Kristian Behrens

K B E H R E N

Jean-Marie Huriot

Christian Michelot

Jacques-François Thisse

Asymmetric trade and agglomeration

Keywords: unilateral trade, asymmetry, agglomeration, geography Classification : Rll, R12, 018

We extend the quadratic utility approach by Ottaviano et al. [19] and by Behrens [1] to the range of intermediate values of transport costs in or der to investigate how asymmetric trade affects the regional distribution of economic activities. Asymmetry in trade is an endogenous result of price competition and transport costs: depending on both the spatial distribution of firms and the value of transport costs, only firms located in one of the two regions can profitably access the foreign market, which gives rise to more complex patterns of trade. We show that the unilateral breaking of autarky gives rise to strong agglomeration forces which lead to the absorption o f the smaller regions' industry. Further, the number of equilibria increases once the structure of trade is endogenously accounted for.

Résumé

 aux valeurs intermédiaires de coûts de transport afin d'examiner la manière dont le commerce asymétrique affecte la répartition régionale de l 'activité économique. L'asymétrie dans le commerce est un résultat endogène de la compétition en prix et de l 'existence de coûts de transport. Suivant la répartition spatiale des finîtes et les valeurs des coûts de transport, seules les firmes installées dans l 'une des deux régions peuvent accéder aux marchés extérieurs, ce qui résulte en des structures d'échanges plus complexes. Nous montrons que l 'abandon unilatéral de l'autarcie donne naissance à de fortes forces d'agglomération, lesquelles mènent à l'absorption de l 'industrie de la petite région. De plus, le nombre d'équilibres augmente dès que la structure des échanges devient endogène.

Introduction

In recent years, an increasing number of authors in international trade theory and spatial economics have emphasized that, in order to understand the workings of trade and agglomeration in a spatial context, we need to move away from traditional trade theory and have to consider models in which production factors are mobile. While the traditional theory of international trade assumes that production factors are immo bile whereas goods are mobile, new trade theory and new economic geography (see Krugman [14] or Wong [START_REF] Wong | International trade in goods and factor mobility[END_REF]) have started to take into account the fact that production factors move between regions and countries in response to economic opportunities.

That this mobility of factors can have a strong qualitative impact on equilibria is known in international trade theory in which, as argued by Wong [START_REF] Wong | International trade in goods and factor mobility[END_REF], it is "shown that in ternational trade in goods only, international capital movement only, and international trade in goods plus capital movement generally lead to different world equilibria". It is less known though that, as argued by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], the mobility of goods plays an equally important role in shaping the space-economy: different assumptions concern ing the mobility o f output also lead to different world equilibria. This aspect is further developed in the present paper.

As shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], agglomeration can arise even if there is no trade between regions. Although this result is interesting by itself, we implicitly assumed that all goods were nontradeable fo r a ll spatial distributions o f firms between the two regions, a scenario we refered to as inconditional autarky. This particular assumption is not more satisfying than the one used in the original model by Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF], in which a ll goods are assumed to be tradeable. The distinction between tradeable and nontradeable is both artificial and blurry, since most goods do not systematically fall in either of those two categories. In the present paper, we complete the basic model by investigating the case in which transport costs take intermediate values. We show that in this case, all goods are only potentially tradeable so that the precise structure of trade and the regional availability of goods depends on both the value o f trade costs and the spatial distribution o f economic activities. We believe this case is empirically important, because both tradeables and nontradeables coexist in each economy (due to the rise of the service industry, nontradeables or imperfectly tradeables play an ever increasing role in modem economies). We also believe this case is theoretically the most interesting one because, instead of assuming in an ad hoc way that goods are either tradeable or nontradeable, we show how economic forces endogenously determine whether certain goods w ill be traded or not.

Before investigating those issues more closely, we ask a question of fact. Are asymmetric structures o f trade o f empirical relevance or is the assumption o f bilateral intraindustry trade between a ll couples o f regions satisfying? First, one should notice that empirically most regions are specialized in the production and export of a restricted range of products to a restricted range of other regions. As pointed out by Mori et al. [START_REF] Mori | Some empirical regularities of spatial economies: A relationship between industrial location and City Size , KIER Working paper[END_REF] for the japanese case, "O f the 125 three-digit manufacturing industries, 38 (resp. [START_REF] Mori | Some empirical regularities of spatial economies: A relationship between industrial location and City Size , KIER Working paper[END_REF] have positive employment in less than 50% (resp. 25%) o f the metro areas". Even the most diversified japanese city Tokyo has 'only' positive employment in 122 of those 125 industries. As further pointed out by Head and Mayer [START_REF] Head | Non-Europe: The Magnitude and causes of market fragmentation in the EU[END_REF] for the case of the EU, while "the m onopolistic competition model predicts positive amounts o f trade between each set o f partners in each industry [...] Greece, Ireland and Portugal have zeros fo r more than 5 percent o f thek industry-partner trade flows". Therefore, the assumption of systematic bilateral trade is questionable in case of developed countries.

Second, the systematic assumption of bilateral trade is also far from being satisfying for developing countries, which ususally have a very different import-export structure than industrialized countries and in which interregional trade is often highly asymmetric (the cases of China and Brazil seem to be rather obvious ones). 1 U nilateral trade relations between regions and even countries seem hence to play an important role fo r certain industrial sectors and should therefore not be neglectedfrom both an empirical and a theoretical po int o f view.

Why do we observe asymmetric patterns of interregional or international trade in [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF] Unfortunately, data on interregional trade flows are almost impossible to obtain for most countries, especially developing ones. Even in case of the OECD countries, only the US and Canada seem to have reliable and rather exhaustive statistics.

certain industries? It is known that unilateral trade can be either a spontaneous market outcome (driven by a subtle interplay of price competition and transportation costs), a result of tariffs and barriers to trade (as was for a long time the case for the japanese market), a result of p o litic a l factors (like restrictions on the mobility of goods and workers in China) or simply another illustration of comparative advantages as devel oped in traditional international trade theory. In this paper, we focus on the spontaneous market outcome only. No matter which factors are at the origin of unilateral trade, we

show that the existence of one-way access to certain regions introduces significant dis tortions into the space-economy by modifying firms' market potential. This point has been emphasized in the recent literature on transportation hubs, in which one explains how economies of transport density create endogenous asymmetries in transport costs and market access, which leads to agglomeration of economic activities at certain hub locations (refer to Konishi [START_REF] Konishi | Formation of hub cities : Advantages and population agglomera tion[END_REF] and to Mori and Nishikimi [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF] for further details).

We believe the case of asymmetric unilateral trade is an important one, both in order to better understand the workings of real space-economies and in order to fill an intellectual gap in the literature. [START_REF] Duncan | Who gains and who loses from unilat eral and concerted trade liberalisation ?[END_REF] We therefore extend, in the present paper, the framework developed by 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] to the case of unilateral trade between regions. To our knowledge, this is the first attempt at describing the workings of a spatial economy in the presence of an endogenously determined structure o f trade (see, however, Mori and Nishikimi [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF] for an alternative approach). We show that as soon as symmetry breaks and regions open themselves to trade, the larger region absorbs the industry o f the smaller one, which confirms the claim Kaldor (1970) made more than thirty years ago. Asymmetry in the patterns of trade leads to asymmetry in the distribution of firms, which further exacerbates the former. This circular aspect is of fundamental importance and drives most of the results developed in this paper.

We further show that as soon as one endogenously accounts for the structure of trade,

2

Fujita et al. [START_REF] Fujita | The evolution of hierarchical urban systems[END_REF] develop a model in which an urban hierarchy in the spirit of Christaller emerges as the result of a process of spatial self-organization in the presence of an exogenously growing population. They show that if we start with a monocentric configuration, the highest order good will only be produced in the central city as the economy develops. Whether this can be interpreted in terms of unilateral trade is not quite clear to us.

the number of potential equilibria becomes larger. The set of equilibria now changes qualitatively with the structure of trade, which renders the investigation quite complex.

The remainder of this paper is organized as follows:

In Section 2, we briefly recall the basic model developed by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], We explain in detail the mechanism of the endogenous determination of trade patterns and

show that interregional trade is first unilateral and then bilateral as the level of transport costs steadily decreases. In Section 3, we extend the previous developments to the case of unilateral trade and derive the equilibrium conditions in a two region setting. In Section 4, we establish our main results which state that when unilateral trade becomes feasible, the larger region always absorbs at least a part o f the industrial sector o f the smaller one. Furthermore, while the larger region benefits from this commercial expansion, it can be detrimental to the smaller region. The jo in t benefits from trade are shown to be not necessarily positive. In Section 5, we discuss the welfare issue in the three possible core-periphery structures under autarky, unilateral trade and bilateral trade. Finally, Section 6 offers some preliminary conclusions and points towards future research directions.

The conditional levels of trade costs

Let us briefly recall the model developed by Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF] and extended by

Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF]. We consider an economy with two regions H and F and two production factors: geographically mobile manufacturing workers which produce a differentiated good under monopolistic competition and increasing returns to scale and immobile agricultural workers which produce a homogeneous good (the numéraire) under con stant returns to scale and perfect competition. We denote by L the mass of mobile and by A the mass of immobile factor. The immobile factor is evenly split between the two regions, while À G [0,1] denotes the share of mobile factor located in region H . Finally, N denotes the mass of varieties (resp. of firms) in the economy. The differentiated good can be transported at no cost in the interior of each region, while shipping a unit of M-good between the two regions requires r > 0 units of the numéraire. Because there are no intraregional transportation costs for M-goods each firm sells its output at least in the market it has chosen to locate in (refer to Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF] for more details).

All agents are assumed to be homogeneous and have the same quadratic utility. As shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], in such a setting the extended (or perceived) demand functions are given by a; and where € := /? -7 is a parameter related to the degree of substitutability between varieties. Using [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF] and the symmetry between firms, the demand a firm located in region r = H , F faces in region s = H ,F is given by

*(i) = [a -(b + c N )p (i) + cP(i)]+, (1)
•^rsiPrs) = \ p, -(b -b ciV)prg + cP9] , (3)
where p r8 is the price charged by a firm located in region r when selling in market s.

Hence, the profit of a representative firm located in region r = H ,F is given by 7T r = p rrX*r iprr) ( ^ + <Mr) + (Pr, ~ l~)x*s (pr s ) -(j)Wr [START_REF] Fujita | When is the economy monocentric: Von Thlinen and Chamberlin unified[END_REF] where x*r and x*8 are given by (3), wr is the wage rate in region r and (f) is fixed labor requirement in terms of mobile production factor L. In accord with empirical evidence, firm's use spatial discriminatory pricing and set a particular price for each market (see, e.g., Greenhut [START_REF] Greenhut | Spatial pricing in the USA, West Germany and Japan[END_REF] and Head and Mayer [START_REF] Head | Non-Europe: The Magnitude and causes of market fragmentation in the EU[END_REF]). Maximizing (4) yields

* a + cPr 1 . . Prr 2 (b + cN) 2Pr {0)
on the home market and on the foreign market, where p r is the cut-off price in region r . The regional priceindices are given by P r -T lrprr(Pr) + T l8psr(Pr), r / s [START_REF] Fujita | Economics o f Agglomeration: Cities, Industrial Location and Regional Growth[END_REF] where t ih = XL/(j> and rip = (1 -A)L/(j> are the mass of firms in each region when labor markets clear. Solving for the equilibrium prices, we obtain [START_REF] Ginsburgh | On existence and stability of spatial equilibria and steady-states[END_REF] 0 )

In Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], we have investigated the equilibrium configurations of the spatial economy when r is so large that no trade occurs no matter how firms are distributed accross regions. As shown, this is the case if and only if

T > Tn --OL.
Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF] have focused on the polar case in which these two values need not and generally will not be the same. Hence, this approach captures the empirical fact that transport costs incurred by firms are not pairwise sym metric between different countries and regions and that these asymmetries are partly an endogenous result of the existing distribution of economic activities.

(ii)
The expressions of t h and t f can be derived from the conditions on r in the optimal pricing decisions of firms. We know from ( 8) and [START_REF] Head | Non-Europe: The Magnitude and causes of market fragmentation in the EU[END_REF] and that both t f and t h are monotonic, convex functions of A on the interval [0,1].

Expression [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF] shows that the larger region has a sort o f endogenous comparative advantage in terms o f profitably exporting to the smaller region. This effect is quite different from the traditional home market effect (refer to Krugman [START_REF] Krugman | Increasing returns and economic geography[END_REF] and Ottaviano

and Thisse [START_REF] Ottaviano | Agglomeration and economic geography[END_REF]) which asserts that, as neatly stated by Head and Ries [START_REF] Head | Increasing returns versus national product differentiation as an explanation for the pattern of U.S.-Canada trade[END_REF], "home de mand plays a crucial role in explaining export performance". In our model, segmented markets imply that the degree of price competition can vary greatly between regions, hence creating sufficient incentives or disincentives for firms to have an export activity.

Condition [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF] partly explains why american and japanese authorities were worried about the Single European Act. Even if external tariff and non-tariff barriers (r in our model) of the EU remained constant, a more integrated and unified Single European Market, by reducing market-segmentation and increasing price competition between regions, could render more difficult the access of producers operating outside the EU.

Some empirical results confirming partly those effects have been established by Head

and Mayer [START_REF] Head | Non-Europe: The Magnitude and causes of market fragmentation in the EU[END_REF] and by Winters and Chang [START_REF] Winters | Regional integration and import prices: an empir ical investigation[END_REF].

Using ( 13) and ( 14), one can check that the two levels of trade costs tf and t h are equal for A = 1/2. In that case

(.6)

The value f plays an important role in determining which structures of trade we have Consider a given and fixed value of transport costs To G [f,ra] and ask the following question: Under which conditions do we observe unilateral trade? It can be seen from Figure 1 that for r G [f, r a] there can never be bilateral trade since there are no values of A such that t f { A) > r and t h ( A) > r simultaneously hold. Hence, there is either autarky or unilateral trade, depending on the spatial distribution o f firm s. We have autarky for all spatial distributions A such that the two conditions t h { A) < r and Tf(A) < r simultaneously hold (i.e. no region can profitably export to the other one given the actual distribution o f firms). In Figure 1, this autarky region is centered around A = 1/2 and delimited by the dashed vertical lines. Rewriting the condition for autarky, using ( 13) and ( 14), yields

(17) rc L rc L
which can of course only be met if r > f . For all values of A outside of this interval, the pattern of trade is unilateral, with the larger region exporting to the smaller one.

Finally, it is important to note how the possibility for unilateral trade evolves as r decreases from a high initial value. Consider a given autarky equilibrium with r > r a and assume that the value of transport costs decreases. Once r G [f, r0], there is the possibility to observe unilateral trade for certain spatial configurations A. The precise value of trade costs r from which on we effectively observe a passage to unilateral trade strongly depends on initial conditions (do we start from a dispersed, partially agglomerated or fully agglomerated autarky equilibrium?). Hence, "history matters" a lot in this case. 3 Suppose, for example, that we start from a fully agglomerated initial equilibrium with A = 1. As soon as r = ra, conditions [START_REF] Head | Increasing returns versus national product differentiation as an explanation for the pattern of U.S.-Canada trade[END_REF] and [START_REF] Ottaviano | On economic geography in economic theory: increasing returns and pecuniary externalities[END_REF] show that the interval for regional autarky is given by 0 < A < 1, which still holds since we consider an agglomerated initial equilibrium. Should r continue to decrease strictly below ra, the new interval for regional autarky is given by 0 < A < 1 so that the actual agglomerated configuration with A = 1 is no longer compatible with autarky.

3 Although mostly associated with Krugman [START_REF] Krugman | Increasing returns and economic geography[END_REF], the term "history matters" can already be found in Eaton and Lipsey [START_REF] Eaton | The introduction of space into the neoclassical model of value theory[END_REF].

Firms located in region H can, due to the decrease in transportation costs, profitably export to the foreign market F . One can further see that, as r continues to decrease towards f , the interval of spatial distributions A compatible with autarky gets smaller and smaller. In the limit, when r = f the 'interval' reduces to A = 1/2 so that only the dispersed configuration remains compatible with regional autarky. As transport costs continue to decrease below f , but remain higher than Ttrade> the passage to bilateral trade becomes possible. Depending on the spatial distribution of economic activities,

we have either unilateral or bilateral trade (note from Figure 1 that autarky is no longer a feasible option, since the conditions t f (A) > rand 77* (A) > r never simultaneously hold). When do we observe bilateral trade? Analogously to the previous case, for this to be possible, the conditions r < tf(A ) and r < t h {A) must simultaneously hold.

Rewriting these conditions yields

1 _ W t -b r ) < A < 2 ^0 -M r c L " tcL which is possible if and only if r < f . It is easy to check that in case r = f , condition (18)
reduces to A = 1/ 2, so that bilateral trade is possible if and only if we start from a dispersed configuration. As r continues to decrease, the interval of values of A compatible with bilateral trade gets larger until finally the condition 0 < A < 1 is met as r = Ttrade-Hence, as soon as r < Ttrade, we are back to inconditional bilateral trade as developed by Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF].

Let us close this section with two remarks. First, it is interesting to notice that the passage from autarky to unilateral trade and the passage from unilateral to bilateral trade are sort of 'asymmetric'. On the one hand, the stronger the initial differences in terms of A between the two regions, the sooner the passage to unilateral trade but the later the passage to bilateral trade. On the other hand, the more symmetric the regions in terms of A, the later the passage to unilateral trade but the sooner the passage to bilateral trade. Hence, it seems that symmetry (resp. asymmetry) in the spatial distribution o f economic activities leads to symmetry ( resp. asymmetry) in the patterns o f trade. The extreme scenario is the one in which we start with full agglomeration A = 1. In that particular case, unilateral trade becomes feasible as soon as r hits ra, but bilateral trade only becomes feasible when r hits Ttrade • If we start, on the contrary, with a dispersed configuration A = 1/2, we never observe unilateral trade. There will be autarky until r < f , from which point we directly switch to bilateral trade. Hence, asymmetry in the spatial distribution of firms, which creates endogenous comparative advantages, is needed for unilateral trade to possibly emerge. Second, as one can see from ( 16), the more differentiated the varieties (the smaller the value of c), the higher f . In the limit case in which varieties are independent (hence as c -* 0), the three critical levels of transport costs ra, f and Ttrade coincide. Therefore, more product differentiation leads to an earlier passage from autarky to unilateral (resp. bilateral) trade.

As shown in this section, unilateral trade can emerge as the joint consequence of decreasing transport costs and asymmetric firm distributions. Each region is hence characterized by a particular critical level of trade costs, above which foreign firms cannot be active in it.

Extending the model to unilateral trade

We develop in this section the expressions we need in order to establish the indirect utility differential in the case of conditional unilateral trade. In what follows, we assume for simplicity, without loss of generality, that H is the large region i.e. that A > 1/ 2. 4 Hence, as one can see from [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF] we have t f > t h • Assume further that t < t < t f so that firms operating in the large region H can be active in the foreign market F , while firms operating in the small region F cannot be active in the foreign market H . Using expressions ( 5) and ( 6), local prices are given by

* a + cPF , * a + cPH . Pf f ~ W + c N ) ^ PHH~ 2(b + c N Y ( 19
)
while export prices are given by

"* _ a + cPF + (b + c N )r a + cPH p« r ----- T + d v " <20)
It is important to notice the asymmetry between p ^F and p FH, which is due to the unilateral trade pattern under which firms in H export to F but not the other way round. Using expressions ( 7), ( 19) and ( 20), we can solve for the equilibrium prices.

Those are given by

★ _ & * _ 2d to-t\ Ph h ~ 2b + c\N' Pfh ~ 2b + cXN W and _ 1 2a + tcX N l 2 a + r[2b + c ( l + X)N] P f f ~2 2b + cN ' -2 2 & id V ( 22
)
Note that because firms in region F do not export to region H , the prices p j jH and Pf h g*ven by ( 21) are naturally independent o f the transport cost parameter r. This does of course not hold for p*FF and p*H F, as can be seen from ( 22). While it is easy to understand that the prices firms located in H set in market F do depend on r, it is more difficult to see a priori why prices firms in F set in their home market also depend on r. This is due to an indirect price competition effect. Due to the substitutability between varieties, firms in F need to adapt their local prices to the prices of imports, which depend themselves on r. Therefore, local prices o f a ll varieties sold in region F depend on the value o f transport costsf no matter whether they are produced locally or imported. Finally, the aggregate price indices in both regions are given by

H 2b + cXN F 2b + cN v '
Expressions (3), ( 21), ( 22) and (23) allow us to derive the individual equilibrium demands of agents, which are given by

_ a(b + cN ) XHH~2b + cXN' X f h ~° (24)
in region H and by and

_ (b + c N )(2 a + crX N ) X pF~ 2(2 b + cN ) and _ ( b + c N ) [2a -r (c(l -X )N + 2b)] Xh f ~ 2(2b + cN ) (25)
Pf h = 2Ph h , Ph f = ^ + ^e + j N ] + ^' (29)
In order to establish the expressions of the indirect utilities in regions H and F , we need the expressions of the equilibrium wages w# and w*F in the two regions. These wages are determined by a bargaining 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. All profits are hence absorbed by the wage bill, which is synonymous with zero profits for firms in the M-industry. The equilibrium wages can be obtained by substituting (26), ( 27), ( 28) and ( 29) into (4) and equating the resulting expression to zero, which yields Let us turn next to the indirect utilities in regions H and F . Since neither prices nor wages are symmetric anymore, the indirect utilities will not be symmetric either. Using the quadratic utility as given by Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF], the indirect utility is given by r N

f rN V * = a I x* (i)d i-^ x * (i)2d i Jo 2 J 0 ~ 2 ( / + ( w* + ( t > o~ f P*(*)®*(»)d*). ( 32
)
Using the symmetry between firms and the relations N = n n + rip , rtn = AL/0 and rip = (1 -\)L/<j>t the indirect utility in region H can be expressed as (recall that

x f h = 0 since there are no trade flows from F to H )

V f f = TIh X*h h --Xj j h - ~ PHH^j + W H + 00-
Using ( 24), ( 28) and (31), straightforward but cumbersome calculus shows that the indirect utility in region H is given by

+ 00 n (A ) -| ( 2€ + A " 0_ i 7) ^e + \ 2-y^+ A I f A | (-1 \ \ L\ [Ha -r) -(1 -A)t70-1I,]2 + U + ( ) J ----40e[2e + !</>-'L]>-----(33)
As one can see from (33), the indirect utility in region H under unilateral trade is given by the sum of two terms. The first term is the indirect utility under autarky (refer to Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF]), to which we add a positive second term corresponding to the benefits of increased market size for the local export industry. This positive effect is due to the fact that exploitation o f scale economies lowers average production costs and increases nominal wages, while prices stay constant in region H . Hence, real wages in the industrial sector rise. As for region F , the indirect utility is given by (recall that Substitution of ( 27), ( 28), ( 29) and ( 30) into (34) yields, after some rather long rearrangements, the following expressions for A±, A2 and A$:

4e[2e + 7 JV]2 2 a(e + 7 N) -r[2e . _ \ \ m ~i (35) ,\»n 1/ \ 2e(<* -t ) -T7 ( l -A)iV + 7 (A + 1 )N] --(e + 7 AJV)-------------- 4 ,\\ 1 1 xx»r2Q!€ + 7'rAJV ¿ 2(A) = (1 -A)N- 4e[2e + jN ] 2 2a[2e + 7JV] 1 2 a t + ■yrXN ~ 2 e [3e + 7(1 -X)N] (36) ^A> = q s r w [ ( £ + (1 -A)w) (2" e + ,rAJV1 -■rA ( l -A ) ^2e(', -T )-rT (1-A)iV] . (37)
Finally, the indirect utility differential is defined as 18 (38) An equilibrium arises at A = 0 if expression (38) is negative, an equilibrium arises at A = 1 if this expression is positive and an interior equilibrium arises at 0 < A < 1 if this expression is equal to zero. The two fully agglomerated equilibria are always stable if they exist, while an interior equilibrium is stable if and only if the slope of the indirect utility differential is negative in a neighborhood of the equilibrium.

As one can already see from (33), ( 35), ( 36) and (37), the analytical expression of the indirect utility differential in the case of asymmetric trade turns out to be a relatively complex function of A. This is further exacerbated by the fact that, for a given and no longer a differentiable one. This is easy to check and due to the fact that the function A V * depends on the extended demand functions, given by (1), which are themselves non-differentiable functions. This non-differentiability requires us to be careful when studying the stability of equilibria, since the sign of the 'derivative' at the equilibrium eventually no longer provides an adequat information concerning stability.

As one can already guess, the interesting cases will be precisely the ones in which the non-differentiability occurs for the current equilibrium value A*.

Equilibria under unilateral trade

As shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], the case of inconditional autarky is neither subject to multiple equilibria nor to catastrophic bifurcations. Those results no longer hold in the case of conditional unilateral trade, where transitions can be catastrophic and where the number of equilibria increases. Hence, the analysis gets more involved because in itia l conditions and history play an important role. In order to tackle this issue, we use a 'standard' approach of new economic geography and appeal to an artifical history (refer to Fujita and Krugman [START_REF] Fujita | When is the economy monocentric: Von Thlinen and Chamberlin unified[END_REF] and Fujita et al. [START_REF] Fujita | The evolution of hierarchical urban systems[END_REF]). We assume we start from an initial equilibrium configuration in which transport costs are so high that we are in the inconditional autarky case (i.e. r > r a) and we examine how the opening to trade of region F , due to a decrease in transport costs, modifies the existing spatial equilibrium.

As shown by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], there are three possible types of equilibria under autarky: full agglomeration if A lies below the sustain point A s, dispersion if A lies above the break point At, and partial agglomeration in between. Further, the interior equilibria are given by when A 8 < A < A b. As argued in Section 2, some asymmetry between regions is needed in order for unilateral trade to be feasible for at least some spatial distributions A. This rules out the case of initial dispersion and leaves us with two possible scenarii:

that of full and that of partial initial agglomeration. In what follows, the subscripts aut, Several remarks are in order. [START_REF] Fujita | The Spatial Economy -Cities, regions and international trade[END_REF] As one can see from (45), the unilateral opening to trade of region F is never profitable to agents in that region if r remains close to r a = a .

Therefore, i f transport costs remain close to ra, unilateral trade does not allow to increase the utility o f agents in region F , even i f goods are highly differentiated (e is large). Roughly speaking, the terms of trade get worse for region F . This suggests that 'partial liberalization' and suppression of only a small part of tariffs and barriers to trade can have a detrimental effect on the opening region, because prices of import varieties remain high while foreign competition degrades local market conditions.

Hence, transport costs, tariffs and barriers to trade must decrease sufficiently in order fo r unilateral trade to possibly have a positive impact on the welfare in the smaller region. This result shows that, as Duncan et al. [START_REF] Duncan | Who gains and who loses from unilat eral and concerted trade liberalisation ?[END_REF] have pointed out, "proximity matters" so that positive impacts of unilateral opening to trade are more likely if the industrialized regions are not too far away from the peripheral ones. If r decreases 5 Recall that the indirect utility is given for mobile M-workers only. Hence, because there is no intersectoral mobility, the utility of immobile A-workers can behave quite differently, especially since the agricultural wage does not depend on considerations of competition in the M-industry. Note also that in case A =l, the indirect utility in the periphery corresponds to the maximal utility a firm could offer to its workers by relocating to the peripheral region.

sufficiently below ra, condition (45) is more likely to hold if e is large i.e. if goods are bad substitutes and the impact on consumer utility is hence strong. This shows that complementary patterns of trade (characterized by low elasticities of substitution)

are more likely to lead to benefits from liberalization than similar patterns of trade (characterized by high elasticities of substitution). In this case, the positive effect on the demand side in region F , going through preference fo r variety due to a larger array o f available goods, can possibly offset the negative effect on the production side, consisting in stronger price competition, profit erosion and wage cuts. Stronger scale economies (hence larger values of < j >) make condition (45) also more likely to hold.

The important point to note is that the opening of the small region to imports from the larger one is not automatically synonymous with increasing utility in the sm all region.

Only i f the terms o f trade are not too detrimental to the smaller region can the agents o f that region benefit from this unilateral opening.

Using (33) and (39), it is readily verified for region H that VHuni X=1

-V* V Haut A e(a -r )2 > 0 , (46)
a=i m z + 'iN ? so that a unilateral export activity is always profitable to agents in that region. This is due to the fact that only a positive effect on wages is at work in region H . We can hence conclude that a sufficient condition for a fully agglomerated initial autarky equilibrium with A = 1 to remain stable during the passage to unilateral trade is given by the reverse of condition (45). Note that this condition is a rather special one, because it states that the stability of the initial equilibrium is guaranteed by a deterioration of the indirect utility in region F . Therefore, this condition is not compatible with a simultaneous increase of the utility levels in both regions (which would be more in line with what one would expect from trade). Note also that it is a priori unclear if the joint benefits from the passage to unilateral trade are positive. We can therefore not exclude the situation in which trade is globally detrimental to the two regions since benefits to region H are not sufficient in order to compensate fo r losses to region F .

In order for the (individual) net benefits from unilateral trade to be positive, we need to know when the condition

( V H uni ~ V *Haut V *Funi ~ V *Faut \ A=1 A=1 A=1
holds. Using ( 44) and ( 46), condition (47) is equivalent to

A=1 > 0 (47)
Because r < a , inspection of condition (48) reveals that it never holds if goods are perfect substitutes (when e -> 0) and that it always holds if goods are independent (when 7 -> 0 and hence e -> /?). It is also easily seen that the stronger the economies of scale, the more likely that trade will be globally beneficial. When scale economies are large (formally when 0 -> +oo) trade will always be globally beneficial; when there are no economies of scale (formally when 0 -* 0) trade will never be globally beneficial. We may hence conclude that the opening of a region to exports from another is a) beneficial to the exporting region, b) not necessarily beneficial to the importing region and c) not necessarily jointly beneficial to the two regions. Only i f goods are sufficiently differentiated and scale economies strong enough can unilateral trade have a positive impact on the regions' jo in t level o f welfare. In that case, even if trade is detrimental to the importing region, there is the possibility for a welfare maximizing planner to use lump-sum transfers in order to compensate the importing region for the losses so that trade is jointly beneficial to all involved parties.

We now establish a necessary and sufficient condition under which full agglome ration is an equilibrium under unilateral trade.

P ro p o sitio n 4.1 (UNILATERAL TRADE AND FULL AGGLOMERATION)

Assume th a t A > 1/2 so th a t H is the large region. Consider the case in which t < r < r a so th a t unilateral trade can occur fo r certain s p a tia l d is trib u tio n s A. Then fu ll agglomeration A = 1 is a s p a tia l e q uilibrium i f and

o nly i f 3a 2(f>Ne + a 2<jryN2 + A a 2e -A ^a€ > { a -r)2 [e(0iV --A) -f 07iV2].
P ro o f. E v alu atin g expressions ( 33) and (34) (using ( 35

□

Proposition 4.2 roughly states the following. If we consider an initial configuration in which full agglomeration is a stable spatial equilibrium for autarky, full agglomeration remains a stable spatial equilibrium for unilateral trade. As we have argued previously in this section, the opening of the small region F to imports from region H is always beneficial to H and eventually beneficial to F . Proposition 4.2 states that even i f unilateral trade is beneficial to the importing region, it w ill never be beneficial enough to attract firms from the larger region to the smaller one. Hence, a fully agglomerated autarky equilibrium cannot be broken by the sole passage to unilateral trade, which highlights again the fact that spatial equilibria can be highly robust to even significant modifications in transportation costs. This is due to the fact that the structural change in the patterns of trade, by creating asymmetries in market access, gives rise to new agglomeration forces that were not active in the case of autarky. Therefore, the decrease in transport costs and the emergence of trade (which could a priori have worked against agglomeration in this scenario) give rise to new agglomeration forces via the passage to unilateral trade, which locks-in the initial equilibrium configuration. One should finally note that Proposition 4.2 only holds for 0 < A < A s. Examples for which full agglomeration is not an equilibrium configuration for unilateral trade when A > A 8

are easy to construct.

In the remainder of this section, we examine how a partially agglomerated initial equilibrium evolves when autarky is broken by the passage to unilateral trade. This issue, concerning the impacts of trade liberalization on the spatial distribution of economic activities and on regional inequalities, has always been and still is a main point of interest both from a theoretical and a policy point of view. These last decades, several large-scale projects of trade liberalization like e.g. the EU, NAFTA and APEC, have raised renewed interest in those questions. O f course, those projects and the resulting trading blocks are usually characterized by bilateral trade. We nevertheless remind the reader that bilateral trade liberalization, consisting in an a priori symmetric decrease of trade costs between two regions, can lead to unilateral patterns of trade.

Especially, as explained in Section 2, symmetric liberalization only leads to bilateral trade if a) the reduction in trade costs is large enough and b) the initial distribution of economic activities is not too asymmetric between the two regions. Both of those points strike us as being empirically fairly implausible. Does a decrease in transport costs, and the eventual opening to trade, lead to regional convergence or divergence? Will the Single Market Program of the EU favor dispersion of economic activities among member states or could we observe an ad ditional polarization of the space-economy? Although there is still no consensus on those issues (despite the results established in new economic geography), the fear of regional divergence has always been on economists' minds. Kaldor wrote in 1970:

"When trade is opened up between them, the region with the more developed industry w ill be able to supply the need o f the agricultural area o f the other region on more favourable terms: with the result that the industrial center o f the second region w ill lose its market and w ill tend to be eliminated" (as quoted in Ottaviano and Thisse [START_REF] Ottaviano | On economic geography in economic theory: increasing returns and pecuniary externalities[END_REF]).

There are many examples of regions or nations that used, or still use, barriers to trade in order to protect their nascent industry against outside competition from a larger neighbor. Even in todays world of 'free trade', trade is often restricted in complex ways. That such protectionnist positions could be theoretically grounded is shown in the remainder of this section.

In order to investigate the issue concerning regional covergence or divergence, we start with an inconditional autarky equilibrium A+ that exhibits partial agglomeration, given by (41), and ask the following question: How does the spatial structure of the economy change when a decrease in transport costs allows firms of the larger region H to enter the peripheral market F I Let us start with an illustrative example: The first one corresponds to complete agglomeration under unilateral trade at A = 1 and the second one to partial agglomeration under autarky at A = A+ > 0.5 (as given by expression (41)). Figure 2 further shows that there are two unstable equilibria, corresponding also to two different patterns of trade. Hence, there are seven equilibria in our model i f we consider the general case in which A € [0,1]. This shows that if we take into account the structure of trade, the number of equilibria increases and changes with that structure. Finally, Figure 2 shows that the indirect utility differential is a non-differentiable function of A. The kink is located at A = Asup, which is strictly greater than the interior equilibrium A+ in our example.

Expression (50) clearly shows that \ 8up is an increasing function of r. Hence, the kink Xsup 'moves to the left' as r decreases. 6 What happens when r decreases [START_REF] Fujita | The evolution of hierarchical urban systems[END_REF] As one can see from (50), A,up decreases with decreasing transport costs, increasing product differentiation, increasing scale economies and increasing consumer preference for the differentiated product. Hence, the passage to trade does not necessarily rely on a sole sufficiently so that \ 8up = A"1 ", i.e. when the industry of the larger region H is able to profitably enter the market of the smaller region F I Our main result, which is in line with the 'kaldorian absorption scenario', is summarized in the next proposition. (51) holds. Rewrite (51) as A < Ai> + / ( r ) , where

Clearly, / is an increasing function of t . Because A < A f, must hold (recall that we assume we have an interior equilibrium in autarky) and since / is increasing with respect to r, condition (51) always holds if / ( f ) > 0. Using the definition of f , given by ( 16), we have 1L >0. Hence, A+ does not change as r decreases below ra, as long as there is no structural change in the model (i.e. as long as the pattern of trade remains stable). This implies Figure 3: Indirect utility differential as autarky breaks that for all 1/2 < A < A+, the relevant indirect utility differential is that of autarky given by AV*aut.

Figure 3 illustrates the passage from autarky to unilateral trade in the case of a partially agglomerated initial equilibrium. While the interior equilibrium A+ is stable under autarky (refer to Figure 2) it becomes unstable once there is unilateral trade, i.e. as soon as \ 8up hits the value A+ of the interior equilibrium. The new stable equilibrium in our example is one of complete agglomeration in which the larger region H has absorbed the whole industrial sector o f the smaller region F .

This example raises the question of whether once the interior equilibrium gets unstable there is always complete agglomeration in the larger region. Because the general analytical expression of the derivative of AV^ni is very complicated, we have no formal proof. Especially, we have not been able to determine whether A F *ni is a monotonous function for A > X8up or not. Numerical simulations suggest that the indirect utility differential is indeed monotonically increasing beyond X8up. Hence, in creasing agglomeration with partially agglomerated equilibria, as depicted in Figure 4, seems not to be a possible issue. More work is called for here.

U tility under different core-periphery structures

As shown by Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF] and by Behrens [START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF], there is not only one but there are three types o f core-periphery structure, depending on the pattern o f trade between the two regions. We can have full agglomeration of all firms in region H under either autarky, unilateral or bilateral trade. Therefore, there are three qualitatively different core-periphery structures in our model. Each core-periphery structure is susceptible to yield a different level of welfare in the periphery, depending on the actual structure of trade in the economy. We can hence investigate whether there is a core-periphery structure that yields a higher level of utility in the periphery than the other coreperiphery structures do. One might be tempted to think that utility in the periphery gradually rises as we first pass from autarky to unilateral trade and then from unilateral to bilateral trade. That things are not that simple and that the degree of openess to trade is not monotonously correlated with the level of welfare in the periphery will be explained in this section.

As shown in Section 4, the indirect utilities in the peripheral region F under a core-periphery structure are given by ( 42) and ( 43) respectively. In order to evaluate the utilities in region F under all three basic patterns of trade, we have to establish the expression of the indirect utility in region F for bilateral trade. Using the results established by Ottaviano et al. [START_REF] Ottaviano | Agglomeration and trade revisited[END_REF], one can show that the indirect utility in the bilateral core-periphery structure is given by Therefore, the slope of V p uni | A=1 is always positive at r = r a = a, which shows that the passage from autarky to unilateral trade (which implies a decrease in r) deteriorates the conditions in the periphery. Suppose th a t A = 1 is a s p a tia l equilibrium for a ll values o f t . Then the

0.5----------1 ----------1 ---------- 0 0.5 1 1.5

□

Proposition 5.1 explicits conditions under which the worst possible situation fo r the periphery is achieved when trade is unilateral. This is reminiscent of the situation in many of todays developing countries, in which the large metropolitan core regions provide all goods and services and in which peripheral regions are unable to provide goods or services to the core in return. Hence, unilateral trade, by lowering the attractivity o f the periphery, is a strong force working against economic development and regional convergence. Locations that allow fo r unilateral access to peripheral markets attract economic activities, which in turn enhances their access capacity and diminishes that o f the periphery.

Note finally that if Proposition 5.1 does not hold, the minimum utility in the peripheral region under a core-periphery structure is just reached at the passage from unilateral to bilateral trade (as is the case in figures 5 and 6). 7 This shows that autarky never corresponds to the worst situation fo r the periphery, at least under a core-periphery structure.

7 Numerical simulations reveal that this case is the most frequently observed one. It occurs whenever the derivative of the indirect utility differential, evaluated at T = r t r a d« is positive under unilateral and negative under bilateral trade.

Conclusions

We have shown in this paper that, despite a priori symmetric transport costs be tween regions, asymmetries in regional access can endogenously appear. This endoge nous asymmetry arises naturally if firms are unevenly distributed in the space-economy, because this uneven distribution is synonymous with different degrees of local price competition in segmented markets. Hence, it is tempting to conclude that transport costs are never symmetric between regions, at least as long as those transport costs remain higher than a certain threshold (given by Ttrade hi our model). This aspect is of fundamental importance and is reminiscent of basic questions addressed in the ever growing literature on hubs and endogenous network formation. As argued by Mori and Nishikimi [START_REF] Mori | Economies of transport density and industrial agglomeration[END_REF], "interregional trade patterns and the structure of the transport network are determined endogenously as a result of the interaction between industrial location behavior and increasing returns in transportation, in particular economies o f transport d e n s ity As shown in this paper, there is no need for economies of transport density in order for market access between regions to be eventually asymmetric; price competi tion among monopolistically competitive firms producing imperfect substitutes can be sufficient in order to create endogenous asymmetries. Of course, economies of trans port density do play a role in creating even more asymmetry in market access. Hence, we may safely conclude that both price competition and economies of transport density work together to create significant distortions in market access in todays differentiated industries. Unfortunately, we are not aware of any empirical work trying to investigate the degree of asymmetry in market access and the impact of this market access on the spatial distribution of economic activities (see, however, Head and Mayer [START_REF] Head | Non-Europe: The Magnitude and causes of market fragmentation in the EU[END_REF] for some interesting developments). More work is called for here.

We have also shown in this paper that asymmetries in trade patterns give rise to strong agglomeration forces, because they introduce a significant distortion into firms' market access capacities. In case firms in region H can access market F but not the other way round, region H offers a significant locational advantage: the overall market-size accessible from region H is larger than that accessible from region F , while price competition is not fiercer. This in turn deteriorates the operating conditions in region F and leads to relocation of (a part of) the industry of region F to region H .

As shown by Theorem 4.4, the unilateral opening to trade of one region leads to the absorption of its industry by the other region. Hence, there is no reason to assume that unilateral trade will lead to regional convergence. Since < 7 is decreasing with respect to £, g is always negative on if it is negative at Straightforward calculus shows that the condition 0(0 < 0 ^ 7k2 + 8 h > 0 automatically holds since A : > 0. Therefore since, by (67), g is decreasing with respect to f , < 0 on ft and hence the partial derivative of / is negative with respect to A on ft.

In order to establish the non-negativity of / on ft, we can show that / is non negative on the right side of the rectangle ft. Since, as shown previously, / is decreasing with respect to A , the result follows. (2 + fc)3

which is clearly non-negative.

Since / is continuous (with respect to A and r), decreasing with respect to A and non-negative on the right side of the rectangle ft, we conclude that / cannot take negative values on ft, which proves our claim.

  where o, b and c are positive coefficients, given by a = ---r-, b = --and c = (2). e + N 7 e + N 7 e(e + N j )

Figure 1 :Figure 1

 11 Figure 1: Conditional values of trade costs th and tf in the intermediate cases

x h f > 0

 0 since there are positive trade flows from H to F ) V*F = a ( n Hx*HF + n F x*F F ) -^( n Hx%F + n F x f F ) -2 \ nH XHF + ^F^ff) + 00 ~ nHPHFxHF ~ nFpFFxFF> which can be rewritten as the sum of three terms (dropping the constant 0o)-Therefore n = ^( A ) + ¿ 2(A) + ¿ 3 (A) njfX%F (a -^x*HF -| nHx*HF -p *HF^ ¿2(A) := nF x*FF -~2X* f f ~ '^211f x*ff ~ PffJ and -A3(A) := Wp -'y(nHx*H F) (nFxFF).

  fixed value of transport costs r, the patterns of trade (autarky, unilateral or bilateral) depend on the spatial distribution A of firms. Therefore, the indirect utility differential has an analytical expression that changes structurally with the spatial distribution o f firm s as we pass from autarky to unilateral trade. As argued in Section 2, for some values of A, given by (17), we have autarky, for the other values we have unilateral trade. Because we assume that H is the large region so that A > 1/2, we can restrict ourselves to A € [1/2,1]. Nevertheless, a fundamental technical difficulty could appear in the model. Since AV* is defined by two different functions on two different domains, nothing ensures a p rio ri that AV* remains a continuous function o f A. Hence, the first natural questions to investigate is the continuity and the differentiability of the global indirect utility differential in the case of unilateral trade. As shown by Behrens [1], in case of autarky the indirect utility differential is given by where n out( a) = i ( 2e + 7 A L<t>-1 a ) [3T e+x2'r¥ + A^. + 0 ° (39) and In case of unilateral trade, the indirect utility differential is given by the difference between expressions (33) and (34). Fortunately, the structural change in the indirect utility differential is located such that our function remains continuous with respect toA. Hence, pathological cases in which an equilibrium might not exist can be excluded (seeGinsburgh et al. [8]). This is summarized by the following proposition.P ro p o sitio n 3.1 (CONTINUITY OF THE INDIRECT UTILITY DIFFERENTIAL)Assume th a t A > 1/2 so th a t H is the large region. For a ll f < r < r aj the g lo b al in d ire ct u tility differential A V * is a continuous function o f A on the in te rv al [1/ 2, 1]. P r o o f . Clearly, both p*8 and x*8 are all continuous functions of A (refer to (1), (5) and (6)). Because the integral operator preserves continuity, we conclude that the indirect utility differential, given by the difference of two continuous functions V *, r = H , F is a continuous function of the firm distribution A. □ Yet, while the indirect utility differential remains a continuous function o f A, it is

  uni and bil refer to variables and expressions under autarky, unilateral and bilateral patterns of trade respectively. Let us begin by investigating how the unilateral opening to trade of region F affects the spatial equilibrium if we start from a fully agglomerated autarky equilibrium with A = 1. Evaluating expressions (34) (using (35), (36) and (37)) and (40) at A of (42) and (43) yields V* V F uni -^ [4ae + 7 N ( a + r)]| , (44) which (recall that r < a ) is positive if and only if (45)

Figure 2

 2 Figure 2 plots the two indirect utility differentials & V *ni and A V *ut on the interval [1/2,1] and illustrates a possible situation in the case of unilateral trade. Because we are in the conditional case, the pattern of trade (autarky or unilateral) depends on the spatial distribution A. Using condition (17), the expression of the indirect utility differential changes at the critical value

P r o p o s itio n 4 . 3 (

 43 ABSORPTION OF THE SMALLER REGION' S INDUSTRY)Assume th a t A > 1/2 so th a t H is the large region. Consider a stable in itia l au tark y equilibrium w ith p a r tia l agglomeration. Suppose th a t the level o f transport costs r decreases so th a t f < r <rp{ A), which allows the firm s o f the larger region H to p ro fita b ly enter the m arket o f the sm aller region F . Then the in itia l equilibrium becomes unstable and we observe more agglom eration in the larger region.Pr o o f . We have to show that the derivative of the indirect utility differential AV*ni in the case of unilateral trade, given by (38), is always positive when evaluated at A = X8tip. Hence, when r decreases so that unilateral trade be comes profitable for firms of the larger region H , the stability of the initial partially agglomerated equilibrium is broken and more agglomeration arises in region H . Due to its length, part of the proof is relegated to appendices B and C. In Appendix B, we establish the analytical expression of the derivative of the indirect utility differential in the unilateral trade case at A = A sup. In Appendix C, we show that this derivative is always positive when the condition decrease in the value of trade costs. For a given and fixed value of transport costs, unilateral trade can emerge as a consequence of technological evolutions in the productive structures: larger production scale and stronger product differentiation can trigger agglomeration and a passage to trade, even if trade costs stay constant and high. As far as we know, this aspect has not been treated in the literature until now. More work is called for here.

  At < Ab -f /(r ), Vr G [f, a], so that condition (51) always holds. □ As shown by Proposition 4.3, (part of the) firms of region F relocate to the larger region H as soon as market F opens to imports from H . Let us restate this fundamental result in the following theorem. T h e o r e m 4 .4 (N O REGIONAL CONVERGENCE THROUGH UNILATERAL TRADE) Consider a stable in itia l autarky equilibrium w ith either fu ll or p a r tia l agglom eration. Suppose th a t the value o f transport costs r decreases so th a t the firm s o f the larger region can p ro fitab ly enter the m arket o f the sm aller one. Then agglom eration in the larger region is non-decreasing so th a t we never observe a process o f dispersion. Hence, in the presence o f m obile production factorsj regional convergence can never result from the u n ilate ral opening to trade o f one region alone. P r o o f . Being a simple reformulation of Propositions 4.2 and 4.3, the proof is omitted. □ Let us close this section with an important final remark. As one can see from expres sion (41), the interior equilibrium A+ is independent of the value of transport costs r .

Figure 5 :Figure 5 , 5 TFigure 6 :

 5556 Figure 5: Evolution of V p under a core-periphery structure with high e

Finally, we have

  highlighted the fact that it is difficult to speak of the core periphery structure of the economy, because there are three core-periphery structures corresponding to three basic patterns of trade. As shown in Section 5, the welfare results for those three configurations are very different. Contrary to what one might expect, the periphery is worst off in case of unilateral trade, because in this situation the terms of trade are the most detrimental. Hence, the passage from autarky to unilateral trade, which can be a forced stage as transport costs decrease, deteriorates significantly the attractivity of the periphery. This result is reminiscent of ancient colonial trade policies, which imposed unilateral patterns of trade between the mother country and the colonies and hence significantly slowed down their economic development. Only when transport costs become sufficiently low and barriers to trade sufficiently negligible will bilateral trade emerge and lead to increasing utility in the periphery. and we have to show that this partial derivative is negative for all values of ? < t < a . Define f := -and k := (63) a 2e so that the condition d f/d A < 0 can be expressed as 4a 2 e2g (0 < 0, (64) where «,(£) = l -( l + fc£)2 + ( l -£ ) 2. (65) Since (A ,t) € ft, we know that £ e [£> l] where £ := -= --_ = ---r-

First, consider the

  partial derivative of / with respect to r. Using (63), this derivative can be written as= a -A ( 1 + k £ h N + 2(1 -0[e(<t>N -A) + ^i V 2]].(68)Clearly, (68) is negative for r = a i.e. for £ = 1. One can further see that for a given value of A , this derivative is at most equal to zero a single time.Therefore, since f ( A 8 , a ) = 0, / is non-negative on the right border of therectangle Q if f( A 8 , f ) > 0.That this holds indeed is shown now.First, note that (59) can be rewritten as which, using the relations (63) yieldsf ( A , 0 = a 2e(L(3 + 2 k )+ A [ l-(l + k t f + ( l-Ç )2] -( l-Ç ) 2 L (l+ 2 k )). (69)The function / , given by (69), has to be evaluated at (>!*,£). Using (61) and(63), A8 can be expressed as A-= Lm r y <™ > Using (69) and (70), the condition f ( A s,£) > 0 is equivalent toL(3 + 2k) + A,[l -(1 + kO 2 + (1 -1)2] -(1 -|)2L ( 1 + 2k) > 0,which, after some calculus, reduces to fc(10fc2 + 26fc 4-15) > 0.(71)Clearly, (71) always holds since k is non-negative. Therefore tt a ^ _ <x2 ekL(10k2 + 26k + 15)

Figure 7 :Figure 7

 77 Figure 7: Positivity of / on ft

We can disregard the case in which we start from a symmetric distribution A=l/2. As mentioned in the previous section, we never observe unilateral trade in that situation. Hence, Behrens[START_REF] Behrens | Trade is not necessary for agglomeration to arise[END_REF] and Ottaviano et al.[START_REF] Ottaviano | Agglomeration and trade revisited[END_REF] completely cover this special case.

(a -r)e + r iH L )-3e <0

respectively. As one can see from (54), the passage from unilateral trade to bilateral trade is always beneficial to the periphery because the second term is non-negative. If a firm was to deviate to the periphery, it would be able to offer its workers a higher level of utility under a bilateral trade pattern than under a unilateral one. This result is easy to understand: competition in region F is not fiercer under bilateral trade than it is under unilateral trade, but bilateral trade allows the periphery to access the market and products of the core. always holds for all f < r < a if we start from complete agglomeration in autarky (i.e. for 0 < A < A8). This is equivalent to showing that the function Let us begin by showing that / is decreasing with respect to A. The partial derivative of / with respect to A is given by A ppendix B

Proof. We show in this appendix that the derivative of A F *ni is always po sitive at A = A8up if we are in the case of an initial partially agglomerated equilibrium. This establishes that the opening of the smaller region to unilateral trade renders the equilibrium unstable and leads to more agglomeration in the larger region.

We evaluate at A = A8up, where X8up is given by (50). Let us start with the derivative of AVtfum at critical value X8up. First note that AVffuni> as given by (33), can be expressed as Using (50) and some more simplifications finally yields Xgup'yN

which is the derivative of A V'^tini evaluated at the critical value.

Let us turn next to A V Funi as given by (34). Its derivative is best evaluated using the relation and ci, given by (76), are constants. By (77) we know that u(Xeup) = 0 so that dAi y (X ,up

which, after some simplifications, can be expressed as

Next, we use (36) in order to evaluate the derivative of A2 . The expression A2 can be written as

where we have redefined u and v as u_(A)

Note that Ci is still given by (76). Hence, the derivative of A2 is given by

Adding expressions (83), ( 86) and (89), we finally get the expression of the derivative of the indirect utility of region F evaluated at X8up, given by ~3<) ] -(90)

It is the sign of expression (90) that determines whether partial agglomeration remains stable as the smaller region is opened to trade. As we show in Ap pendix C, the sign of ( 90) is always positive when we start from initial partial agglomeration, so that the equilibrium always becomes unstable. □

Proof. We derive in this appendix a sufficient condition under which the deriva tive of the indirect utility differential is always positive at A = X8up if we start from an autarkic situation in which there are stable interior equilibria.

Since by definition

it is sufficient to show that the first term is positive and the second negative.

Let us start with the first term and let us show that this term is always positive.


Économiques Umr5118

Louis De Mesnard Juin
email: louis.de-mesnard@u-bourgogne.fr

LABORATOIRE D'ANALYSE ET DE TECHNIQUES

Keywords: JEL classification. C63, C67, D57. KEYWORDS. Input-Output, Demand, Supply, Change, RAS, Biproportion

About the criteria

Following [START_REF] Leontief | Studies in the structure of the American economy[END_REF]), Carter (1967, 1970) and [START_REF] Vaccara Beatrice | Changes over time in input-output coefficients for the United States[END_REF] that have examined the stability of technical coefficients, Bon (1986)1 tries to evaluate the comparative stability of the coefficients of the demand-driven model [START_REF] Leontief | Qualitative input-output relations in the economic system of the United States[END_REF] and of the supply-driven model [START_REF] Ghosh | Input-output approach to an allocation system[END_REF], in the framework of a national economy. Starting from the idea that the model that has the more stable coefficients over time is the more valid, he uses an indirect but simple method 2: the output of each sector is forecast under the base of each model and then it is compared to the true value of the output. The model that produces the best forecast is the better for this sector but one model can be the best for one given sector and the alternative model can be the best for another sector 3. I name this method the criteria of output 1

Exactly the same methodology applied to other countries than US (e.g., UK, Japan, Italy, Turkey) can be seen in (Bon, 1993(Bon, , 1996a(Bon, , 1996b(Bon, , 1997(Bon, , 2000a)); all these papers are reprinted in (Bon, 2000b).

2 It is not the aim of this paper to discuss the respective merits and dismerits of these two polar models. For an introduction see [START_REF] Miller | Input-Output Analysis, Foundations and Extensions[END_REF] and for a complete discussion, see [START_REF] Oosterhaven | On the Plausibility of the Supply-Driven Input-Output Model[END_REF]Oosterhaven ( ,1989Oosterhaven ( , 1996)), [START_REF] Miller | Stability of Supply Coefficients and Consistency of Supply-Driven and Demand-Driven Input-Output Models: a Comment[END_REF], Gruver (1989), [START_REF] Rose | On the Plausibility of the Supply-Driven Input-Output Model: Empirical Evidence on Joint Stability[END_REF].

3 Do not confuse with the discussion conducted by Bon (1984) about the comparative merits and dismerits --in a multiregional input-output framework --of a column coefficient model, a row coefficient model, and a Leontief-Strout gravity model when the economy is assumed to be demand-driven: only the row coefficient model is consistent, the other violate the conditions of productivity.

coincidence for forecasts. In this paper, after recalling this method in details, I will explain its drawbacks then I will expose an alternative method that is not affected by these drawbacks. factor (demand and value-added, respectively). It is a pity because the evolution of the exogenous factor could hide the evolution of the structure. It is even possible to compute what is the final demand vector (respectively the added-value vector) that allow the best matching as possible, that is:

II

x* = rfx*

x* = ( I -A ^1r

=> ( i -A T ' r = ( i -A ) _1r ^[ ( i -A T ' -a -A ) -1] r = o
This is a simple homogenous system. If |(I-A *)-1 -(I-A )-11 = 0 then there is a non-trivial solution (and if not, the trivial solution is f* = 0). As the solution is parametric (at least, any is a function of one of the final demands, say f[ ), there is an infinite set of vectors f* that are solutions of the problem of output coincidence for forecasts! Fortunately, the method of output coincidence for forecasts is not the only possible to do the job. At least another is possible, that has not the above weakness.

III. Another method

It is preferable to take a look at the structure itself to see what model is the best: the stability of technical and allocation coefficients over time could be also a good criterion. Assuming that a direct comparison of the stability of technical coefficients and of allocation coefficients is possible, one can decide what are the more stable, the technical coefficients or the allocation coefficients of each sector, and one can decide if the model is demand-driven or supply-driven for this sector. This analysis is not affected by the above critic, namely the mixing of the structural effect and the effect of the exogenous factor.

Unfortunately, the direct comparison of coefficients cannot be done so simply. As it is well known, when technical coefficients are assumed to be stable (A* = A), allocation coefficients cannot (B* = x* -1 A x* * B), and conversely, except in a very special case, the absolute joint stability, that is the homothetical variation of the gross output of all sectors (Chen and Rose, 1986 and[START_REF]The absolute and relative Joint Stability of Input-Output Production and Allocation Coefficients[END_REF]: x* = kx and A* = A imply that B*=i* Ax*=B. So, one has to use a more sophisticated method than the direct comparison of the stability of column or row coefficients, that is the biproportional filter (Mesnard, 1990a(Mesnard, and b, 1994(Mesnard, , 1997)). See in Mesnard (1990aMesnard ( , 1997) ) why it is more suitable to use a biproportion (that is a generalization of RAS) instead of another criterion of projection, as the orthogonal projection.

In addition to provide the projected matrix that is close to the original matrix Z, under the respects of the margins of Z*, biproportion guarantees that coefficients are positive in the projected matrix K(Z, Z*) if they are in the original matrix Z.

To do the job, some variants can be used, as projecting Z* to Z in a reverse

P i = m
, for ail i, and qj = , for all j. This cannot be solved analytically but only iteratively. However, it is demonstrated that biproportion is a very safe operation: the solution of Stone's RAS --another biproportional algorithm -has a unique and convergent solution (Bacharach, 1970) and any algorithm, the above, RAS or any other, lead to the same solution (Mesnard, 1994).

Then the result is compared to Z* by computing the Frobenius norm of column or row vectors of the difference matrix Z* -K(Z, Z*), divided by the margin of Z* to obtain a percentage of variation6: declared as not demand-driven, nevertheless one cannot say that it is supply-driven (and allocation coefficients) are stable, then it is false to say that the model is demand-driven (respectively supply-driven) -even one can suspect that it is -but it is true to say that it is not supply-driven (respectively demand-driven). Here, the logic does not lead to accept a model directly -only to suspect that it works but they authorize to reject its alternative (Mesnard, 1997).

for column j and, a, = j If for any sector i one has o f > o f , then the row is more stable than the column: the sector is conversely if erf < of). Following the rules of logic, if technical coefficients (respectively computation, or as giving to both Z and Z* the same margins: it is not the aim of this paper to develop this point (Mesnard, 1998).

6

Other types of indices can be build.

I will apply both methods, the output coincidence for forecasts and the biproportional filter to

France, for the period 1980-1997 7. I have adopted the grand total of each table as output (vector x) and not the distributed production 8, so the output of a column is equal to the output of a row, the account of each sector is at equilibrium and both technical coefficients and allocation coefficients are consistent. The tables are aggregated into 9 sectors 9. I have made them square by simply removing the following column sectors: T25 Trade and T38 Non market services.

IV. Application

Tables 1 to 3 about here

The tables used are price-corrected (all are at the base price of 1980). The table of 1980 is "definitive", the table of 1997 is "temporary".

8

In the French accounting system, in addition to the distributed production, the grand total of a column includes the imports, customs duty, commercial margins, VAT; the distributed production is equal to the total of the intermediate buyings plus the added-value and some transfers; so, as added-value, I take the difference between the grand total and the intermediate buyings. The grand total of a row includes the total of the intermediate sales, the final consumption, the gross formation of fixed capital, the variation of stocks and the exportations; so, as final demand, I take the difference between the grand total and the intermediate sales.

9 About the stability of aggregated coefficients over time, see [START_REF] Per | The stability of input-output coefficients[END_REF].

Tables 4 and5 give the inverse matrices for technical and allocation coefficients, while table 6 indicates the result of the biproportional projection of Z (year 1980) on Z* (year 1997).

Tables 4 to 6 about here

The results are not exactly comparable, but there are three cases of divergence toward a supply driven model for the method of output coincidence for forecasts: Minerals, Trade, Transport and Telecommunications. For these three cases, the biproportional filter indicates that the concerning sectors are not supply driven (and one can suspect that they are demand-driven), when the method of output coincidence for forecasts indicates that they are.

There is a clear bias in favor of supply-driven sectors with the method of output coincidence for forecasts. There is also one divergence toward a demand driven model for Financial Services. However, I insist on the fact that the results of the method of output coincidence for forecasts could have been very different with any other final demand or added-value vector (and particularly with another definition of these aggregates).

Tables 7 to 8 about here

V. Conclusion

The purpose of the method of "output coincidence for forecasts" is to determine what sectors are demand-driven and what sectors are supply driven in an input-output framework. The output of each sector is forecast under the base of each alternative model and the model that produces the best forecast is the better for this sector. This method mixes two effects, a structural effect -choosing between demand and supply driven models -and the effect of an exogenous factor -final demand or added-value-. Depending of the exact value of final demand or added-value, coincidence can be obtained or not for a given sector, so the choice between a demand-driven model and a supply-driven model is affected by the final-demand or by the added-value. This makes the choice faulty in a general way: if one decide that the behavior of economic agents (here, the sectors) can de determined, this behavior cannot be dependent of an exogenous factor. In other terms, the behavior determined by die method of output coincidence for forecasts is not an absolute behavior, but only a "functional" behavior, dependent of the exact value of the exogenous factors: behavior(z') =f(/~i, ...,/", vi, ..., v"),

where fj and vj denote the final-demand and the added-value of sector j. This is annoying because this "functional" behavior is not generally applicable.

The alternative method that is proposed -the biproportional filter -has not these drawbacks.

As generalization of the direct comparison of the variations of technical and allocation

coefficients -what allows to focus the measure of change on the exchange structure itself -it is a direct method, not an indirect method as the method of output coincidence for forecasts, without any interference of any exogenous factor as final demand or added-value.

The application for France, 1980-1997, indicates that the method of output coincidence for forecasts creates a bias in favor of the supply-driven model.

VI. Bibliographical references

.

  The weakness of the method of output coincidence for forecasts: mixing between the exogenous factor effect and the structural effect Assume that we have two years or two countries (or regions), the second denoted by a star in superscript. Denote x and x* the two output vectors, either at two different dates, either in two different countries (or regions) of space, Z and Z* the two flow matrices that correspond to them, denote A = Z x-1and A*=Z*(x*) 1 the two technical coefficient matrices, B = x-1 Z and B* = (x*) 1 Z* the two allocation coefficient matrices deduced from Z and Z*; denote f* the final demand vector for the second year or the second and v* the added-value vector for the second year or the second countries. At equilibrium, the forecast output is given by d\* -(I-A )-1 f* and ix* = v* (I-B )-1. Both x^ and x* will be compared to the true value x*. This comparison is done sector by sector: if dx* -x* < sx* -x* then the sector / is declared as more column-stable than row-stable, and conversely 4. Note that matrices Z and Z* have to be square. This is a very simple way to perform a comparative evaluation of the alternative models but it has a main drawback. When you compare outputs, you introduce the final demand for the demand-driven model, or the added-value for the supply-driven model, so you mix two different things: 1) the structure (the structure of production for the demand-driven model or the structure of allocation for the supply-driven model), and 2) the effect of the exogenous One could have done reverse forecasts also: dx = (I -A*)'1 f and sx = v (I -B*)''.

  When you compare technical or column coefficients, you remove the effect of the variation of the margins of columns; when you compare allocation or row coefficients, you remove the effect of the variations of row margins. With the biproportional filter, the idea consists into removing the effect of the variation of both types of margins. To perform this, matrix Z can be equipped with the margins of Z* by a biproportion: Z = K(Z, Z*) 5: K(Z, Z*) = P Z Q , with 5 Note that matrices Z and Z* have not to be square, what is an advantage to take into account of some sectors (for example, Trade in French accounting: it has only a column but not a row). K denotes the biproportional operator, what gives to Z the margins of Z*, the result K(Z, Z*) being the closer as possible to Z.

Bacharach, Michael 1970 .

 1970 Biproportional Matrices and Input-Output Change. Cambridge University Press, Cambridge. Bon, Ranko 1977. "Some conditions of macroeconomic stability of multiregional input-output models", Economic Analysis, 16, 1-2: 65-87. _______ 1984. "Comparative stability analysis of multiregional input-output models: column, row, and Leontief-Strout gravity coefficient models", Quarterly Journal of Economics, 99, 4: 791-815. _______ 1986. "Comparative stability analysis of demand-side and supply-side input-output models", International Journal of Forecasting, 2, 2: 231-235; reprinted under the title "Comparative stability analysis of demand-side and supply-side input-output models: the case of the US" in: BON, R. 2000. Economic structure and maturity. Collected Papers in input-output modelling and applications, Ashgate, Adelshot. _______ 1993. "Comparative stability analysis of demand-side and supply-side input-output models: the case of U.K.", Applied Economics, 25, 1: 75-79; reprinted in: BON, R. 2000. Economic structure and maturity. Collected Papers in input-output modelling and applications, Ashgate, Adelshot. _______ 1996a. "Comparative stability analysis of demand-side and supply-side input-output models: the case of Japan, 1960-1990", Applied Economic Letters, 3: 349-354; reprinted in: BON, R. 2000. Economic structure and maturity. Collected Papers in input-output modelling and applications, Ashgate, Adelshot. _______ 1996b. "Comparative stability analysis of demand-side and supply-side input-output models: the case of Italy", Rivista Internationale de Scienze Economische e Commerciali, 43, 3: 669-679; reprinted in: BON, R. 2000. Economic structure and maturity. Collected Papers in input-output modelling and applications, Ashgate, Adelshot. _______ 1997. "Comparative stability analysis of demand-side and supply-side input-output models: toward an index of economic maturity", Proceedings of the 44th North American Meetings of the Regional Science Association International, Buffalo, New-York, November 6-9, 1997; reprinted in: BON, R. 2000. Economic structure and maturity. Collected Papers in input-output modelling and applications, Ashgate, Adelshot. _______ 2000a. "Comparative stability analysis of demand-side and supply-side input-output models: the case of Turkey, 1973-1990, in: BON, R. 2000. Economic structure and maturity. Collected Papers in input-output modelling and applications, Ashgate, Adelshot. _______ 2000b. Economic structure and maturity. Collected Papers in input-output modelling and applications, Ashgate, Adelshot. Carter, Anne P. 1967. "Changes in the structure of the American economy, 1947-148 and 1962. Review of Economics and Statistics, 49: 209-224. _______ 1970. Structural change in the American economy, Harvard University Press, Cambridge, Mass. Chen, Chia-Yon And Adam Rose 1986. "The Joint Stability Of Input-Output Production And Allocation Coefficients," Modeling And Simulation 17, 251-255,

Table 1 .

 1 The aggregation scheme

		Agri	Energy Minerals	Manu	Buildings	Trade	T ransport	Services Financial
	1980	culture...		facturing			and	Services
							Telecom.
	A griculture...	1.522538 0.002751 0.004774 0.029356 0.007405 0.256677 0.006146 0.011097 0.003969
	Energy	0.063111 1.470860 0.178983 0.069641 0.070377 0.085416 0.176085 0.028532 0.023749
	Minerals	0.014176 0.015634 1.401103 0.076307 0.201235 0.027559 0.010360 0.009952 0.012472
	Manufacturing	0.138945 0.060580 0.085809 1.340108 0.249892 0.140368 0.092993 0.122074 0.061860
	Buildings	0.003247 0.017171 0.004581 0.003052 1.003235 0.004967 0.005211 0.005033 0.045361
	Trade	0.002689 0.001665 0.008272 0.003612 0.005417 1.025267 0.009150 0.013587 0.008308
	T ransport and	0.018118 0.022288 0.059615 0.036057 0.051156 0.039959 1.106398 0.039047 0.044054
	T elecommunications
	Services	0.053302 0.048847 0.069435 0.079998 0.153004 0.116824 0.087480 1.162317 0.124790
	Financial Services	0.002846 0.002277 0.004950 0.005099 0.015605 0.006727 0.003553 0.003415 1.030017
			Table4. (I-A )"1 for 1980

Table 5

 5

	. (I-B )-1 for 1980

Table 8 .

 8 Method of the biproportional filter