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Regional Convergence and International Integration
Journal of Urban Economics 48, 286᎐306 Ž2000. doi:10.1006rjuec.1999.2167, available online at http:rrwww.idealibrary.com on
Regional Convergence and International IntegrationU Philippe Monfort† and Rosella Nicolini IRES and Department of Economics, Uni¨ ersite´ Catholique de Lou¨ ain, Place Montesquieu 3, 1348 Lou¨ ain-La-Neu¨ e, Belgium E-mail: [email protected] Received November 25, 1998; revised August 10, 1999 We analyze the geographic concentration of economic activities within the framework of a two country᎐four region model. Trade between both regions and countries entails transaction costs which are differentiated according to the interregional or international nature of the flows. Allowing for regional migration of the population, the model configures the equilibria of this system as key parameters change. The results obtained suggest that both types of transaction costs affect the incentive for firms to cluster geographically. Consequently, processes of integration between nations are identified as factors possibly favoring the emergence of regional economic agglomerations. 䊚 2000 Academic Press Key Words: international trade; agglomeration; economic geography.
1. INTRODUCTION Recent contributions in economic theory have shown how some central international trade issues, for instance the effects of trade liberalization on the wealth of nations, could be interestingly revisited by adopting an economic geography approach Žsee, for instance, Fujita and Thisse w4x for an extensive survey of this literature .. Countries are then considered as regions between which the mobility of goods andror factors is more or less fluid Žsee, for instance, Krugman and Venables w8x or Matsuyama w10x.. The two fields indeed largely overlap if not focus on the same object Ž‘‘Who produces what, where?’’ to quote Krugman w7x. and it would probably be quite limiting to try to draw a clear line of demarcation between them. In a certain number of cases, however, regional and national levels should be distinguished as they might at the same time coexist and differ * We gratefully acknowledge Jacques Thisse and Xavier Wauthy for helpful comments. The usual disclaimer applies. This research is also part of a research program supported by the Belgian Program on Interuniversity Poles of Attraction ŽPAI N P4r01.. † Author to whom correspondence should be addressed. 286 0094-1190r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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in their characteristics. In particular, the concept of nation seems appropriate if one wants to allow for institutional peculiarities, for goods or factors flow impediments related to trade policy, or for factor immobility related to cultural or linguistic barriers for which national boundaries are relevant geographic features. This is especially clear when one examines the differences between the regional agglomeration processes in the United States on the one hand and in Europe on the other. While in the former case, models considering an area composed of a set of regions seem indeed opportune, the analysis of the latter probably requires consideration of both interregional and international trade and factor flows. In particular, international labor mobility in Europe remains extremely low compared to the interstate one in the U.S. Žsee, for instance, Decressin and Fatas ` w1x1 .. Accordingly, although some kind of agglomeration has undoubtedly taken place in Europeᎏone clearly sees the existence of a European core-periphery pattern Žpossibly centered in Belgium ŽNewhouse w11x. ᎏyet a large number of European industries cannot be considered as concentrated in a particularly country. In fact, the concentration of economic activities is a phenomenon which might be more likely detected inside each European country. Moreover, the in-depth integration process that Europe has witnessed since World War II has been accompanied by a tendency of European economies to converge but, for the last few years, regional disparities seem to be growing across European regions, which suggests that it is inside countries that the agglomeration of economic activities is now taking place ŽEC w3x; Puga w12x.. The objective of this paper is therefore to build a framework taking such features into account and to deepen our understanding of how international economic integration affects the process of agglomeration within trading countries. To this end, we extend the Krugman w6x framework to a two country᎐four region model in which each country consists of two regions between which national labor is mobile. Labor is, on the other hand, immobile between nations. Although extreme, this assumption is intended to account for differences in the extent of interregional and international factor mobility. As far as goods are concerned, both interregional and international trade entail transaction costs but their magnitude might differ according to the interregional or international feature of the trade flows. The extent and the nature of trading costs are indeed likely to be quite different whenever goods are shipped to another country rather to another region of the same country. While the latter form of trade merely entails transportation costs, 1 Eurostat displays an EU-12 average rate of migration of 2.9 per thousand in 1991 while, for the same period, the U.S. rate of geographical mobility is 2.9% ŽU.S. Bureau of the Census..
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the former also includes costs related to institutional factors like trade policy, customs duties and formalities, or adaptation to foreign legislation standards, which also act as other impediments to international trade. Moreover, it is also clear that the process of European integration affected mostly international transaction costs while leaving interregional trading costs unchanged. If we want to discuss the impact of integration between nations on a country’s regional outcome, it is important to explicitly account for possible differences in the two kinds of transaction costs. The choice to investigate these questions within a Krugman w6x type of framework is of course disputable as, for instance, it does not allow for intersectoral labor mobility, which would give to the analysis a more pronounced international trade flavor. Nevertheless, the capacity of this type of model to capture the essence of an agglomeration process in a very simple way makes it worthwhile to show how it can be generalized by incorporating both regions and nations. The results of this analysis suggest that variations in international transaction costs are able to modify the structure of the countries’ spatial equilibria and more specifically that the agglomeration of national economic activities proves to be enhanced by the process of international integration. Consequently, even if, as underlined by Puga w12x, the EU integration process is not likely to yield an economic geography comparable to that of the U.S. because of a relative international factor immobility, it is nevertheless potentially able to generate or increase regional disparities inside European countries, which seems in line with the stylized facts mentioned above. The issue of the relationship between a country’s openness to trade and the internal geographic distribution of its activities is also addressed in Krugman and Elizondo w9x. In their framework, however, openness to international trade fosters the dispersion of economic activities because the centrifugal forces hinge on rent and commuting costs while, in ours, they rest on the existence of an immobile consumers fringe. They indeed focus on metropolization phenomena in developing countries, both a topic and a context for which their assumptions concerning the centrifugal forces are relevant. In our case, we intend rather to elaborate on the emergence of regional disparities in a framework suitable for capturing European-like contexts. In this prospect, the alternative of focusing on dispersion forces due to local markets is worthwhile to investigate and is actually shown to reverse the conclusions of Krugman and Elizondo concerning the effect of international economic integration on the countries’ spatial equilibria. The paper is organized as follows: Sect. 2 develops a model which allows convenient introduction of selected ingredients of location and international trade theory. Sect. 3 examines the characteristics of the price system and the structure of the spatial equilibrium for different levels of transac-
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tion costs. Sect. 4 explicitly derives the mechanisms through which the spatial equilibrium is affected by both interregional and international transaction costs. Finally, Sect. 5 concludes. 2. A TWO-COUNTRY AND FOUR-REGION MODEL We consider a world with two countries and two regions in each country. In the first country Žhome., we distinguish region 1 and 2, while the second country Žforeign. is composed of regions 3 and 4. Following Krugman w6x, there are two sectors of production: agriculture and manufacturing. Agriculture is a sector tied to the land and is therefore geographically immobile while manufacturing is an increasing returns sector that can be located anywhere. 2.1. Consumers᎐Workers Consumers in the two countries are identical and share the same utility function expressed as: U s CM CA1y ,
Ž 1.
where CA is the consumption of agricultural goods and CM is the consumption of the manufactures aggregate. According to this specification, consumers devote a share of their expenditure to manufactured goods. The manufactures aggregate CM is defined by: n
CM s
Ý
r Ž y1 .
c kŽ y1.r
,
ks1
where n is the number of potential products and Žwith ) 1. is the elasticity of substitution among those goods. The population of each country is normalized at 1, of which a fraction 1 y are peasants and are manufacturing workers. World population therefore sums up to 2. Workers and peasants are the only factors of production. They are assumed to be sector specific, peasants for agriculture and workers for manufacturing. Moreover, we assume that workers move across regions but not across countries while peasants are completely immobile. In each of the four regions, we therefore have a fixed peasants supply equal to Ž1 y .r2 while the countries’ workers supplies meet the conditions: L1 q L 2 s , L3 q L4 s , where L i denotes the workers supply of region i.
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2.2. Agriculture and Manufacturing Technology Agriculture is perfectly competitive. It is a constant returns to scale sector and we normalize the unitary labor requirement to 1. Trade of the agricultural good is supposed to be costless between both regions and countries, which implies that agricultural prices and peasants’ wages are equalized across the whole area. The agricultural good is therefore used as the numeraire. The production of manufactures features economies of scale and the labor requirement for an individual manufactured good Ž k . is defined as: LM k s  q ␥ x k , where  corresponds to a fixed cost, ␥ to the constant marginal cost of production, and x k to the quantity produced. The manufactures market structure is characterized by monopolistic competition following Dixit and Stiglitz w2x, under which each firm produces a single differentiated good. 2.3. Transaction Costs Interregional and international trade of manufactured goods is assumed to incur transaction costs. These are formalized as ‘‘Samuelsonian iceberg transportation costs’’ for which a fraction Ž1 y . of one unit shipped to another region is lost in the transport Ž - 1.. In a broad sense, ’s value not only captures the extent of transport costs associated with interregional or international trade, but also accounts for any kind of trade barrier which affects a region’s openness to trade.2 In this perspective, extreme high values of represent fully integrated markets while extreme low values of correspond to autarchy. We assume that transaction costs differ according to the interregional or international nature of the trade flow. We can indeed expect international transaction costs to incorporate a larger fraction of institutional trade impediments than interregional transaction costs, which should mainly represent transport costs. Moreover, by distinguishing both types of transaction costs, we can analyze the effect of an international integration process Žwhich is simply formalized as a decrease in the international transaction costs. on the outcome of the model. Let i denote transaction costs associated with trade between regions of a same country while e represents those entailed by international trade flows. We adopt two simplifying assumptions. On the one hand, interre2 Since we do not introduce a public sector in the model, the trade barriers considered here should be viewed as those generating no rents or tax revenue Žfor instance, safety and health regulations, industrial standards, etc...
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gional transaction costs are supposed to be identical in both countries. On the other hand, international transaction costs are assumed to be independent of the regions involved, i.e., the cost incurred by trading with the other country is the same whatever the foreign region the good is exported to. 2.4. Size of the Regional Manufacturing Industry Firms are symmetric as far as individual demand and technology is concerned. Given the previous definitions, one easily ascertains Žsee, for instance, Krugman w6x. that the elasticity of the total demand facing any firm is independent of the transaction costs and simply corresponds to . The profit-maximizing pricing behavior of firms located in region i is then: pi s
ž /␥ y1
i s 1, 2, 3, 4.
wi
Ž 2.
The price set in a particular region is a constant mark-up over the regional wage rate Ž wi . which implies that: pi pj
s
wi
i , j s 1, 2, 3, 4.
wj
Free entry leads to complete erosion of pure profits so that the usual zero-profit condition applies with the corollary that the equilibrium output level Ž x . is the same for each firm in any region. In particular, we have: xi s x s
 Ž y 1. ␥
i s 1, 2, 3, 4.
Full employment of the labor force then allows us to specify the number of firms active in a particular region as: ni s
Li

i s 1, 2, 3, 4
Ž 3.
The equilibrium number of firms is proportional to the region’s workers population and we therefore have: ni nj
s
Li Lj
i , j s 1, 2, 3, 4.
2.5. Equilibrium Conditions in Goods and Labor Markets In shipping to another region or country, the price of a local manufactured product is increased by the relevant transaction costs so that to acquire one unit of a region i’s manufactured good, consumers of region j
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must spend pir i if region i and j are located in the same country or pire if located in different countries. Given the utility function Ž1., the consumers price index of manufactured goods is given for each region by: P1 s
n1 p1y 1
P2 s n1
P3 s n1
P4 s n1
p1
ž / ž / ž /
q n2
e
i
q
n 2 p1y 2
1y
q n2
e
p1
q n3
ž /
1y
i
p1
1y
p2
1y
q n3
p2
ž / ž /
q n3
1y
ž / ž /
q n4
e
p3
1y
q n4
e
1y
q
e
p2
p3
n 3 p1y 3
1y
e
q n3
p3
q n4
p4
ž / ž / ž /
1y
,
e
p4
1y
i
1r Ž1y .
,
e
p4
1y
1r Ž1y .
,
i
1r Ž1y .
1y
ž /
1r Ž1y .
q
n 4 p41y
.
On the manufactured goods market, the equilibrium condition requires production of region i to be equal to the world demand. Since, at zero profit, each firm produces the same quantity x, we have: 4
ni x s
Ý ni js1
pi
ž / Pj
y
Yj 1 Pj
,
Ž 4.
where 䢇 䢇 䢇 䢇
s 1 if i s j, s i if j is a region belonging to the same country as i, s e if j refers to a region of the other country, Yj is the income of region j.
On the other hand, the equilibrium condition on the world agricultural market is given by:
Ž 1 y . w Y1 q Y2 q Y3 q Y4 x s 2 Ž 1 y . , the LHS of the equality corresponding to world consumption and the RHS to world production of the agricultural good. Finally, the income of each region corresponds to the revenues of peasants and manufacturing workers. The wage rate of peasants being the numeraire, we have: Yi s
1y 2
q wi L i
i s 1, 2, 3, 4.
Ž 5.
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Note that, by the price setting rule Ž2. and the expression giving the equilibrium number of firms as a function of the region’s workers population Ž3., we have wi L i s n i pi x. We can now express the whole system in terms of the relative regional sizes and prices. Let us define relative prices as ¨ s p 2rp1 , ¨ U s p4rp 3 , and ¨ s p 3rp1. Similarly, we have z s n 2rn1 and zU s n 4rn 3 . Noting that n 2 p 2 s z.¨ .n1 p1 , n 3 p 3 s ŽŽ1 q z .rŽ1 q zU .. ¨ .n1 p1 , and n 4 p4 s ŽŽ1 q z .rŽ1 q zU .. zU .¨ U .¨ .n1 p1 and using the definitions of the regional incomes Ž5., the equilibrium condition on the agricultural good markets is rewritten: 2 Ž 1 y . q Ž n1 p1 q n 2 p 2 q n 3 p 3 q n 4 p4 . x s 2, which, after manipulation, leads to: n1 p 1 x s
2 ⌬
,
where ⌬ s 1 q z.¨ q ŽŽ1 q z .rŽ1 q zU ...¨ q ŽŽ1 q z .rŽ1 q zU ... zU .¨ U .¨ . The equilibrium condition for the manufactured goods can now be expressed in the following way. Let t i s iy1 F 1 and t e s ey1 F 1 and consider, for instance, condition Ž4. for region 1. This can be rewritten as: xs
py 1 Ž 1 y . r2 q n1 p 1 x
P11y q
q
py 1 t e Ž 1 y . r2 q n 3 p 3 x
P31y
py 1 t i Ž 1 y . r2 q n 2 p 2 x
P21y q
py 1 t e Ž 1 y . r2 q n 4 p4 x
P41y
Ž 6. The other regions’ equilibrium conditions are similarly transformed. Expressing each product n i pi Ž i s 2, 3, 4. in terms of n1 p1 and making use of relative prices and sizes, we finally obtain the following system: 3 a1 q t i a2 q t e a3 q t e a4 s 1,
Ž 7.
t i a1 q a2 q t e a3 q t e a4 s ¨ ,
Ž 8.
t e a1 q t e a2 q a3 q t i a4 s ¨ ,
Ž 9. U
t e a1 q t e a2 q t i a3 q a4 s Ž ¨¨ . , 3
Ž 10 .
As usual, one of the four equations is redundant if one makes use of the Walras law.
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MONFORT AND NICOLINI
with: a1 s
a2 s
a3 s
a4 s
1 q Ž Ž 1 y . r2 . 2⌬ 1 q t i z¨ 1y q t e Ž Ž 1 q z . r Ž 1 q zU . . ¨ 1y q t e Ž Ž 1 q z . r Ž 1 q zU . . zU Ž ¨ U ¨ .
1y
z¨ q Ž Ž 1 y . r2 . 2⌬ t i q z¨ 1y q t e Ž Ž 1 q z . r Ž 1 q zU . . ¨ 1y q t e Ž Ž 1 q z . r Ž 1 q zU . . zU Ž ¨ U ¨ .
1y
Ž Ž 1 q z . r Ž 1 q zU . . ¨ q Ž Ž 1 y . r2 . 2⌬ t e q t e z¨ 1y q Ž Ž 1 q z . r Ž 1 q zU . . ¨ 1y q t i Ž Ž 1 q z . r Ž 1 q zU . . zU Ž ¨ U ¨ .
1y
Ž Ž 1 q z . r Ž 1 q zU . . zU ¨ U ¨ q Ž Ž 1 y . r2 . 2⌬ t e q t e z¨
1y
q t i Ž Ž 1 q z . r Ž 1 q zU . . ¨ 1y q Ž Ž 1 q z . r Ž 1 q zU . . zU Ž ¨ U ¨ .
1y
,
,
,
.
Coefficients a i represent the equilibrium shipment of a particular firm Žwhatever its localization. to region’s i market. The first equation of the system above states that one unit produced in region 1 is consumed at the rate of a1 in region 1, t i a2 in region 2, and t e a3 and t e a4 in region 3 and 4, respectively. 3. SPATIAL EQUILIBRIA This section is devoted to a preliminary examination of the different types of spatial configuration obtained for various levels of interregional and international transaction costs. In the absence of migration costs, real wage differentials are the sole determinant in the decision of mobile workers to move from one region to the other. We therefore define the relative real wage for the home and foreign country, respectively Ž and U ., as the ratios of region 2 to region 1 real wages and region 4 to region 3 real wages:
s¨ U
s¨
P1
ž / ž / P2
U
P3 P4
,
Ž 11 .
.
Ž 12 .
Real wages are driven by the combination of backward and forward linkages. On the one hand, relative prices Žand wages. ¨ , ¨ U , and ¨ are determined by the system Ž7᎐10.. If, in a given country, a region features a larger proportion of the manufacturing industry and workers than the other, the firms localized in that region have access to a large market without incurring transaction costs. This allows better exploitation of their economies of scale, which pushes that region’s relative price and wage up
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Žthis is the so-called large market effect which is a centripetal force.. At the same time, firms compete to serve the local and immobile consumers, which tends to depress the core region’s price and wage Žthis corresponds to the competition effect which is a centrifugal force.. These are the backward linkages at work in the model. On the other hand, the manufacturing price index is negatively related to the number of firms installed in a given region, which constitutes a forward linkage and acts as an additional centripetal force reinforcing the large market effect. 3.1. Nature of the Spatial Equilibria A spatial equilibrium is a set of prices and wages associated with a particular geographic distribution of the manufacturing industry for which migration flows are nil. It is said to be stable if, when perturbed by an infinitesimal shock, the system responds by a tendency to return to the original equilibrium. More formally, since z describes the distribution of manufacturing in the home economy, a stable equilibrium is such that, in the home economy, either s 1 with ⭸r⭸ z strictly negative or z s 0 Žor ⬁. with - 1 Žor ) 1.. Similar conditions must simultaneously hold in the foreign economy. If we consider the particular case of a homogeneous distribution of workers across regions, we know by symmetry that every region offers the same real wage. This means that this distribution is always a spatial equilibrium which can be stable or unstable. Simulations Žwhose results are summarized in Table 1 displayed in the Appendix. yielded four different types of configuration. Figures 1, 2, 3, and 4 represent the home and foreign relative real wage Ž and U , respectively. as a function of the distribution of the manufacturing industry within each country Ž f 1 and f 3 are the home and foreign proportions of
FIG. 1. Unstable symmetric equilibrium Ž i s .8, e s .8..
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MONFORT AND NICOLINI
FIG. 2. Stable symmetric equilibrium Ž i s .3, e s .2..
the country’s manufacturing workers located in region 1 and region 3, respectively. f 1 and f 3 contain the same information as z and zU but are more easily used for graphical representations as they are bounded by 0 and 1.. The figures correspond to the reported values of i and e , while s .3 and s 4. The first configuration Žtype I. corresponds to an unstable symmetric equilibrium and is generally obtained for low values of both interregional and international transaction costs Žsee Table I in the Appendix.. If, without loss of generality, we consider only the range of z and zU between 0 and 1, thereby defining regions 2 and 4 to be the less populated inside each country, the stable equilibrium corresponding to type I configuration
FIG. 3. Multiple equilibria Ž i s .6, e s .2..
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297
FIG. 4. Interdependent regional equilibria Ž i s .5, e s .5..
is therefore z s zU s 0 Žwe of course have a mirror stable equilibrium with z s zU s ⬁.. The second case Žtype II. is obtained for higher values of transaction costs. It corresponds to the stable symmetric equilibrium z s zU s 1 for which the manufacturing industry is homogeneously distributed across regions. For intermediate transaction costs, two other types of configuration may emerge. Type III configuration displays the case in which there are multiple equilibria since both the homogeneous distribution and the full regional concentration of activities are sustainable. The associated stable equilibria are z s zU s 0, z s zU s 1, z s 1 and zU s 0 or z s 0 and zU s 1. Finally, the type IV shows that there is a possibility for the distribution of manufacturing workers in one country to affect the nature of the equilibrium in the other country. Indeed, for a complete concentration of workers in region 3 Žor 4. of the foreign country, the symmetric spatial equilibrium of the home country is unstable. Alternatively, for a homogeneous distribution in the foreign country, the symmetric home equilibrium appears stable. This suggests that the characteristics of the countries’ regional equilibria can be interdependent, as the agglomeration of activities in one country affects the likelihood of agglomeration in the other country. It also corresponds to another type of multiple equilibria since the combinations of both homogeneous and heterogeneous distributions of manufacturing in the two countries are stable.4 The corresponding stable equilibria are z s zU s 0 or z s zU s 1. 4
Intuitively, this possibility of interdependence in the agglomeration process obtains for particular cases in which the country that agglomerates its industry sees its term of trade decrease. This implies that the competition on the other country’s markets is fiercer, which can make the dispersion of economic activities no longer sustainable.
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MONFORT AND NICOLINI
CONCLUSION 1. Simulations suggest that the stable symmetric equilibrium dominates for both high interregional andror international transaction costs while for low le¨ els of transaction costs, the regional agglomeration of acti¨ ities is either sustainable or is the only stable equilibrium. As far as interregional transaction costs are concerned, the mechanism is similar to the one described in Krugman w6x where a fall in the level of interregional transaction costs increases the strength of the large market effect compared to that of the competition effect and therefore fosters the agglomeration of the manufacturing industry in a particular region. The next section shows how this argument generalizes when one incorporates international transaction costs. 4. AGGLOMERATION AND OPENNESS TO TRADE This section analyzes how international transaction costs affect the structure of the spatial equilibria when allowing for interregional labor migration inside countries. To this end, we first consider the role played by interregional transaction costs. For a particular country and a particular level of international transaction costs, the sequence of configurations can generally be represented by Fig. 5 which displays the share of manufacturing in each region as a function of the extent of interregional transaction costs. For high transaction costs Žlow values of i . the stable symmetric equilibrium Žtype II. is obtained. As transaction costs fall, agglomeration becomes sustainable Žtype III, IV., and the critical level of i at which this change occurs is denoted by i a . Eventually, the symmetric equilibrium becomes unstable, in which case the unique outcome is the agglomeration
FIG. 5. Region i’s share of domestic manufacturing industry Ž f i ..
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
299
of manufacturing in a particular region. The level at which regional symmetry breaks is denoted i s .5 We now check how these critical values are affected by a change in the extent of international transaction costs. 4.1. Symmetry Breaking We first examine how the range of interregional transaction costs over which the symmetric equilibrium proves stable varies with the extent of international transaction costs. The configurations featuring stable symmetric equilibria are of type II, III, or IV. Configurations of type II and IV imply that, for z s 1, z s zU s 1 is the only spatial equilibrium Žas defined above. in which case, by symmetry, wages and prices equalize across countries Ž ¨ s 1..6 We focus on the domestic relative real wage Ž . but the analysis readily extends to the foreign economy. For the symmetric equilibrium to be stable Žunstable ., the derivative of with respect to z when evaluated at z s zU s 1 should be negative Žpositive.. The critical value i s should therefore be that for which ⭸ Ž1, 1.r⭸ z s 0. For z s zU s 1, we have ¨ s 1 and P1 s P2 so that, by use of Ž11.:
⭸ Ž 1, 1 . ⭸z
s
⭸¨ ⭸z
q
⭸ Ž P1rP2 . ⭸z
.
The Appendix shows that there is a unique value i s for which ⭸ Ž1, 1.r⭸ z s 0:
Ž 1 y . Ž 1 y . y 1 y 2 Ž 2 y 1 . ey1 is s Ž1 q . Ž1 q . y 1
½
1r Ž y1 .
5
.
For i F i s , the symmetric equilibrium is stable so that dispersion of the manufacturing industry either is the only stable equilibrium Žconfiguration of type II. or is sustainable Žconfigurations of type III or IV.. On the contrary, for i ) i s , the symmetric equilibrium is unstable Žconfiguration of type I. and the regional concentration of manufacturing is the only stable equilibrium. Note that for sufficiently high values of e , i s - 0, which means that configuration of type I prevails for any level of interregional transaction costs. 5 One can check with the help of Table 1 in the Appendix that this sequence, corresponding to the one systematically obtained in the context of a closed economy Žsee Puga w12x., does not necessarily obtain here. In particular, for high values of international transaction cost, the sequence may not show a range over which multiple equilibria appear or even exclusively feature agglomeration of economic activities. 6 As in Krugman w5x, international trade between monopolistic competitive industries of countries identical in terms of their population size leads to factor price equalization.
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MONFORT AND NICOLINI
It is now trivial to assess the effect of a change in the extent of international transaction costs on the symmetry breaking critical value i s .
⭸ i s ⭸e
s
½
Ž 1 y . Ž 1 y . y 1 y 2 Ž 2 y 1 . ey1 Ž1 q . Ž1 q . y 1
=
y2 Ž 2 y 1 . ey2
Ž1 q . Ž1 q . y 1
Ž2y .r Ž y1 .
5
,
which, provided i s ) 0, is always negative. Configuration of type III leaves open the possibility of spatial equilibria for which the regional distribution of the manufacturing industry differs from one country to the other Žfor instance, z s 1 and zU s 0.. In such a case, the model does not deliver an analytical solution for ¨ .7 Numerical simulations obtained for a large number of parameter values nevertheless confirmed a negative relationship between i s and e even in this asymmetric countries case. CONCLUSION 2. A decrease in the le¨ el of international transaction costs Ž ⭸e ) 0. unambiguously reduces the range of interregional transaction costs for which the dispersion of the manufacturing industry can be a stable spatial equilibrium. 4.2. Sustainable Agglomeration We now turn to the analysis of the critical value at which agglomeration in a particular region becomes sustainable Ž i a .. Let us start with a situation in which the domestic economy is characterized by the full geographic concentration of its industry, i.e., z s 0. We then consider the case of a single firm that contemplates the opportunity to delocalize in region 2 and compares its payoff with the one obtained by the firms remaining in region 1. In order to attract workers in region 2, the delocalizing firm must concede higher wages than in region 1 in order to compensate workers for the fact that they must import Žand pay transaction costs on. the goods they buy in region 1. This compensation equalizes real wages across regions 1 and 2 and it is therefore such that: ¨s
w2 w1
s
P2
ž / P1
.
Ž 13 .
This in fact corresponds to the framework presented in Krugman w5x in which there is no explicit solution for the term of trade as soon as the trading economies differ in size. 7
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
301
The sales value of a typical firm corresponds to the share of income spent by consumers of each region on this particular product. For a firm installed in region 1, this writes: V1 s
n
p1
Ž1y .
Y1 q
ž / P1
p1
Ž1y .
ž / P2
Y2 t i q
q
p1
ž / ž /
Ž1y .
Y3 t e
P3
p1
Ž1y .
Y4 t e ,
P4
while for region 2, we have: V2 s
n
p2
ž / P1
Ž1y .
Y1 t i q
p2
ž / P2
Ž1y .
Y2 q
q
p2
ž / ž /
Ž1y .
Y3 t e
P3
p2
Ž1y .
P4
Y4 t e .
For delocalization to be profitable, the sales differential must be sufficient to compensate for the higher wages, and in turn the higher fixed costs, that the delocalizing firm must support, i.e., V2rV1 ) ¨ . Following Krugman w6x, this condition is synthesized by defining a function s V2rV1 ⭈ ¨ y1 . Delocalization is then profitable if and only if ) 1. If, in both countries, the manufacturing industry is entirely located in one region Žregion 1 for the domestic economy, region 3 for the foreign economy. so that z s zU s 0, symmetry implies that ¨ s 1 and we can rewrite as:
s 12 c rŽ y1. Ž 1 q . c q Ž 1 y . cy1 ,
Ž 14 .
q te where c s t1i q t e . This is a functional form similar to the one displayed in Krugman w6x except that our expression incorporates a two-dimensional transaction costs term. In particular, one easily sees that for small values of c, approaches ŽŽ1 y .r2. c Ž1y Ž1y ..rŽ y1. which is greater than 1 if is not too large.8 In such a case, we have Fig. 6, the Žtraditional. graphical representation for the function Ž c .. For c F c1 , delocalization from the large market is profitable so that concentration is not sustainable, while for values of c above c1 , regional agglomeration of the manufacturing sector is a stable equilibrium. 8
Otherwise, remains inferior to 1 for all c so that agglomeration is always sustainable.
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MONFORT AND NICOLINI
FIG. 6.
Ž c ..
The relationship between c and both i and e is unambiguously positive. Indeed, ⭸ cr⭸ i s ŽŽ y 1.rŽ1 q ey1 ..ey2 ) 0 and ⭸ cr⭸e s ŽŽ y 1.Ž1 y iy1 .rŽ1 q ey1 . 2 . iy2 ) 0. Note also that in the neighborhood of c1 , ⭸r⭸ c - 0 so that:
⭸ i a ⭸e
s
⭸ i ⭸e
- 0. s1
Once again, if in coherence with configurations of type III, we allow the concentration of home manufacturing industry in region 1 Ž z s 0. to be combined with a homogeneous distribution of manufacturing between regions 3 and 4 Ž zU s 1., we no longer have an analytical solution for ¨ , which makes the analysis of much more difficult than for the symmetric countries case. However, simulations confirm the result obtained above of a positive relationship between and both i and e in the neighborhood of s 1. CONCLUSION 3. A decrease in the le¨ el of international transaction unambiguously increases the range of interregional transaction costs for which the agglomeration of the manufacturing industry pro¨ es sustainable. 4.3. Summary What can we say about the effects of an international integration process on the structure of spatial equilibria inside the trading countries? A decrease in the level of international transaction costs has a negative impact on both i s and i a . This means that the more the economy opens to trade, the narrower the range of interregional transaction costs for which Ži. a homogeneous distribution of the manufacturing industry across
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
303
the country’s regions can be considered as a candidate equilibrium Ž i s . and Žii. the agglomeration of manufacturing in a particular region is not sustainable Ž i a .. Consequently, both lower interregional and international transaction costs foster concentration of the manufacturing industry as they increase the likelihood of a spatial equilibrium characterized by agglomeration. In particular, openness to trade works against con¨ ergence as the international integration process exacerbates the agglomeration forces at work inside countries. The intuition behind this result is the following. The localization choice is driven by the opposing large market Žcentripetal . and competition Žcentrifugal . effects. The incentive for a firm to locate in the periphery stems from the perspective of being sheltered from another producers’ competition when serving the local population. If interregional transaction costs are low, competition from firms located in the large market is severe so that remaining in the periphery is not profitable. Now, if international transaction costs are low, competition in the periphery comes from foreign producers and similarly makes this location less attractive than if this local market was sheltered from international competition. As already mentioned, this result contrasts with Krugman and Elizondo w9x who emphasize a prodispersion effect of openness to trade. This is because they assume the centrifugal forces to be based on rent and commuting costs. International integration therefore guarantees a larger access to export markets so that the localization on a large domestic market becomes less important. As we showed here, the existence of centrifugal forces based on an immobile local consumers fringe is likely to balance or even reverse this type of conclusion concerning the effect of international trade on the internal agglomeration process. 5. CONCLUSIONS In this paper, we analyzed the process of geographical concentration of economic activities in a framework incorporating both interregional and international trade. The distinction between regions and nations was formalized in a two country᎐four region model in which we allowed for workers’ mobility between the regions of a particular country but not across countries. This setting seems indeed relevant for a number of cases for which international migration barriers remain unimportant. At the same time, integration and liberalization of trade between nations is a striking feature of the world economic scene of the post-war period. The question we therefore attempted to address was the following: What is the effect of international markets integration on the characteristics of the countries’ regional equilibria?
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MONFORT AND NICOLINI
To this end, we developed an extension of Krugman w6x in which trade between both regions and countries entails transaction costs that are differentiated according to the interregional or international nature of the flows. The results obtained suggest that both types of transaction costs affect the incentive for firms to concentrate geographically. In particular, a decrease in interregional andror international transaction costs was shown to reinforce the tendency for some national economic activities to cluster on a particular location of the country. In this perspective, movements toward integration and international trade liberalization could be considered as factors possibly favoring the emergence of regional economic agglomeration inside countries. This finding is coherent with the stylized facts established for Europe: although the convergence process that one observes between European countries since the 1960s is still at work today, it has been stopped and even reversed for several years at the level of European regions ŽEC w3x; Puga w12x.. One possible interpretation of such evolution is that it is inside countries that regional disparities are growing. According to the results obtained in the present paper, this evolution could partly be explained by the integration and trade liberalization processes which have been taking place under the construction of the European common market. APPENDIX A. Simulations This appendix reports the results of the simulations mentioned in Sect. 3. As an illustration of the impact of interregional and international transaction costs on the nature of spatial equilibrium, Table 1 displays a typology in which the different types of configuration are mapped in the i y e space. Configurations allowing for agglomeration as a stable equilibrium Žtype I, III, IV. dominate for low transaction costs in general Žhigh ’s. while, for high levels of transaction costs Žlow ’s., the homogeneous distribution of activities is stable Žtype II, III, IV.. B. Symmetry Breaking Critical Value Ž i s . This appendix derives the critical value Ž i s . at which a global symmetric equilibrium Ž z s zU s 1. breaks and becomes unstable. As explained in Sect. 3, i s is the solution of the following equation:
⭸ Ž 1, 1 . ⭸z
s
⭸¨ ⭸z
q
d Ž P1rP2 . dz
s 0.
Ž 15 .
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
305
TABLE 1 Configuration Types
s 4, s .3
e
.99 .89 .79 .69 .59 .49 .39 .29 .19 .09
I I I I II II II II II II .09
I I I I II II II II II II .19
I I I I IV II II II II II .29
I I I I IV II II II II II .39
I I I I I IV II II II II .49
I I I I I I I I III III .59
I I I I I I I I I I .69
I I I I I I I I I I .79
I I I I I I I I I I .89
I I I I I I I I I I .99
i
A change in z implies a change in ¨ and ¨ . We therefore use the first two equations of the system Ž7᎐10.: E1 ' a1 q t i a2 q t e a3 q t e a4 y 1 s 0, E2 ' t i a1 q a2 q t e a3 q t e a y ¨ s 0. Applying the implicit function theorem, we have:
⭸ E1 ⭸z ⭸ E2 ⭸z
dz q dz q
⭸ E1 ⭸¨ ⭸ E2 ⭸¨
d¨ q d¨ q
⭸ E1 ⭸¨ ⭸ E2 ⭸¨
d ¨ s 0, d ¨ s 0.
This system is solved for d¨ rdz and d ¨ rdz. In particular, for z s zU s ¨ s ¨ U s ¨ s 1, we have: d¨ dz
s
Ž 1 y t i .Ž y 1 q 2 t e q t i q t i .
Ž 1 y y 2 t e q 4 t e q 4 t e2 y 2 t i q 4 t i q 2 t e t i q 4 t e t i q t i2 q mti2 . Ž 16 .
d¨ dz
s
Ž 1 y t i .Ž y 1 q 2 t e q t i q t i . 2 Ž 1 y y 2 t e q 4 t e q 4 t e2 y 2 t i q 4 t i q 2 t e t i q 4 t e t i q t i2 q m2i .
.
Ž 17 .
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MONFORT AND NICOLINI
As far as the ratio P1rP2 is concerned, we have: d Ž P1rP2 . dz
s
⭸ Ž P1rP2 . ⭸z
q
⭸ Ž P1rP2 . ⭸ ⭸¨
⭸z
q
⭸ Ž P1rP2 . ⭸ ¨ ⭸¨
⭸z
.
For z s zU s ¨ s ¨ U s ¨ s 1 and using Ž16. and Ž17., we obtain: d Ž P1rP2 . dz
s
ti y 1
Ž y 1. Ž 1 q 2 t e q ti .
.
Ž 18 .
Plugging Eqs. Ž18. and Ž16. into Eq. Ž15., we have the following condition:
⭸ Ž 1, 1 . ⭸z
s 0 m y1 q q y 2 q 2 q 2 t e y 4 t e qt i q t i y t i y 2 t i y 2 t i s 0,
with t i s Ž i . y1 and t e s Že . y1. This equation admits a unique root in i which is: y1 Ž 1 y . Ž 1 y . y 1 y 2 Ž 2 y 1 . Ž e . is s Ž1 q . Ž1 q . y 1
½
1r Ž y1 .
5
.
REFERENCES 1. J. Decressin and A. Fatas, ` ‘‘Regional Labour Market Dynamics in Europe,’’ CEPR Discussion Paper No. 1085 Ž1994.. 2. A. K. Dixit and J. E. Stiglitz, Monopolistic competition and optimum product diversity, American Economic Re¨ iew, 67, 297᎐308 Ž1997.. 3. European Commission, ‘‘Economic Evaluation of the Internal Market, European Economy’’ Ž1996.. 4. M. Fujita and J. F. Thisse, Economics of agglomeration, Journal of the Japanese and International Economies, 10, 339᎐378 Ž1996.. 5. P. Krugman, Scale economies, product differentiation and the pattern of trade, American Economic Re¨ iew, 70, 950᎐959 Ž1980.. 6. P. Krugman, Increasing returns and economic geography, Journal of Political Economy, 99, 483᎐499 Ž1991.. 7. P. Krugman, On the relationship between trade theory and location theory, Re¨ iew of International Economies, 1, 110᎐122 Ž1993.. 8. P. Krugman and A. Venables, Globalisation and the inequality of nations, The Quarterly Journal of Economics, CX, 857᎐880 Ž1995.. 9. P. Krugman and R. Livas Elizondo, Trade policy and the Third World metropolis, Journal of De¨ elopment Economics, 49, 137᎐150 Ž1996.. 10. K. Matsuyama, ‘‘Why Are There Rich and Poor Countries?: Symmetry-Breaking in the World Economy,’’ NBER Working paper No. 5697 Ž1996.. 11. J. Newhouse, Europe rising regionalism, Foreign Affairs, 76, 67᎐84 Ž1997.. 12. D. Puga, The rise and the fall of regional inequalities, European Economic Re¨ iew, 43, 303᎐335 Ž1999..
Regional Convergence and International IntegrationU Philippe Monfort† and Rosella Nicolini IRES and Department of Economics, Uni¨ ersite´ Catholique de Lou¨ ain, Place Montesquieu 3, 1348 Lou¨ ain-La-Neu¨ e, Belgium E-mail: [email protected] Received November 25, 1998; revised August 10, 1999 We analyze the geographic concentration of economic activities within the framework of a two country᎐four region model. Trade between both regions and countries entails transaction costs which are differentiated according to the interregional or international nature of the flows. Allowing for regional migration of the population, the model configures the equilibria of this system as key parameters change. The results obtained suggest that both types of transaction costs affect the incentive for firms to cluster geographically. Consequently, processes of integration between nations are identified as factors possibly favoring the emergence of regional economic agglomerations. 䊚 2000 Academic Press Key Words: international trade; agglomeration; economic geography.
1. INTRODUCTION Recent contributions in economic theory have shown how some central international trade issues, for instance the effects of trade liberalization on the wealth of nations, could be interestingly revisited by adopting an economic geography approach Žsee, for instance, Fujita and Thisse w4x for an extensive survey of this literature .. Countries are then considered as regions between which the mobility of goods andror factors is more or less fluid Žsee, for instance, Krugman and Venables w8x or Matsuyama w10x.. The two fields indeed largely overlap if not focus on the same object Ž‘‘Who produces what, where?’’ to quote Krugman w7x. and it would probably be quite limiting to try to draw a clear line of demarcation between them. In a certain number of cases, however, regional and national levels should be distinguished as they might at the same time coexist and differ * We gratefully acknowledge Jacques Thisse and Xavier Wauthy for helpful comments. The usual disclaimer applies. This research is also part of a research program supported by the Belgian Program on Interuniversity Poles of Attraction ŽPAI N P4r01.. † Author to whom correspondence should be addressed. 286 0094-1190r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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in their characteristics. In particular, the concept of nation seems appropriate if one wants to allow for institutional peculiarities, for goods or factors flow impediments related to trade policy, or for factor immobility related to cultural or linguistic barriers for which national boundaries are relevant geographic features. This is especially clear when one examines the differences between the regional agglomeration processes in the United States on the one hand and in Europe on the other. While in the former case, models considering an area composed of a set of regions seem indeed opportune, the analysis of the latter probably requires consideration of both interregional and international trade and factor flows. In particular, international labor mobility in Europe remains extremely low compared to the interstate one in the U.S. Žsee, for instance, Decressin and Fatas ` w1x1 .. Accordingly, although some kind of agglomeration has undoubtedly taken place in Europeᎏone clearly sees the existence of a European core-periphery pattern Žpossibly centered in Belgium ŽNewhouse w11x. ᎏyet a large number of European industries cannot be considered as concentrated in a particularly country. In fact, the concentration of economic activities is a phenomenon which might be more likely detected inside each European country. Moreover, the in-depth integration process that Europe has witnessed since World War II has been accompanied by a tendency of European economies to converge but, for the last few years, regional disparities seem to be growing across European regions, which suggests that it is inside countries that the agglomeration of economic activities is now taking place ŽEC w3x; Puga w12x.. The objective of this paper is therefore to build a framework taking such features into account and to deepen our understanding of how international economic integration affects the process of agglomeration within trading countries. To this end, we extend the Krugman w6x framework to a two country᎐four region model in which each country consists of two regions between which national labor is mobile. Labor is, on the other hand, immobile between nations. Although extreme, this assumption is intended to account for differences in the extent of interregional and international factor mobility. As far as goods are concerned, both interregional and international trade entail transaction costs but their magnitude might differ according to the interregional or international feature of the trade flows. The extent and the nature of trading costs are indeed likely to be quite different whenever goods are shipped to another country rather to another region of the same country. While the latter form of trade merely entails transportation costs, 1 Eurostat displays an EU-12 average rate of migration of 2.9 per thousand in 1991 while, for the same period, the U.S. rate of geographical mobility is 2.9% ŽU.S. Bureau of the Census..
288
MONFORT AND NICOLINI
the former also includes costs related to institutional factors like trade policy, customs duties and formalities, or adaptation to foreign legislation standards, which also act as other impediments to international trade. Moreover, it is also clear that the process of European integration affected mostly international transaction costs while leaving interregional trading costs unchanged. If we want to discuss the impact of integration between nations on a country’s regional outcome, it is important to explicitly account for possible differences in the two kinds of transaction costs. The choice to investigate these questions within a Krugman w6x type of framework is of course disputable as, for instance, it does not allow for intersectoral labor mobility, which would give to the analysis a more pronounced international trade flavor. Nevertheless, the capacity of this type of model to capture the essence of an agglomeration process in a very simple way makes it worthwhile to show how it can be generalized by incorporating both regions and nations. The results of this analysis suggest that variations in international transaction costs are able to modify the structure of the countries’ spatial equilibria and more specifically that the agglomeration of national economic activities proves to be enhanced by the process of international integration. Consequently, even if, as underlined by Puga w12x, the EU integration process is not likely to yield an economic geography comparable to that of the U.S. because of a relative international factor immobility, it is nevertheless potentially able to generate or increase regional disparities inside European countries, which seems in line with the stylized facts mentioned above. The issue of the relationship between a country’s openness to trade and the internal geographic distribution of its activities is also addressed in Krugman and Elizondo w9x. In their framework, however, openness to international trade fosters the dispersion of economic activities because the centrifugal forces hinge on rent and commuting costs while, in ours, they rest on the existence of an immobile consumers fringe. They indeed focus on metropolization phenomena in developing countries, both a topic and a context for which their assumptions concerning the centrifugal forces are relevant. In our case, we intend rather to elaborate on the emergence of regional disparities in a framework suitable for capturing European-like contexts. In this prospect, the alternative of focusing on dispersion forces due to local markets is worthwhile to investigate and is actually shown to reverse the conclusions of Krugman and Elizondo concerning the effect of international economic integration on the countries’ spatial equilibria. The paper is organized as follows: Sect. 2 develops a model which allows convenient introduction of selected ingredients of location and international trade theory. Sect. 3 examines the characteristics of the price system and the structure of the spatial equilibrium for different levels of transac-
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289
tion costs. Sect. 4 explicitly derives the mechanisms through which the spatial equilibrium is affected by both interregional and international transaction costs. Finally, Sect. 5 concludes. 2. A TWO-COUNTRY AND FOUR-REGION MODEL We consider a world with two countries and two regions in each country. In the first country Žhome., we distinguish region 1 and 2, while the second country Žforeign. is composed of regions 3 and 4. Following Krugman w6x, there are two sectors of production: agriculture and manufacturing. Agriculture is a sector tied to the land and is therefore geographically immobile while manufacturing is an increasing returns sector that can be located anywhere. 2.1. Consumers᎐Workers Consumers in the two countries are identical and share the same utility function expressed as: U s CM CA1y ,
Ž 1.
where CA is the consumption of agricultural goods and CM is the consumption of the manufactures aggregate. According to this specification, consumers devote a share of their expenditure to manufactured goods. The manufactures aggregate CM is defined by: n
CM s
Ý
r Ž y1 .
c kŽ y1.r
,
ks1
where n is the number of potential products and Žwith ) 1. is the elasticity of substitution among those goods. The population of each country is normalized at 1, of which a fraction 1 y are peasants and are manufacturing workers. World population therefore sums up to 2. Workers and peasants are the only factors of production. They are assumed to be sector specific, peasants for agriculture and workers for manufacturing. Moreover, we assume that workers move across regions but not across countries while peasants are completely immobile. In each of the four regions, we therefore have a fixed peasants supply equal to Ž1 y .r2 while the countries’ workers supplies meet the conditions: L1 q L 2 s , L3 q L4 s , where L i denotes the workers supply of region i.
290
MONFORT AND NICOLINI
2.2. Agriculture and Manufacturing Technology Agriculture is perfectly competitive. It is a constant returns to scale sector and we normalize the unitary labor requirement to 1. Trade of the agricultural good is supposed to be costless between both regions and countries, which implies that agricultural prices and peasants’ wages are equalized across the whole area. The agricultural good is therefore used as the numeraire. The production of manufactures features economies of scale and the labor requirement for an individual manufactured good Ž k . is defined as: LM k s  q ␥ x k , where  corresponds to a fixed cost, ␥ to the constant marginal cost of production, and x k to the quantity produced. The manufactures market structure is characterized by monopolistic competition following Dixit and Stiglitz w2x, under which each firm produces a single differentiated good. 2.3. Transaction Costs Interregional and international trade of manufactured goods is assumed to incur transaction costs. These are formalized as ‘‘Samuelsonian iceberg transportation costs’’ for which a fraction Ž1 y . of one unit shipped to another region is lost in the transport Ž - 1.. In a broad sense, ’s value not only captures the extent of transport costs associated with interregional or international trade, but also accounts for any kind of trade barrier which affects a region’s openness to trade.2 In this perspective, extreme high values of represent fully integrated markets while extreme low values of correspond to autarchy. We assume that transaction costs differ according to the interregional or international nature of the trade flow. We can indeed expect international transaction costs to incorporate a larger fraction of institutional trade impediments than interregional transaction costs, which should mainly represent transport costs. Moreover, by distinguishing both types of transaction costs, we can analyze the effect of an international integration process Žwhich is simply formalized as a decrease in the international transaction costs. on the outcome of the model. Let i denote transaction costs associated with trade between regions of a same country while e represents those entailed by international trade flows. We adopt two simplifying assumptions. On the one hand, interre2 Since we do not introduce a public sector in the model, the trade barriers considered here should be viewed as those generating no rents or tax revenue Žfor instance, safety and health regulations, industrial standards, etc...
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
291
gional transaction costs are supposed to be identical in both countries. On the other hand, international transaction costs are assumed to be independent of the regions involved, i.e., the cost incurred by trading with the other country is the same whatever the foreign region the good is exported to. 2.4. Size of the Regional Manufacturing Industry Firms are symmetric as far as individual demand and technology is concerned. Given the previous definitions, one easily ascertains Žsee, for instance, Krugman w6x. that the elasticity of the total demand facing any firm is independent of the transaction costs and simply corresponds to . The profit-maximizing pricing behavior of firms located in region i is then: pi s
ž /␥ y1
i s 1, 2, 3, 4.
wi
Ž 2.
The price set in a particular region is a constant mark-up over the regional wage rate Ž wi . which implies that: pi pj
s
wi
i , j s 1, 2, 3, 4.
wj
Free entry leads to complete erosion of pure profits so that the usual zero-profit condition applies with the corollary that the equilibrium output level Ž x . is the same for each firm in any region. In particular, we have: xi s x s
 Ž y 1. ␥
i s 1, 2, 3, 4.
Full employment of the labor force then allows us to specify the number of firms active in a particular region as: ni s
Li

i s 1, 2, 3, 4
Ž 3.
The equilibrium number of firms is proportional to the region’s workers population and we therefore have: ni nj
s
Li Lj
i , j s 1, 2, 3, 4.
2.5. Equilibrium Conditions in Goods and Labor Markets In shipping to another region or country, the price of a local manufactured product is increased by the relevant transaction costs so that to acquire one unit of a region i’s manufactured good, consumers of region j
292
MONFORT AND NICOLINI
must spend pir i if region i and j are located in the same country or pire if located in different countries. Given the utility function Ž1., the consumers price index of manufactured goods is given for each region by: P1 s
n1 p1y 1
P2 s n1
P3 s n1
P4 s n1
p1
ž / ž / ž /
q n2
e
i
q
n 2 p1y 2
1y
q n2
e
p1
q n3
ž /
1y
i
p1
1y
p2
1y
q n3
p2
ž / ž /
q n3
1y
ž / ž /
q n4
e
p3
1y
q n4
e
1y
q
e
p2
p3
n 3 p1y 3
1y
e
q n3
p3
q n4
p4
ž / ž / ž /
1y
,
e
p4
1y
i
1r Ž1y .
,
e
p4
1y
1r Ž1y .
,
i
1r Ž1y .
1y
ž /
1r Ž1y .
q
n 4 p41y
.
On the manufactured goods market, the equilibrium condition requires production of region i to be equal to the world demand. Since, at zero profit, each firm produces the same quantity x, we have: 4
ni x s
Ý ni js1
pi
ž / Pj
y
Yj 1 Pj
,
Ž 4.
where 䢇 䢇 䢇 䢇
s 1 if i s j, s i if j is a region belonging to the same country as i, s e if j refers to a region of the other country, Yj is the income of region j.
On the other hand, the equilibrium condition on the world agricultural market is given by:
Ž 1 y . w Y1 q Y2 q Y3 q Y4 x s 2 Ž 1 y . , the LHS of the equality corresponding to world consumption and the RHS to world production of the agricultural good. Finally, the income of each region corresponds to the revenues of peasants and manufacturing workers. The wage rate of peasants being the numeraire, we have: Yi s
1y 2
q wi L i
i s 1, 2, 3, 4.
Ž 5.
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
293
Note that, by the price setting rule Ž2. and the expression giving the equilibrium number of firms as a function of the region’s workers population Ž3., we have wi L i s n i pi x. We can now express the whole system in terms of the relative regional sizes and prices. Let us define relative prices as ¨ s p 2rp1 , ¨ U s p4rp 3 , and ¨ s p 3rp1. Similarly, we have z s n 2rn1 and zU s n 4rn 3 . Noting that n 2 p 2 s z.¨ .n1 p1 , n 3 p 3 s ŽŽ1 q z .rŽ1 q zU .. ¨ .n1 p1 , and n 4 p4 s ŽŽ1 q z .rŽ1 q zU .. zU .¨ U .¨ .n1 p1 and using the definitions of the regional incomes Ž5., the equilibrium condition on the agricultural good markets is rewritten: 2 Ž 1 y . q Ž n1 p1 q n 2 p 2 q n 3 p 3 q n 4 p4 . x s 2, which, after manipulation, leads to: n1 p 1 x s
2 ⌬
,
where ⌬ s 1 q z.¨ q ŽŽ1 q z .rŽ1 q zU ...¨ q ŽŽ1 q z .rŽ1 q zU ... zU .¨ U .¨ . The equilibrium condition for the manufactured goods can now be expressed in the following way. Let t i s iy1 F 1 and t e s ey1 F 1 and consider, for instance, condition Ž4. for region 1. This can be rewritten as: xs
py 1 Ž 1 y . r2 q n1 p 1 x
P11y q
q
py 1 t e Ž 1 y . r2 q n 3 p 3 x
P31y
py 1 t i Ž 1 y . r2 q n 2 p 2 x
P21y q
py 1 t e Ž 1 y . r2 q n 4 p4 x
P41y
Ž 6. The other regions’ equilibrium conditions are similarly transformed. Expressing each product n i pi Ž i s 2, 3, 4. in terms of n1 p1 and making use of relative prices and sizes, we finally obtain the following system: 3 a1 q t i a2 q t e a3 q t e a4 s 1,
Ž 7.
t i a1 q a2 q t e a3 q t e a4 s ¨ ,
Ž 8.
t e a1 q t e a2 q a3 q t i a4 s ¨ ,
Ž 9. U
t e a1 q t e a2 q t i a3 q a4 s Ž ¨¨ . , 3
Ž 10 .
As usual, one of the four equations is redundant if one makes use of the Walras law.
294
MONFORT AND NICOLINI
with: a1 s
a2 s
a3 s
a4 s
1 q Ž Ž 1 y . r2 . 2⌬ 1 q t i z¨ 1y q t e Ž Ž 1 q z . r Ž 1 q zU . . ¨ 1y q t e Ž Ž 1 q z . r Ž 1 q zU . . zU Ž ¨ U ¨ .
1y
z¨ q Ž Ž 1 y . r2 . 2⌬ t i q z¨ 1y q t e Ž Ž 1 q z . r Ž 1 q zU . . ¨ 1y q t e Ž Ž 1 q z . r Ž 1 q zU . . zU Ž ¨ U ¨ .
1y
Ž Ž 1 q z . r Ž 1 q zU . . ¨ q Ž Ž 1 y . r2 . 2⌬ t e q t e z¨ 1y q Ž Ž 1 q z . r Ž 1 q zU . . ¨ 1y q t i Ž Ž 1 q z . r Ž 1 q zU . . zU Ž ¨ U ¨ .
1y
Ž Ž 1 q z . r Ž 1 q zU . . zU ¨ U ¨ q Ž Ž 1 y . r2 . 2⌬ t e q t e z¨
1y
q t i Ž Ž 1 q z . r Ž 1 q zU . . ¨ 1y q Ž Ž 1 q z . r Ž 1 q zU . . zU Ž ¨ U ¨ .
1y
,
,
,
.
Coefficients a i represent the equilibrium shipment of a particular firm Žwhatever its localization. to region’s i market. The first equation of the system above states that one unit produced in region 1 is consumed at the rate of a1 in region 1, t i a2 in region 2, and t e a3 and t e a4 in region 3 and 4, respectively. 3. SPATIAL EQUILIBRIA This section is devoted to a preliminary examination of the different types of spatial configuration obtained for various levels of interregional and international transaction costs. In the absence of migration costs, real wage differentials are the sole determinant in the decision of mobile workers to move from one region to the other. We therefore define the relative real wage for the home and foreign country, respectively Ž and U ., as the ratios of region 2 to region 1 real wages and region 4 to region 3 real wages:
s¨ U
s¨
P1
ž / ž / P2
U
P3 P4
,
Ž 11 .
.
Ž 12 .
Real wages are driven by the combination of backward and forward linkages. On the one hand, relative prices Žand wages. ¨ , ¨ U , and ¨ are determined by the system Ž7᎐10.. If, in a given country, a region features a larger proportion of the manufacturing industry and workers than the other, the firms localized in that region have access to a large market without incurring transaction costs. This allows better exploitation of their economies of scale, which pushes that region’s relative price and wage up
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295
Žthis is the so-called large market effect which is a centripetal force.. At the same time, firms compete to serve the local and immobile consumers, which tends to depress the core region’s price and wage Žthis corresponds to the competition effect which is a centrifugal force.. These are the backward linkages at work in the model. On the other hand, the manufacturing price index is negatively related to the number of firms installed in a given region, which constitutes a forward linkage and acts as an additional centripetal force reinforcing the large market effect. 3.1. Nature of the Spatial Equilibria A spatial equilibrium is a set of prices and wages associated with a particular geographic distribution of the manufacturing industry for which migration flows are nil. It is said to be stable if, when perturbed by an infinitesimal shock, the system responds by a tendency to return to the original equilibrium. More formally, since z describes the distribution of manufacturing in the home economy, a stable equilibrium is such that, in the home economy, either s 1 with ⭸r⭸ z strictly negative or z s 0 Žor ⬁. with - 1 Žor ) 1.. Similar conditions must simultaneously hold in the foreign economy. If we consider the particular case of a homogeneous distribution of workers across regions, we know by symmetry that every region offers the same real wage. This means that this distribution is always a spatial equilibrium which can be stable or unstable. Simulations Žwhose results are summarized in Table 1 displayed in the Appendix. yielded four different types of configuration. Figures 1, 2, 3, and 4 represent the home and foreign relative real wage Ž and U , respectively. as a function of the distribution of the manufacturing industry within each country Ž f 1 and f 3 are the home and foreign proportions of
FIG. 1. Unstable symmetric equilibrium Ž i s .8, e s .8..
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MONFORT AND NICOLINI
FIG. 2. Stable symmetric equilibrium Ž i s .3, e s .2..
the country’s manufacturing workers located in region 1 and region 3, respectively. f 1 and f 3 contain the same information as z and zU but are more easily used for graphical representations as they are bounded by 0 and 1.. The figures correspond to the reported values of i and e , while s .3 and s 4. The first configuration Žtype I. corresponds to an unstable symmetric equilibrium and is generally obtained for low values of both interregional and international transaction costs Žsee Table I in the Appendix.. If, without loss of generality, we consider only the range of z and zU between 0 and 1, thereby defining regions 2 and 4 to be the less populated inside each country, the stable equilibrium corresponding to type I configuration
FIG. 3. Multiple equilibria Ž i s .6, e s .2..
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
297
FIG. 4. Interdependent regional equilibria Ž i s .5, e s .5..
is therefore z s zU s 0 Žwe of course have a mirror stable equilibrium with z s zU s ⬁.. The second case Žtype II. is obtained for higher values of transaction costs. It corresponds to the stable symmetric equilibrium z s zU s 1 for which the manufacturing industry is homogeneously distributed across regions. For intermediate transaction costs, two other types of configuration may emerge. Type III configuration displays the case in which there are multiple equilibria since both the homogeneous distribution and the full regional concentration of activities are sustainable. The associated stable equilibria are z s zU s 0, z s zU s 1, z s 1 and zU s 0 or z s 0 and zU s 1. Finally, the type IV shows that there is a possibility for the distribution of manufacturing workers in one country to affect the nature of the equilibrium in the other country. Indeed, for a complete concentration of workers in region 3 Žor 4. of the foreign country, the symmetric spatial equilibrium of the home country is unstable. Alternatively, for a homogeneous distribution in the foreign country, the symmetric home equilibrium appears stable. This suggests that the characteristics of the countries’ regional equilibria can be interdependent, as the agglomeration of activities in one country affects the likelihood of agglomeration in the other country. It also corresponds to another type of multiple equilibria since the combinations of both homogeneous and heterogeneous distributions of manufacturing in the two countries are stable.4 The corresponding stable equilibria are z s zU s 0 or z s zU s 1. 4
Intuitively, this possibility of interdependence in the agglomeration process obtains for particular cases in which the country that agglomerates its industry sees its term of trade decrease. This implies that the competition on the other country’s markets is fiercer, which can make the dispersion of economic activities no longer sustainable.
298
MONFORT AND NICOLINI
CONCLUSION 1. Simulations suggest that the stable symmetric equilibrium dominates for both high interregional andror international transaction costs while for low le¨ els of transaction costs, the regional agglomeration of acti¨ ities is either sustainable or is the only stable equilibrium. As far as interregional transaction costs are concerned, the mechanism is similar to the one described in Krugman w6x where a fall in the level of interregional transaction costs increases the strength of the large market effect compared to that of the competition effect and therefore fosters the agglomeration of the manufacturing industry in a particular region. The next section shows how this argument generalizes when one incorporates international transaction costs. 4. AGGLOMERATION AND OPENNESS TO TRADE This section analyzes how international transaction costs affect the structure of the spatial equilibria when allowing for interregional labor migration inside countries. To this end, we first consider the role played by interregional transaction costs. For a particular country and a particular level of international transaction costs, the sequence of configurations can generally be represented by Fig. 5 which displays the share of manufacturing in each region as a function of the extent of interregional transaction costs. For high transaction costs Žlow values of i . the stable symmetric equilibrium Žtype II. is obtained. As transaction costs fall, agglomeration becomes sustainable Žtype III, IV., and the critical level of i at which this change occurs is denoted by i a . Eventually, the symmetric equilibrium becomes unstable, in which case the unique outcome is the agglomeration
FIG. 5. Region i’s share of domestic manufacturing industry Ž f i ..
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
299
of manufacturing in a particular region. The level at which regional symmetry breaks is denoted i s .5 We now check how these critical values are affected by a change in the extent of international transaction costs. 4.1. Symmetry Breaking We first examine how the range of interregional transaction costs over which the symmetric equilibrium proves stable varies with the extent of international transaction costs. The configurations featuring stable symmetric equilibria are of type II, III, or IV. Configurations of type II and IV imply that, for z s 1, z s zU s 1 is the only spatial equilibrium Žas defined above. in which case, by symmetry, wages and prices equalize across countries Ž ¨ s 1..6 We focus on the domestic relative real wage Ž . but the analysis readily extends to the foreign economy. For the symmetric equilibrium to be stable Žunstable ., the derivative of with respect to z when evaluated at z s zU s 1 should be negative Žpositive.. The critical value i s should therefore be that for which ⭸ Ž1, 1.r⭸ z s 0. For z s zU s 1, we have ¨ s 1 and P1 s P2 so that, by use of Ž11.:
⭸ Ž 1, 1 . ⭸z
s
⭸¨ ⭸z
q
⭸ Ž P1rP2 . ⭸z
.
The Appendix shows that there is a unique value i s for which ⭸ Ž1, 1.r⭸ z s 0:
Ž 1 y . Ž 1 y . y 1 y 2 Ž 2 y 1 . ey1 is s Ž1 q . Ž1 q . y 1
½
1r Ž y1 .
5
.
For i F i s , the symmetric equilibrium is stable so that dispersion of the manufacturing industry either is the only stable equilibrium Žconfiguration of type II. or is sustainable Žconfigurations of type III or IV.. On the contrary, for i ) i s , the symmetric equilibrium is unstable Žconfiguration of type I. and the regional concentration of manufacturing is the only stable equilibrium. Note that for sufficiently high values of e , i s - 0, which means that configuration of type I prevails for any level of interregional transaction costs. 5 One can check with the help of Table 1 in the Appendix that this sequence, corresponding to the one systematically obtained in the context of a closed economy Žsee Puga w12x., does not necessarily obtain here. In particular, for high values of international transaction cost, the sequence may not show a range over which multiple equilibria appear or even exclusively feature agglomeration of economic activities. 6 As in Krugman w5x, international trade between monopolistic competitive industries of countries identical in terms of their population size leads to factor price equalization.
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MONFORT AND NICOLINI
It is now trivial to assess the effect of a change in the extent of international transaction costs on the symmetry breaking critical value i s .
⭸ i s ⭸e
s
½
Ž 1 y . Ž 1 y . y 1 y 2 Ž 2 y 1 . ey1 Ž1 q . Ž1 q . y 1
=
y2 Ž 2 y 1 . ey2
Ž1 q . Ž1 q . y 1
Ž2y .r Ž y1 .
5
,
which, provided i s ) 0, is always negative. Configuration of type III leaves open the possibility of spatial equilibria for which the regional distribution of the manufacturing industry differs from one country to the other Žfor instance, z s 1 and zU s 0.. In such a case, the model does not deliver an analytical solution for ¨ .7 Numerical simulations obtained for a large number of parameter values nevertheless confirmed a negative relationship between i s and e even in this asymmetric countries case. CONCLUSION 2. A decrease in the le¨ el of international transaction costs Ž ⭸e ) 0. unambiguously reduces the range of interregional transaction costs for which the dispersion of the manufacturing industry can be a stable spatial equilibrium. 4.2. Sustainable Agglomeration We now turn to the analysis of the critical value at which agglomeration in a particular region becomes sustainable Ž i a .. Let us start with a situation in which the domestic economy is characterized by the full geographic concentration of its industry, i.e., z s 0. We then consider the case of a single firm that contemplates the opportunity to delocalize in region 2 and compares its payoff with the one obtained by the firms remaining in region 1. In order to attract workers in region 2, the delocalizing firm must concede higher wages than in region 1 in order to compensate workers for the fact that they must import Žand pay transaction costs on. the goods they buy in region 1. This compensation equalizes real wages across regions 1 and 2 and it is therefore such that: ¨s
w2 w1
s
P2
ž / P1
.
Ž 13 .
This in fact corresponds to the framework presented in Krugman w5x in which there is no explicit solution for the term of trade as soon as the trading economies differ in size. 7
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
301
The sales value of a typical firm corresponds to the share of income spent by consumers of each region on this particular product. For a firm installed in region 1, this writes: V1 s
n
p1
Ž1y .
Y1 q
ž / P1
p1
Ž1y .
ž / P2
Y2 t i q
q
p1
ž / ž /
Ž1y .
Y3 t e
P3
p1
Ž1y .
Y4 t e ,
P4
while for region 2, we have: V2 s
n
p2
ž / P1
Ž1y .
Y1 t i q
p2
ž / P2
Ž1y .
Y2 q
q
p2
ž / ž /
Ž1y .
Y3 t e
P3
p2
Ž1y .
P4
Y4 t e .
For delocalization to be profitable, the sales differential must be sufficient to compensate for the higher wages, and in turn the higher fixed costs, that the delocalizing firm must support, i.e., V2rV1 ) ¨ . Following Krugman w6x, this condition is synthesized by defining a function s V2rV1 ⭈ ¨ y1 . Delocalization is then profitable if and only if ) 1. If, in both countries, the manufacturing industry is entirely located in one region Žregion 1 for the domestic economy, region 3 for the foreign economy. so that z s zU s 0, symmetry implies that ¨ s 1 and we can rewrite as:
s 12 c rŽ y1. Ž 1 q . c q Ž 1 y . cy1 ,
Ž 14 .
q te where c s t1i q t e . This is a functional form similar to the one displayed in Krugman w6x except that our expression incorporates a two-dimensional transaction costs term. In particular, one easily sees that for small values of c, approaches ŽŽ1 y .r2. c Ž1y Ž1y ..rŽ y1. which is greater than 1 if is not too large.8 In such a case, we have Fig. 6, the Žtraditional. graphical representation for the function Ž c .. For c F c1 , delocalization from the large market is profitable so that concentration is not sustainable, while for values of c above c1 , regional agglomeration of the manufacturing sector is a stable equilibrium. 8
Otherwise, remains inferior to 1 for all c so that agglomeration is always sustainable.
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MONFORT AND NICOLINI
FIG. 6.
Ž c ..
The relationship between c and both i and e is unambiguously positive. Indeed, ⭸ cr⭸ i s ŽŽ y 1.rŽ1 q ey1 ..ey2 ) 0 and ⭸ cr⭸e s ŽŽ y 1.Ž1 y iy1 .rŽ1 q ey1 . 2 . iy2 ) 0. Note also that in the neighborhood of c1 , ⭸r⭸ c - 0 so that:
⭸ i a ⭸e
s
⭸ i ⭸e
- 0. s1
Once again, if in coherence with configurations of type III, we allow the concentration of home manufacturing industry in region 1 Ž z s 0. to be combined with a homogeneous distribution of manufacturing between regions 3 and 4 Ž zU s 1., we no longer have an analytical solution for ¨ , which makes the analysis of much more difficult than for the symmetric countries case. However, simulations confirm the result obtained above of a positive relationship between and both i and e in the neighborhood of s 1. CONCLUSION 3. A decrease in the le¨ el of international transaction unambiguously increases the range of interregional transaction costs for which the agglomeration of the manufacturing industry pro¨ es sustainable. 4.3. Summary What can we say about the effects of an international integration process on the structure of spatial equilibria inside the trading countries? A decrease in the level of international transaction costs has a negative impact on both i s and i a . This means that the more the economy opens to trade, the narrower the range of interregional transaction costs for which Ži. a homogeneous distribution of the manufacturing industry across
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
303
the country’s regions can be considered as a candidate equilibrium Ž i s . and Žii. the agglomeration of manufacturing in a particular region is not sustainable Ž i a .. Consequently, both lower interregional and international transaction costs foster concentration of the manufacturing industry as they increase the likelihood of a spatial equilibrium characterized by agglomeration. In particular, openness to trade works against con¨ ergence as the international integration process exacerbates the agglomeration forces at work inside countries. The intuition behind this result is the following. The localization choice is driven by the opposing large market Žcentripetal . and competition Žcentrifugal . effects. The incentive for a firm to locate in the periphery stems from the perspective of being sheltered from another producers’ competition when serving the local population. If interregional transaction costs are low, competition from firms located in the large market is severe so that remaining in the periphery is not profitable. Now, if international transaction costs are low, competition in the periphery comes from foreign producers and similarly makes this location less attractive than if this local market was sheltered from international competition. As already mentioned, this result contrasts with Krugman and Elizondo w9x who emphasize a prodispersion effect of openness to trade. This is because they assume the centrifugal forces to be based on rent and commuting costs. International integration therefore guarantees a larger access to export markets so that the localization on a large domestic market becomes less important. As we showed here, the existence of centrifugal forces based on an immobile local consumers fringe is likely to balance or even reverse this type of conclusion concerning the effect of international trade on the internal agglomeration process. 5. CONCLUSIONS In this paper, we analyzed the process of geographical concentration of economic activities in a framework incorporating both interregional and international trade. The distinction between regions and nations was formalized in a two country᎐four region model in which we allowed for workers’ mobility between the regions of a particular country but not across countries. This setting seems indeed relevant for a number of cases for which international migration barriers remain unimportant. At the same time, integration and liberalization of trade between nations is a striking feature of the world economic scene of the post-war period. The question we therefore attempted to address was the following: What is the effect of international markets integration on the characteristics of the countries’ regional equilibria?
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MONFORT AND NICOLINI
To this end, we developed an extension of Krugman w6x in which trade between both regions and countries entails transaction costs that are differentiated according to the interregional or international nature of the flows. The results obtained suggest that both types of transaction costs affect the incentive for firms to concentrate geographically. In particular, a decrease in interregional andror international transaction costs was shown to reinforce the tendency for some national economic activities to cluster on a particular location of the country. In this perspective, movements toward integration and international trade liberalization could be considered as factors possibly favoring the emergence of regional economic agglomeration inside countries. This finding is coherent with the stylized facts established for Europe: although the convergence process that one observes between European countries since the 1960s is still at work today, it has been stopped and even reversed for several years at the level of European regions ŽEC w3x; Puga w12x.. One possible interpretation of such evolution is that it is inside countries that regional disparities are growing. According to the results obtained in the present paper, this evolution could partly be explained by the integration and trade liberalization processes which have been taking place under the construction of the European common market. APPENDIX A. Simulations This appendix reports the results of the simulations mentioned in Sect. 3. As an illustration of the impact of interregional and international transaction costs on the nature of spatial equilibrium, Table 1 displays a typology in which the different types of configuration are mapped in the i y e space. Configurations allowing for agglomeration as a stable equilibrium Žtype I, III, IV. dominate for low transaction costs in general Žhigh ’s. while, for high levels of transaction costs Žlow ’s., the homogeneous distribution of activities is stable Žtype II, III, IV.. B. Symmetry Breaking Critical Value Ž i s . This appendix derives the critical value Ž i s . at which a global symmetric equilibrium Ž z s zU s 1. breaks and becomes unstable. As explained in Sect. 3, i s is the solution of the following equation:
⭸ Ž 1, 1 . ⭸z
s
⭸¨ ⭸z
q
d Ž P1rP2 . dz
s 0.
Ž 15 .
REGIONAL CONVERGENCE AND INTERNATIONAL INTEGRATION
305
TABLE 1 Configuration Types
s 4, s .3
e
.99 .89 .79 .69 .59 .49 .39 .29 .19 .09
I I I I II II II II II II .09
I I I I II II II II II II .19
I I I I IV II II II II II .29
I I I I IV II II II II II .39
I I I I I IV II II II II .49
I I I I I I I I III III .59
I I I I I I I I I I .69
I I I I I I I I I I .79
I I I I I I I I I I .89
I I I I I I I I I I .99
i
A change in z implies a change in ¨ and ¨ . We therefore use the first two equations of the system Ž7᎐10.: E1 ' a1 q t i a2 q t e a3 q t e a4 y 1 s 0, E2 ' t i a1 q a2 q t e a3 q t e a y ¨ s 0. Applying the implicit function theorem, we have:
⭸ E1 ⭸z ⭸ E2 ⭸z
dz q dz q
⭸ E1 ⭸¨ ⭸ E2 ⭸¨
d¨ q d¨ q
⭸ E1 ⭸¨ ⭸ E2 ⭸¨
d ¨ s 0, d ¨ s 0.
This system is solved for d¨ rdz and d ¨ rdz. In particular, for z s zU s ¨ s ¨ U s ¨ s 1, we have: d¨ dz
s
Ž 1 y t i .Ž y 1 q 2 t e q t i q t i .
Ž 1 y y 2 t e q 4 t e q 4 t e2 y 2 t i q 4 t i q 2 t e t i q 4 t e t i q t i2 q mti2 . Ž 16 .
d¨ dz
s
Ž 1 y t i .Ž y 1 q 2 t e q t i q t i . 2 Ž 1 y y 2 t e q 4 t e q 4 t e2 y 2 t i q 4 t i q 2 t e t i q 4 t e t i q t i2 q m2i .
.
Ž 17 .
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MONFORT AND NICOLINI
As far as the ratio P1rP2 is concerned, we have: d Ž P1rP2 . dz
s
⭸ Ž P1rP2 . ⭸z
q
⭸ Ž P1rP2 . ⭸ ⭸¨
⭸z
q
⭸ Ž P1rP2 . ⭸ ¨ ⭸¨
⭸z
.
For z s zU s ¨ s ¨ U s ¨ s 1 and using Ž16. and Ž17., we obtain: d Ž P1rP2 . dz
s
ti y 1
Ž y 1. Ž 1 q 2 t e q ti .
.
Ž 18 .
Plugging Eqs. Ž18. and Ž16. into Eq. Ž15., we have the following condition:
⭸ Ž 1, 1 . ⭸z
s 0 m y1 q q y 2 q 2 q 2 t e y 4 t e qt i q t i y t i y 2 t i y 2 t i s 0,
with t i s Ž i . y1 and t e s Že . y1. This equation admits a unique root in i which is: y1 Ž 1 y . Ž 1 y . y 1 y 2 Ž 2 y 1 . Ž e . is s Ž1 q . Ž1 q . y 1
½
1r Ž y1 .
5
.
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