

_* _ 2o + [2(& + cnR) + cnr]TR _* , tr ----2[26 + c(nr + njj)] ~P " + T (13)
Clearly, ( 13) is increasing with respect to tr, which shows that protectionnist trade policies decrease the competitiveness of foreign firms in country C. One can further see that prices [START_REF] Head | Non-Europe: The Magnitude and causes of market fragmentation in the EU[END_REF] are decreasing with respect to the local mass of firms n r . Hence, the larger region imports goods cheaper from the ROW than the smaller region does (because local equilibrium prices of substitute varieties are lower). Neither of those two competition effects is directly present in the Dixit-Stiglitz-Iceberg framework.

Finally, individual demands in region r for imports from the ROW are given by . _

(6 + cN)[2a -(2b + cnr )TR] . XRr~ 2[2 b + c(nr + n R)] ' y '
It is easy to check that by construction x*HF -XpH = 0, because there is no interregional trade in country C. Further, since country C firms do not export, we also

have x*H R = x*F R = 0.
Under which conditions on (r, tr, tir) is a trade structure with regional autarky and imports from the rest of the world sustainable? Following the same approach as in Behrens [START_REF] Behrens | Asymmetric trade and agglomeration[END_REF], one can show that

. ~ m _ 2 a + c t i r t r ( . T -Tf(A) • 2 b + c[nH(A) + n R) ( 15
)
and

r > f/f(A) := 0, 2a + < ^^--■ (16) 2 b -h c [u f ( A) + t i r ]
must both hold in order for firms located in region r = H , F not to sell in the other region. Conditions [START_REF] Krugman | Increasing returns and economic geography[END_REF] and ( 16) hold for all values of A G [0,1] if and only if

R _ 2 a + c u r t r 2 b + criR
Further, for ROW firms to profitably export to regions H and F , both

T R ~ 2fe + cnH(A) " d n (Ji) (18)
must hold. One can check that conditions [START_REF] Krugman | Globalization and the inequality of nations[END_REF] ensure that demands for imports as well as import prices net of external trade costs are non-negative for a given spatial distribution A. Hence, firms of the ROW export to country C no matter the value of A if and only if [START_REF] Ottaviano | Growing locations: industry location in a model of endogenous growth[END_REF] where N c = t i h + t i f is the total mass of firms in country C. In the remainder of this section we assume that conditions ( 17) and ( 19) hold.

Equilibrium wages in country C are determined as in Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF] by a bar gaining process in the monopolistically competitive industry, in which firms compete for workers by offering higher wages until no additional firm can profitably enter the market. Hence, there are no pure profits and all operating profits are absorbed by the wage bill. Substituting ( 11) -( 14) into ( 5) and equating the resulting expression to zero, the equilibrium wage in region r is given by Given the quadratic utility problem (V q ) , the symmetry between firms and the structure of trade, the indirect utility in region r = H ,F can be expressed as

+ w* + a;o -nrp*rx*r -n Rp*Rrx*Rr. ( 21
)
Substituting ( 11) -( 14) and ( 20

) V* = a ( n rx*r + n Rx*Rr) -^( n rxfr + nRx% ) -^( n rx*r + nRx* R r ) 2 A V *(X):=V *h (X)-V*f (X), ( 20
) 22
In order to investigate the impact of an international trade liberalization on the internal geography of country C, we evaluate the sign of the derivative of the indirect utility differential [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF] at the symmetric equilibrium A = 1/ 2. Some long calculations show that is the break-point of the closed economy under regional autarky. If [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF] does not hold, the symmetric configuration is unstable and, as shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], either full or partial agglomeration of firms in one of the two regions is a stable spatial equilibrium.

It is of interest to note that if country C is closed (which corresponds to t ï r = 0 ), the sign of the derivative at the symmetric equilibrium depends on the sign of A \ > -A as shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], Some closer inspection of condition [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF]

R n R <f>[4b + c N c + 2c t ir ] -2c(Ab -A ) + 5c<jmR s u c h t h a t ( f u l l o r p a r t i a l ) a g g l o m e r a t i o n i s a s t a b l e e q u i l i b r i u m f o r a l l r R < T r w h i l e d i s p e r s i o n i s t h e o n l y s t a b l e e q u i l i b r i u m f o r a l l t r > t # .

Proof. Since t r is positive, the first two cases are obvious since they imply that expression ( 23) is either always positive or always negative. In those cases the nature of the spatial equilibrium is the same for all values of external trade costs Koenig-Soubeyran [START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF] and Montfort and van Ypersele [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF]). As one can see, the condition for dispersion

(Ab -A ) + | <jynR < 0 ( 25
)
is more likely to hold if the mass of immobile factor in country C is large when compared to the mass of firms in the rest of the world and if the degree of scale economies (¡> is not too large. Condition (25) can be interpreted as a relative-size effect.

If A is large so that A* -A < 0, dispersion prevails if the economy is closed (refer to Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF]). Since the second term in expression ( 25) is positive, we see that the opening of the economy to international trade (hence u r > 0) weakens the centrifugal force created by the immobile population. As stated by Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF],

"openness to trade works against convergence as the international integration process exacerbates the agglomeration forces at work within countries". In the presence of international trade, dispersion no longer weakens price competition sufficiently since local price competition is strengthened by foreign competitors. Hence, the centripetal forces generated by the large market are stronger, which explains why agglomeration is more likely to arise. Turning to he condition

(^4ft -A) + \ <jyn,R -^-[46 + cNc + 2cnR] > 0, I ¿C
we see that it is more likely to hold if N c is small and if scale economies 0 are significant. Hence, trade liberalization is more likely to lead to regional disparities in autarkic countries when those countries are small when compared to the rest of the world and when their industries exhibit significant scale economies.

In this section, we have shown that international trade liberalization is likely to increase the regional disparities in developing countries characterized by poor internal infrastructures. This result is in accordance with the ones usually obtained in the literature but opposes those of Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF]. We explain later in Section 5 more thoroughly how all those seemingly different results might fit together. But let us first turn next to the analysis of the impacts of international trade liberalization on the internal geography of developped countries.

Trade liberalization in developped countries

In the preceeding section, we assumed that developping countries are characte rized by high values of internal transport costs and no interregional trade. In this section, we focus on the polar case of a developped country characterized by low values of internal transport costs and bilateral interregional trade. Assume hence that r is sufficiently small so that firms in country C always serve both regions no matter the internal distribution of economic activity. We also assume again, without loss of generality, that firms in country C do not sell to the ROW, while the rest of the world sells to both region H and region F (refer to Appendix A for a justification). Under those assumptions on the structure of trade, expressions ( 7), ( 8) and ( 9) yield the equilibrium price index

oAT + (6 + cN) [n"T + jir tr ]

Pr = ------26T dV ------ This result shows that if there are imports from the ROW, internal transport costs must be sufficiently low for domestic firms to operate in both regions. In case internal transport costs are too high, interregional trade breaks down since consumers switch from locally imported to internationally imported varieties. We believe this aspect could explaining why many developing countries are characterized by low volumes of interregional trade but comparatively high volumes of imports from the ROW.

Condition [START_REF] Puga | Agglomeration and economic development: Import substitution vs. trade liberalisation[END_REF] ensures that regions H and F of country C always trade with each other. We still need to explicit the conditions that ensure that consumers in region H and in region F have a positive demand for imports from the ROW. This is the case if and only if

^ r , ^ 2o + c(l -\)NC T , 0(Vl S r « m := 2b + cNc • <29) and R 2a + c\Nc r , , n £ r?(A ) := 2b + cN c ' <30)
simultaneously hold for all A € [0,1]. Some straightforward calculus yields [START_REF] Winters | Regional integration and import prices: an empir ical ionvestigation[END_REF] which is simply condition [START_REF] Ottaviano | Growing locations: industry location in a model of endogenous growth[END_REF] as established in Section 3.

In the remainder of this section, we assume that conditions (28) and (31) hold.

Using ( 26) with ( 7) and ( 8), the equilibrium prices a firm located in region r = H ,F charges in its markets are given by The indirect utility differential is again given by [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF]. Or course, the conditions for an equilibrium and its stability remain the same as before. Substituting expressions [START_REF] Wolf | Patterns ofintra-and inter-state trade NBER Working Paper no 5939[END_REF] to ( 38) into (39), the indirect utility in region r can be expressed as As in Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF], the indirect utility differential is a linear function of A, so that either dispersion or full agglomeration are the only spatial equilibria. As one can see from expression (40), A = 1/2 is always a spatial equilibrium which can be stable or unstable, depending on the sign of C\Tr + C2 -f C$t. If C\tr -f C2 4-C3 T > 0, full agglomeration of all firms in one of the two regions is the only stable equilibrium, while dispersion is the only stable equilibrium otherwise. Since C\ is unambigously positive, we have the following fundamental result. and Montfort and van Ypersele [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF]). This is somewhat surprising because, as shown by Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF] as well as by Fujita and Thisse [START_REF] Fujita | Economics o f Agglomeration: Cities, Industrial Location and Regional Growth[END_REF], the two modeling approaches usually yield the same qualitative results. Note also that, on the other hand, our results confirm those of Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF], despite the fact that those authors use different dispersion forces. Further, our results obtained in section 3 conform to those established by Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF], Crozet and Koenig-Soubeyran [START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF] and Montfort and van Ypersele [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF]. How come? In the next section, we offer a first attempt at tying together those seemingly contradictory results.

Does the way trade costs are modeled matter?

As shown in sections 3 and 4, trade liberalization is susceptible to have different impacts on the internal geography of liberalizing countries, depending crucially on their internal structure of trade. For analytical convenience, this result has been estab lished for two extreme cases only: the one in which firms do not sell interregionally at all and the one in which firms always sell interregionally. 7 Note that the results we obtained in the case with bilateral interregional trade are the opposite of those estab lished in a Dixit-Stiglitz-Iceberg framework, where it is shown that trade liberalization 7 Unfortunately, the three region setting seems to be too complicated to yield any clear analytical results in the asymmetric cases. For results under unilateral trade in a two region setting refer to Behrens [START_REF] Behrens | Asymmetric trade and agglomeration[END_REF].

leads to increasing regional divergence. How come that both models yield such dif ferent results, despite very similar underlying assumptions in terms of agglomeration and dispersion forces? Further, our results confirm those of Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF],

who conclude that international trade liberalization leads to regional convergence in the liberalizing country. How come that both models yield the same results, despite very different assumptions on dispersion forces? In this section, we argue that the way trade costs are modeled matters. In particular, one should distinguish between tariff and non-tariff components because their impact upon the spatial configuration of the economy is susceptible to be different, at least when there are more than two regions.

The differences (in the case of Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF], Crozet and Koenig-Soubeyran [START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF] and Montfort and van Ypersele [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF]) as well as the similarities (in the case of Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF]) in results are most likely driven by the particular way trade costs are modeled in the Dixit-Stiglitz-Iceberg approach. 8 Recall that in this approach trade costs take Samuelsons' multiplicative iceberg form . This assumption, which implies that trade costs are de facto equivalent to an ad valorem ta riff, is often made because alternative specifications of "transportation costs can undermine the constant demand elasticity that is one o f the crucial sim plifying assumptions o f the D ixit-Stiglitz model99 (Krugman [START_REF] Krugman | The Role of Geography in Development[END_REF]). Yet, as argued by Fujita and Thisse [START_REF] Fujita | Economics o f Agglomeration: Cities, Industrial Location and Regional Growth[END_REF], "such a result conflicts with research in spatial pricing theory in which demand elasticity varies with distance [...] although the iceberg cost is able to capture the fact that shipping is resource-consuming, such a modeling strategy implies that any increase in the m ill price is accompanied with a proportional increase in transport costs, which often seems unrealistic [...] this is enough to cast doubt on the generality o f the results derived under the iceberg assumption99. [START_REF] Fujita | The Spatial Economy -Cities, regions and international trade[END_REF] We argue, in the remainder of this section, 8 One should nevertheless keep in mind that our model differs in two respects from the Dixit-Stiglitz-Iceberg approach. First, transport costs are additive instead of m ultiplica tive. Second, preferences are quasi-linear (with quadratic subutility) instead of Cobb-Douglas (w ith CES subutility). Hence, differences in results could be driven by the fact that demand elasticities vary in our approach, while they are constant under a Dixit-Stiglitz-Iceberg spec ification. Because trade costs and demand elasticities are strongly intertwined through firms' mark-up strategy, this issue is difficult to settle. More work is called for here. 9 It is surprising that this issue has only attracted little attention. Indeed, it is usually 22 that the way transport and trade costs affect prices in both approaches could explain the differences in results.

Assume that r € (0,1) is the iceberg coefficient in the Dixit-Stiglitz-Iceberg model. Hence, if one unit of any variety is shipped between regions H and F , only a fraction r of this unit arrives at its destination. Denote by prr the mill price in region r = H, F . The delivered price in region s is then given by _ Prr Prsr

The share of trade costs (pr8 -prr) ¡prs in delivered prices is given by 1 -r , which is independent of the mill price. Hence, no matter the mill price, the share of trade costs is constant, which implies that it is always equally profitable to ship the good between both regions. Roughly speaking, whether mill prices increase or decrease, firms' profit margins on interregionally sold goods remain constant.

Consider now the quadratic setting with linear trade costs r > 0. The delivered price in region s, as given by [START_REF] Fujita | When is the economy monocentric: Von Thünen and Chamberlin unified[END_REF], is equal to Prs -P8S + 2 * Note that if there are no transport costs, the firm sells at the "mill price" p*s (and not p*r). Hence, in this case the share of trade costs in delivered prices is given by (Prs -P88 )/Pr8, Which finally yields P r 8 ~Ps* = ___Z___ (4X) Prs 2 p ss -f-T Clearly, expression (41) is increasing as the mill price in region s decreases. Hence, if mill prices decrease while interregional trade costs remain constant (there is no reason aknowledged that iso-elastic demands are very specific. As noted by Winters and Chang [START_REF] Winters | Regional integration and import prices: an empir ical ionvestigation[END_REF], if demand "is iso-elastic, the mark-up is constant, so if marginal costs are fixed, the impact

effect is to pass on the whole tariff cut to consumers. Less convex (e.g. linear) demand curves generate incomplete pass-through because falling prices reduce the elasticity of demand and thus increase the optimal mark-up

why both should a priori vary simultaneously), firms' profit margins on interregional buisness are gradually wiped out (this can also be seen from the evolution of delivered prices net of trade costs, given by p* 8 -T =p*8-r/2). This result is important and it is well known in international economics that it plays a fundamental role in several industries. As argued by Henderson et al. [START_REF] Henderson | Geography and development[END_REF], quite modest changes in prices (due to increasing liberalization or rapidly changing supply and demand conditions) can completely wipe out profit margins in industries in which either prices are already relatively low or in which imported intermediate inputs account for a large share of total value added. This profit erosion may in turn prove strong enough to incite firms to relocate in order to save on trade costs, which can actually counterbalance the effect of an initial decline in those costs.

How do those differences possibly explain the diverging results? Consider the following hypothetical situation. Assume we are in a Dixit-Stiglitz-Iceberg world and assume further that the current equilibrium is one of complete dispersion. Clearly, in that case p h h = P f f by symmetry. What happens precisely as external barriers to trade gradually dissolve as t r decreases? First, as explained by Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF], price competition in both region H and region F gets fiercer, leading to a decrease in the equilibrium prices Phh and pff-Hence, delivered prices decrease but, as argued previously, the share of trade costs in those prices remains constant. Therefore, trade liberalization has no impact on the relative profitability of country C firms* interregional buisness. To put it somewhat differently, there is no internal trade cost distortion. Yet, increasing price competition due to imports decreases the initial advantage firms enjoyed from being physically separated. Hence, the market size effect comes to dominate, which leads firms to agglomerate in whichever region has a small initial advantage. As explained by Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF], trade liberalization exacerbates the agglomeration forces at work in country C.

Assume now that we are in a quadratic world with linear trade costs and that the initial equilibrium is one of complete agglomeration in, say, region H . As t r decreases, so do the mill prices p h h and p ff , for the same reasons as before. Yet, as argued previously, because mill prices decrease the share of trade costs in delivered prices increase. Hence, it becomes relatively less profitable for firms in the core to ship goods to the periphery, because their profit margins are eroded. This implies that, contrary to the previous case, there is an internal trade cost distortion. Firms in the core suffer from increased price competition due to imports and since profit margins on interregional trade decrease this competition is even stronger in the periphery.

Because the relative dissipative effect of internal distance increases due to external competition, firms eventually find it profitable to relax competition in the core by relocating to the periphery and increasing their profit margins on local markets. Hence, we observe an externally driven redispersion of economic activities in country C, as established by Proposition 4.1. this share gradually rises from approximately 36.21% to 42%. This implies that a progressive trade liberalization with the ROW decreases profitability on interregional trade by up to 5.79%. Hence, firms find it eventually profitable to relocate from the core H to the periphery F in order to capture part of those lost 5.79% .10

Why do the results derived in Section 3 under autarky confirm those of Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF]? In the particular case of autarky, we have

A P *
Prs -Ps &nd p8S -2 > so that the share of trade costs in interregional prices is constant and equal to 1/2 (this particular value being linked to the quadratic framework). Hence, in that case, the share of trade costs in delivered prices behaves as in the Dixit-Stiglitz-Iceberg framework.

International trade liberalization therefore leads to increasing regional divergence, for the same reasons as those established by Montfort and Nicolini [START_REF] Montfort | Regional convergence and international integra tion[END_REF], Crozet and Koenig-Soubeyran [START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF] and Montfort and van Ypersele [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF].

As argued so far, our results fit nicely with those of the Dixit-Stiglitz-Iceberg ap proach once one takes into account the way international trade liberalization affects (or does not affect) the share of trade costs in delivered prices. How do our results relate to those of Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF]? In their specification, Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF] focus on urban issues and model dispersion forces in terms of land-rent/commuting costs.

As argued before, a progressive international trade liberalization keeps unchanged the share of trade costs in delivered prices. Nevertheless, the absolute value of trade costs in the economy decreases as mill prices go down. This in turn implies that commuting costs (which remain constant) get relatively more important than interregional trade costs in the economy. Or, as recently shown by Tabuchi and Thisse [START_REF] Tabuchi | Regional specialization and transport costs[END_REF], when "trans port costs are small (resp. large) with respect to commuting costs, the only stable equilibrium involves dispersion (resp. agglomeration)". Hence, as argued by Crozet and Koenig-Soubeyran [START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF], in such a framework a decrease in international trade costs weakens the agglomeration forces while leaving the dispersion forces largely 10 Notice that in our example very low levels of international trade costs t r < 0 . 0 5 lead to a breakdown of interregional trade from F to H. Hence, as shown by Behrens [START_REF] Behrens | Asymmetric trade and agglomeration[END_REF], firms in the core can no longer profitably relocate to the periphery since in that case they cut themselves completely off from the larger market in H .

untouched.

We believe our results show that conclusions on regional convergence or diver gence due to international trade liberalization should be interpreted very carefully, because they are susceptible to change with the assumptions made on trade costs and dispersion forces. In particular, it seems that decreasing trade costs can have very dif ferent impacts on the spatial distribution of economic activities, depending on whether we consider decreases in tariffs or in transport costs. As shown previously, decreases in ad valorem tariffs favor the agglomeration of economic activities, while decreases in transport costs and non-tariff barriers favor dispersion. Those results might explain why the european integration process initially led to stronger regional divergence, be cause most of the european integration prior to EC-92 consisted in reductions in and abolition of tariff barriers only. It also suggests that the future eastern expansion of the EU to ten new member states should focus on both the development of infrastructures and the abolition of tariffs in order to promote regional convergence and trade.

Some concluding remarks

International trade liberalization and trade policy clearly affect the regional equi librium within a liberalizing country. As shown in this paper, the precise effect of a progressive opening to international trade on the regional distribution of economic activities depends much on the internal structure of trade. At least two cases should be distinguished. As shown in Section 3, trade liberalization in developing countries with poor internal infrastructures can lead to regional divergence. This result is essen tially driven by the fact that increasing international integration, by increasing local price competition, exacerbates the large market effect within the liberalizing country.

Hence, since interregional trade costs remain large, local firms care more about mar ket size, which triggers a process of cumulative regional divergence that leads to a core-periphery structure. The most visible results of such cumulative processes of "ex plosive" urban and regional growth can be found in the giant Third World metropolises.

Although many authors are tempted to agree with Krugman and Livas [START_REF] Krugman | Trade policy and the Third World Metropolis[END_REF] and con sider that those giant cities and regional imbalances are an unintended by-product of import-substituting industrialization policies applied during the 1960's and 70's, more recent policies of trade liberalization do not seem to reverse the trend. Indeed, according to projections of the United Nations (1994), urban and regional growth is likely to contiune and 26 cities could exceed a population of 10 millions in 2025, the majority of which are located in the currently developing world (see Fujita and

Thisse [START_REF] Fujita | Economics o f Agglomeration: Cities, Industrial Location and Regional Growth[END_REF]). We believe that interregional integration within developing countries, by improving infrastructures and unifying local markets, is a necessary condition for subsequent international integration to lead to balanced regional development. Indeed, as shown in Section 4, trade liberalization in developed countries with good internal infrastructures can lead to regional convergence. This result is essentially driven by the fact that a decrease in international trade costs, by decreasing equilibrium prices, leads to a relative increase in interregional trade costs. Hence, firms are more shel tered from local competition, which can lead to redispersion of economic activities.

We believe those results are fundamental to understand, especially with respect to the future eastern extension of the EU. Indeed, they suggest that an increasing integration with the EU is likely to widen the gap of regional disparities within the new member countries in case their internal trade costs are too high. Hence, integration with the EU should be accompanied by a parallel regional integration within those countries.

As argued in this paper, the nature of the results is likely to hinge on the way we model trade costs. Trade liberalization is susceptible to have different impacts on liberalizing countries, depending on both the nature of trade costs and the degree of national integration. Strangely enough, those points seem to have been missed until now. We believe that, in the end, only empirical investigations will allow to discriminate between those alternative sets of results. Until this work gets done, we should be careful when chosing a modeling strategy for transport and trade costs.

decreasing average production costs and increasing profits (and hence wages).

Using the zero-profit condition, the new equilibrium wages are given by in region r = H ,F . Note that revenue on the world market (the third term) is independent of the internal firm distribution A. Hence which, since what matters in the indirect utility differential ( 22) is the difference t in -wF, establishes that this indirect utility differential is not modified. □ Proposition 6.1 proves that condition (42) may be violated while our discussion on the internal structure of country C remains valid. Hence, from now on, we simply assume that (19) holds, so that there axe imports from the R O W to country C . Proposition 6.1 states the important result that the internal geography of country C depends on the impact of the R O W on that country, but not on the impact of C on the ROW. Hence, it is the incoming component of international trade that shapes the internal geography of the country, while the outgoing component has no effect. This result can be interpreted in terms of a "small country assumption" and does no longer hold if we assume that there are regional differences in accessing world markets (as in Crozet and Koenig-Soubeyran [START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF]). Those regional differences in accessing world markets cannot be explained in the modeling framework we use here. As long as we are interested in how international trade shapes the internal geography of a country, we can focus on the incoming component only and neglect the outgoing component.

Hence it does not matter whether the country has only an import activity in the industry or if it also exports part of its local production.

Wh -Wff + b and wF = wF 4-b

In this appendix, we provide a numerical illustration of the two cases discussed in sections 3 and 4. Assume in both cases that a = j3 = 1,7 = 0.5, L = 5 and 0 = 1.2. Let us start with the case of increasing agglomeration in a developing country. Assume that t i r = 5, A = 16, L = 5, r = 1.2 and t r = 0.32. One can easily check that conditions ( 17) and ( 19) hold and that the unique stable equilibrium for those parameter involves dispersion.

Consider now a trade liberalization and assume that the external level of trade costs decreases to t r = 0.2. The new equilibrium configuration involves full agglomeration of all firms in one of the two regions.

Turn now to the case of decreasing agglomeration in a developed country.

Assume that t i r = 5, A = 30, L = 5, r = 0.3 and t r = 0.32. One can easily check that conditions ( 28) and ( 31) hold and that the unique stable equilibrium for those parameter values involves full agglomeration. Consider now a trade liberalization and assume that the external level of trade costs decreases to t r = 0.2. The new equilibrium configuration involves dispersion of firms among the two regions.

One can see that the trade liberalization from t r = 0.32 to t r = 0.2 has a different impact in both cases. The justifications are provided by propositions 3.1 and 4.1 respectively.

  [/]+ denotes the positive part of / , where a, b and c are positive coefficients

  r,s = H ,F , r ^ s T r s ~ \ \ tr if r -H ,F , $ = R or r = R , s = H ,F .

t h e n t h e r e e x i s t s a u n i q u e v a l u e

  reveals that there are three possible cases, summarized in the following proposition. P ro p o s itio n 3.1 (TRADE LIBERALIZATION AND AGGLOMERATION) Consider a country with high internal transport costs r > r ^ so that there is no interregional trade. I f where 12 b<j) -f cNc(j> Ah : = ---------(24) (Ab -A) + -(jmR < 0 then dispersion is the only stable equilibrium with international trade for all values of t r . I f (At -A) -f f in R -+ cNc 4-2c t i r ] > 0 then (full or partial) agglomeration is a stable equilibrium with international trade for all values of t r . I f d> (Ab -A) + -(jmR > 0 and (Ab -A) + -<jmR --[46 + cNc + 2cnR] < 0 r * _ 2a ___________2(Af, -A) + 5<fmR__________

t r and implies

  either dispersion or (full or partial) agglomeration of economic activities. Case three is also obvious since the equation c\Tr ■ + c 2 = 0 with t r > 0 is linear and has a unique solution if c\ < 0 and C2 > 0. Note finally that the case Ci > 0 and C2 < 0 is impossible and can hence be ruled out. □ Proposition 3.1 shows that i f dispersion is an equilibrium under autarky and i f the dispersion forces are not too strong, the international opening to trade leads to (fu ll or p a rtia l) agglomeration o f economic activities within the liberalizing country. Some straightforward analysis shows that Tr is decreasing with respect to c, which implies that stronger product differentiation leads to symmetry breaking fo r higher values o f external trade costs. Those results are in accord with the ones established in an alternative Dixit-Stiglitz-Iceberg setting (seeMontfort and Nicolini [21],Crozet and

  shows that the equilibrium prices of firms in R selling in region r = H ,F are given by the trade cost pass-through of country C firms is larger than for firms in the rest of the world. Further, the pass-through is incomplete and any decrease in r twists the terms of trade in favor of country C firms. Those results are in accordance with empirical results by Winters and Chang[START_REF] Winters | Regional integration and import prices: an empir ical ionvestigation[END_REF], who have shown that Spain's accession to the EU in 1986 had similar effects with respect to non-EU countries like the US or Japan. Because we are interested in the internal geography of country C only, prices pRR pertaining to the world market are of no interest. 6 Using (26) and (32) -(33) with (1), we get the intraregional demands * -(t + CW >12 (a2 t* N ) + ,>OTl1 -0 + < " > as well as the interregional demands* (b + cN) [2a + c[nrT + t i r t r ] -(2b + c N ) t ] cN)-----------= X" ~f c ' {36)Further, demands for imports from the ROW to region r = H , F are given by_* _Q> + cN) [2a + c[n,T + tir t r \ -(2b +cN ) tr ] and the zero-profit condition, as in Section 3, the equilibrium wage in region r = H , F is given by (38) ^ = + (t>nr) X* rrP *r + (f + ^n ^j Xrs\PrS Ñote that (38) does not include any revenues from export activities, because we assume that firms in country C do not export to the ROW. Given the quadratic utility problem (V q ), the symmetry between firms and the structure of trade, the indirect utility in region r = H ,F is given by V* = a (n rx*r + n"x*r + nRx*Rr) -| (nrx?r + n ,x fT + n Rx%r) -| {nrx*r + n,x*r + nRx* R r ) 2 + w* + < j >0 -nrp*rx*r -n sp*rx*r -nRp*Rrx*Rr. (39)

  \Np*rr -n s--n R-I ----In rp* + n» ( ^-i ) +n*(p*rr~ y ) ] -C ^4 -+ I T [ ( l + ^r) P* '+ ( * + **') (Ks " i ) ] n rp fT -n s (p*T -0 (p*r + -n R (p*r -y ) (p*r + y ) ]. -( b + cN ) Some tedious calculations yield AK*<A) -~ 1)tc'T * + + C iT ]' (40) where C\, C2 and C3 are constants given by Ci = 2nRC(t>(ScN + 46) > 0 , C2 = 4a0(66 + 4cN) > 0 and C3 = 4>Ncc2N -12062 -30c2 iV2 -2 A (?N -46,4c -1206ciV.

Proposition 4 . 1 (

 41 TRADE LIBERALIZATION AND DISPERSION)Consider a country w ith low in te rn al transport costs r < r f rade, so th a t interregional trade is always bilate ral. Assume th a t the value o f external trade costs decreases. Then dispersion o f economic activities w ithin the country can be sustained as an eq uilibrium for a larger range o f param eter values. P r o o f. Since C\ > 0, we see that dispersion is more likely as the level of international trade costs t r decreases. □ Note that, on the one hand, our results with interregional trade, obtained in a quadratic utility framework, are opposed to those established in a Dixit-Stiglitz-Iceberg framework (see Montfort and Nicolini[START_REF] Montfort | Regional convergence and international integra tion[END_REF], Crozet and Koenig-Soubeyran[START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF]

Figure 1 :Figure 1

 11 Figure 1: Share of interregional trade costs in prices as function of t r

Strictly speaking, we can investigate the impacts of a modification in tir as long as we keep the total mass of firms N constant. As shown by Tabuchi and Thisse[START_REF] Tabuchi | Regional specialization and transport costs[END_REF], by conveniently weighting the quadratic terms in the utility function, we can obtain parameters that are independent of industry size N .

This amounts to making a kind of "small country" assumption.

Appendix A

In this appendix, we prove that our results do not dependent on the fact that firms in country C cannot export to the ROW . Using expression [START_REF] Puga | The rise and fall of regional inequalities[END_REF] with [START_REF] Fujita | When is the economy monocentric: Von Thünen and Chamberlin unified[END_REF], the condition for firms in country C to profitably export to the ROW is given by 2 a

T ft ^ oiTl---(42) 2b + cnR

It is easy to check that (42) applies to both the autarky case of Section 3 and the bilateral trade case of Section 4. This is due to the fact that condition (42) does not depend on the internal geography of country C (and hence on the structure of interregional trade). Clearly, this is a strong assumption that depends on the chosen modeling framework. Indeed, we could expect that a larger agglomeration of firms gives rise to the development of infrastructures and services which allow firms located in the larger region to access the external market more easily and at lower cost (see, e.g., Behrens et al. [START_REF] Behrens | An economic geography model of trade[END_REF]). 11

The following proposition establishes that the internal geography of coun try C is independent of its firms' export activities. It is established for the bilateral trade case only. A strictly similar development applies to the case with regional autarky and is hence omitted.

Proposition 6.1 (INTERNAL INDEPENDENCE FROM EXPORTS)

Assume there is interregional trade in country C for all A G [0,1]. Assume further that both regions H and F have the same access to world markets. [START_REF] Fujita | Economics o f Agglomeration: Cities, Industrial Location and Regional Growth[END_REF] Those considerations are at the heart of the theories of hubs and economies of transport density, in which it is argued that an increase in the mass of local firms decreases costs of accessing foreign markets, since demand for transport services leads to the development of trunk routes and large scale infrastructures. Clearly, those mechanisms play a crucial role in many countries. As stated by Mori and Nishikimi [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF], "On average, transport costs from Japan to a non-hub port in Southeast Asia is [START_REF] Montfort | Integration, regional agglomeration and international trade[END_REF].6% higher than to a hub port in the same region". As shown by Crozet and Koenig-Soubeyran [START_REF] Crozet | Trade liberalization and the internal geography of countries[END_REF], in case regions have different external trade costs, the border region with an advantage in exporting to the ROW usually attracts a larger share of the industry.

29

Then the internal geography of country C is independent of whether firms in country C have an export activity or not. P r o o f . Suppose that country C can sell to the rest of the world while there is internal bilateral trade between regions H and F . One can check that the price indices ( 26) remain the same, while P ^ in ( 27) is now given by

Due to market segmentation, all equilibrium prices, given by ( 32), ( 33) and .

which are non-negative if and only if condition (42) holds. All other demands remain unchanged. Naturally, as soon as country C is able to export to the rest of the world, equilibrium profits and hence wages in regions H and F rise. This is due to the fact that firms produce under increasing returns to scale and have access to an overall larger market when there are exports, which leads to