Regional integration, international liberalisation and the dynamics of industrial agglomeration

Journal of Economic Dynamics & Control 48 (2014) 265–287

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Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Regional integration, international liberalisation and the dynamics of industrial agglomeration Pasquale Commendatore a, Ingrid Kubin b, Carmelo Petraglia c,n, Iryna Sushko d,e a

University of Naples Federico II, Italy Vienna University of Economics and Business Administration, Austria University of Basilicata, Italy d Institute of Mathematics, National Academy of Sciences of Ukraine, Ukraine e Kyiv School of Economics, Ukraine b c

a r t i c l e i n f o

abstract

Article history: Received 21 November 2013 Received in revised form 5 May 2014 Accepted 17 July 2014 Available online 30 July 2014

This paper presents a 3-region footloose-entrepreneur new economic geography model. Two symmetric regions are part of an economically integrated area (the Union), while the third region represents an outside trade partner. We explore how the spatial allocation of industrial production and employment within the Union is affected by changes in two aspects of trade liberalisation, regional integration and globalisation, conditional to the skill endowment and the market size of the outside region. Our main contribution pertains to the analysis of the local and global dynamics of the specified factor mobility process. We show that significant parameter ranges exist for which an asymmetric distribution of economic activities is one of the possible long-run outcomes which may allow a smooth transition to agglomeration (in contrast to the NEG typical catastrophic scenario). In addition, we show that multistability is pervasive and that some attractors are Milnor attractors. Both results reinforce the NEG narrative on the importance of initial conditions for the long-run location of industrial activity. & 2014 Elsevier B.V. All rights reserved.

JEL classification: C62 F12 F2 R12 Keywords: Industrial agglomeration New economic geography Footloose entrepreneurs Local and global dynamics Bifurcation scenarios

1. Introduction New Economic Geography (NEG) models do not typically account for the presence of regions other than the ones involved in the economic integration process. Nevertheless, a vast body of empirical evidence reveals the ongoing long-term parallel trends of increasing regional integration and globalisation. The EU is a part of this phenomenon: on the one hand, within-EU integration has become more important over the last decades and, on the other hand, the EU as a whole has gained greater exposure to the world economy (Foster et al., 2013). The analytic structure of NEG models is intrinsically complex, therefore many NEG models are actually confined to the analysis of two regions, aiming to predict the impact of stronger integration on industrial agglomeration in a given economically integrated area (e.g., EU regions). However, the understanding of agglomeration and dispersion forces

n Corresponding author: Department of Mathematics, Computer Science and Economics, University of Basilicata, Viale dell'Ateneo Lucano 10, 85100, Potenza, Italy. Tel.: +39 971 206119; fax: +39 971 205416. E-mail address: [email protected] (C. Petraglia).

http://dx.doi.org/10.1016/j.jedc.2014.07.011 0165-1889/& 2014 Elsevier B.V. All rights reserved.

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stemming from stronger exposure of the integrated area to the rest of the world (e.g., EU integration into the world economy) requires a more general set up including (at least) an “outside” region. Scholars dealing with 3-region NEG models – see, among others, Paluzie (2001), Krugman and Livas Elizondo (1996), and Brülhart et al. (2004) – typically explore how the spatial distribution of economic activities in a given home country is affected by international trade liberalisation. On the other hand, as pointed out by Behrens (2011), a large part of this literature underplays the role of regional integration. That is, one relevant aspect of economic integration – globalisation – is studied, while the second one – regional integration – is left out of the picture. Commendatore and Kubin (2013) consider a footloose entrepreneur (FE) model (Forslid and Ottaviano, 2003) with three identical regions separated by symmetric trade barriers. The mobility hypothesis for the skilled workers or “entrepreneurs” is such that their decision to move to one of the three regions depends on a comparison between the real income gained in a region and the weighted average of the incomes in all regions. They analyse the impact of regional integration on industrial agglomeration by studying the local stability properties of the long-term stationary equilibria, while Commendatore et al. (2014) look at the global stability properties of the same model. Their results show significant differences with respect to the symmetric 2-region case. In particular, their results reveal the occurrence of stable asymmetric equilibria which do not exist in the 2-region counterpart. Moreover, they detect complex/strange two-dimensional attractors that cannot emerge in a 2-region NEG model, which is typically one-dimensional. Their analysis, however, still considers a unique economic area exploring the effects of further regional integration on the distribution of industrial activities within that same integrated area. In this contribution, instead, inspired by the case of the EU, we aim to explore how the spatial allocation of industrial production and employment within an economically integrated area (the Union) is affected by changes in both aspects of trade liberalisation: regional integration and globalisation. Where the latter aspect can be explored only by admitting the existence of an “outside” region that can be, alternatively, interpreted as “the rest of the World”, which is typically different from and comparatively less integrated than the regions belonging to the Union. Our main objective is to study the effects of higher integration within the Union (reduced internal transport costs), and those due to higher economic integration of the Union as a whole with the rest of the World (reduced external transport costs). Furthermore, motivated by the changing picture of the main trade partners of the EU, we study the impact of both aspects of integration under alternative assumptions on the industrialisation level of the Union's trade partners (outside countries, trade areas or, without any specification, “regions”). In particular, we will show that integration with less industrialised regions will make agglomeration of industrial activity within the Union less likely. In addition, we also analyse the effects of international integration under alternative assumptions about the size of the outside region. For many parameter values, trade liberalisation ultimately leads to agglomeration of economic activity within the Union. However, the pattern of the transition to agglomeration depends upon the size of the outside region. When integrating with a small outside region, catastrophic agglomeration will be observed; instead when integrating with a large outside region, the transition path to full agglomeration will be smooth. We depart from the existing multi-region NEG models in three other relevant respects. In contrast with most previous contributions, we assume that unskilled workers are immobile both domestically and internationally. This assumption makes our model closer to the reality of the EU where labour mobility plays a relatively unimportant role as compared to other economically integrated areas such as the US (Gáková and Dijkstra, 2008). On the other hand, we will maintain that the interregional mobile factor is human capital embodied in skilled workers (Forslid and Ottaviano, 2003). A second important departure is the specification of our model in discrete time. This represents an easy way to account for delays in the dynamic process (that are obviously involved in firm relocations). Finally, we try to fill a relevant gap in the NEG literature: the lack of explicit dynamic analysis. This is a particularly relevant issue as many core results of the NEG depend on the properties of dynamic processes, such as multiple equilibria, change in stability properties, the nature of the basins of attraction. We carefully analyse the emerging bifurcation scenarios – detecting a typical sequence – and show that coexistence of equilibria is much more pervasive than in standard NEG models. We show that in some cases – due to the complex structure of the basins of attraction – it is even impossible to predict the long-run spatial distribution of economic activity. The remainder of the paper is structured as follows. Section 2 provides some stylised facts on the case of EU which inspired our work and reviews the main findings of the literature most related to our contribution. Section 3 presents the general framework of the model including the definition and properties of short-run and long-run equilibria. Section 4 presents results on local and global dynamics of the model. Section 5 concludes.

2. Stylised facts and related literature The starting point for our analysis are three stylised facts:


Trade barriers among European regions have been lowered by the long-term process of EU integration.
Globalisation has produced greater exposure of the EU to the world economy, leading to higher dependency of the Union (and of each Member State) on final demand outside the EU.


The deeper integration into the world economy of the EU is currently characterised by an increasing weight of big, less industrialised trade partners.

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Fig. 1. The index of EU Economic Integration (1957–2001); Notes: The index is defined for the EU-6 founding members; highest score possible for regional integration: 100; 1957¼0. Source: Dorrucci et al. (2002).

The strengthening of the EU internal integration is a well-documented fact. Fig. 1 outlines the evolution of economic integration within the EU from 1957 to 2001 based on the composite index developed by Dorrucci et al. (2002). This is a numerical composite index based on scores attributed to each single event of European integration grouped according to Balassa (1961) five main stages of regional integration: (a) Free Trade Area (FTA) where internal tariffs and quotas are abolished for imports from area members; (b) Customs Union (CU): a FTA setting up tariffs and quotas for trade with nonmembers; (c) Common Market (CM): a CU where restrictions on factor movements as well as non-tariff barriers to trade are abolished; (d) Economic Union (EUN): a CM with a significant degree of coordination of national policies and harmonisation of relevant domestic laws; and (e) Total Economic Integration (TEI), an EUN with all relevant economic policies conducted at a supra-national level. Looking at Fig. 1 one can identify three sub-periods. The first period goes from March 1957 (Treaty of Rome) to July 1968 (completion of the CU) and is characterised by faster integration as, by the end of this period, more than half of the overall institutional integration process had been already completed. In the late 1960s, the EU was indeed much more than a CU, having already some genuine characteristics of subsequent Balassa stages. The second period (between the early 1970s and the mid-1980s) is characterised by sluggish integration, with the noteworthy exception of the European Monetary System start in March 1979. In the third period, the creation of the CM and the Monetary Union has led to considerable acceleration in regional integration. As a result, the EU/euro area in early 2000s could already be classified somewhere between an EUN and a TEI. The 2004 and 2007 EU enlargements to Eastern countries have then pushed this process forward. Turning to the second stylised fact, note that the EU launched in 2006 its new “Global Europe” strategy aiming at further integrating the EU into the world economy (see for a discussion of the institutional progress, Kleimann, 2013). One visible consequence of increased globalisation is that production taking place in an integration area relies more heavily on foreign final demand. Such a process has been at the core of European countries' recent economic performances since the exponential expansion of emerging economies such as China, India and Brazil has provided an increasing demand for others' countries products. In a detailed study, Foster et al. (2013) calculate the EU value added due to foreign final demand in % of GDP for the 1995–2011 period1 and point out that since the mid-1990s the dependency of the EU economy on foreign demand has significantly increased: in 1995 9.9% of GDP of the EU-27 was produced to satisfy – directly and indirectly – foreign demand abroad, while this share has increased to almost 15% in 2011.2 Interestingly, such an increasing trend has continued during the years of the great recession. 1 The value added created in an economy due to demand for final products in other economies – the so-called “value added exports (VAX)” is described in Johnson and Noguera (2012). Foster et al. (2013) calculate the VAX for the set of countries included in the WIOD database. 2 The same holds for employment. EU employment due to foreign demand in % of total employment has risen from 9.3% in 1995 to 11.6% in 2011.

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Fig. 2. EU-27 value added due to foreign demand by partner (% GDP). Source: Foster et al. (2013) elaboration on WIOD database.

The trend of increasing dependency of EU income level upon foreign demand is far from being at rest. According to EC (2013), “over the next two years, 90% of world demand will be generated outside the EU. That is why it is a key priority for the EU to open up more market opportunities for European business by negotiating new Free Trade Agreements with key countries. If we were to complete all our current free trade talks tomorrow, we would add 2.2% to the EU's GDP or €275 billion. This is equivalent of adding a country as big as Austria or Denmark to the EU economy. In terms of employment, these agreements could generate 2.2 million new jobs or additional 1% of the EU total workforce.”3 Turning to our third stylised fact, note that the figures for the value added due to foreign demand disaggregated by trading partners reveal striking regional disparities (see Fig. 2): China's share increased from 3.3% in 1995 to more 11.1% in 2011 at the expense of Japan (8.1% in 1995 and 3.4% in 2011) and the US (24.5% in 1995 compared to 18.4% in 2011). A recent OECD study (Woo, 2012) shows that these changes in the trading partners involve also a change in the technology level. Based on growth accounting, Woo (2012, p. 15) uncovers a considerable technology gap between China and the US and Japan. Labour productivity in China as measured by output per worker is 16% of that of US workers in 2007 and the total factor productivity level as a measure of technology (or overall efficiency) in China are 25% of the US counterpart. Japan's labour productivity (total factor productivity) is 69% (54%) of the respective US value. These facts imply relevant research questions to be answered in multi-region NEG models. That is, how are both agglomeration and dispersion forces that drive agglomeration of economic activities modified once deeper integration within the Union and within the world economy is considered? And how relevant are the size and the industrialisation level of the trade partners? So far a small strand of literature has developed 3-region models within the NEG literature. Inspired by the debate on the role of protectionist policies in the development of striking regional inequalities during the Spanish industrialisation process, Paluzie (2001) proposes a standard Core-Periphery model accommodating for the presence of a third region. She considers a world economy consisting of two domestic regions and one external economy, with labour being mobile only

3

For an overview of the most important forthcoming and on-going free trade negotiations see EC (2013).

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domestically. In line with Krugman (1991), the centripetal forces that produce agglomerations are represented by the interaction of economies of scales, market size and transport costs, while centrifugal forces that tend to weaken agglomerations are the pull of a dispersed rural market. The main result in the model is that a reduction in the external trade cost strengthens the agglomerative forces in the home country with two regions. In other words, external trade liberalisation is expected to increase regional inequalities in the country that opens up to trade. Similar results are put forward by Alonso-Villar (2001) and Monfort and Nicolini (2000). Krugman and Livas Elizondo (1996) obtain the opposite result, in the same context of a model with three regions, two domestic and one external, where the domestic dispersion force is due to land rent and commuting costs and it is thus exogenous and independent of trade costs. The authors study the impact of trade liberalisation on the distribution of economic activities within the home country and conclude that opening up external trade favours dispersion of economic activity between the two internal regions. It is claimed in the paper that such a result explains the rise of large metropolis in developing countries (Mexico City is the case discussed by the authors) and its progressive loss of importance after the implementation of trade liberalisation policies. Brülhart et al. (2004) and Crozet and Koenig Soubeyran (2004) introduce more geographical structure into the analysis, as they assume that one of the home regions is a border region, i.e. that it has lower transport cost with respect to the outside region than the other home region. Also in these frameworks, a reduction of the international transport cost favours agglomeration in the 2-region home country. Brülhart et al. (2004) present a FE model – a 3-region version of Pflüger (2004) – where two of the three regions are relatively more integrated between them. The aim of the authors is to track how the economies of these two regions are affected by an opening towards the third region. The real world case that motivate this work is the 2004 EU enlargement, which integrated ten Central and Eastern European countries (the third region) fully into the EU's internal market. The research question is then linked to the implications for the distribution of economic activities in the incumbent EU countries of the improved access to and from the third region. In the model, the production of the manufactured good requires one unit of human capital and a variable amount of labour. Human capital is mobile between the two regions in one country but immobile with respect to the third region. The sectorial location is determined endogenously through the interplay of agglomeration and dispersion forces. External market opening has a bearing on several spatial forces. Forces related to better access to foreign export markets and cheaper imports enhance the locational attraction of the border region. Conversely, forces related to import competition from foreign firms enhance the locational attraction of the interior region. The interplay of these forces in the non-linear setup of the model can lead to a variety of equilibria. The main results are such that the range of parameter values, for which domestic manufacturing agglomerates in only one region, increases as external trade costs fall. The same result obtains if – given constant external trade costs – the foreign country gets bigger, i.e., the larger the outside economy, ceteris paribus, the greater the probability that domestic manufacturing agglomerates in one region. Hence, the size of the third region matters for the results. With the exception of Brülhart et al. (2004) and Crozet and Koenig Soubeyran (2004), in the above-mentioned contributions, labour mobility is the dynamic process bringing agglomeration about. This is the case of Krugman and Livas Elizondo (1996) for Mexico and Paluzie (2001) for Spain. In all these cases labour mobility is plausible whereas this is not the case for the EU. As shown by Gáková and Dijkstra (2008), labour mobility between the regions of the EU at NUTS 2 level is relatively low. This seems to be a common feature in the EU as it applies to both the old and the new member states, irrespective of their economic development or the openness of their labour market. In particular, the analysis provided by Gáková and Dijkstra (2008) shows that the share of working age residents moving in another EU region represents, on average, less than 1% of the EU's working age population (vs. 2% in the US).4 As migration in Europe is rather weak, as far as the EU is concerned, the mobility of unskilled workers does not really appear to play the role of an adjustment process to wage differential among countries (Siebert, 1997; Obstfeld and Peri, 1998; Puga, 2002). On the other hand, firm mobility has been achieving an increasing role since the EU enlargement to the Eastern European countries from the mid-1990s. The contributions whose main results have been summarised earlier in this section share the common feature of addressing only one part of the issues at hand as they only analyse the effects of a closer integration of the home economy with the rest of the world, independently of the transportation costs within the home economy itself. Nevertheless, confining our attention to the EU, two parallel trends have been taking place in the last decades – gaining momentum with the EU enlargement to the Eastern European countries – and provide the first two stylised facts for our theoretical framework. Moreover, the higher importance gained by China at the expenses of the US provides a motivation for studying the effects of globalisation conditional upon the size and the output composition of the external commercial partners. 3. General framework The theoretical framework used in this paper reformulates the discrete time 3-region FE model studied in Commendatore and Kubin (2013) and Commendatore et al. (2014). In this framework, we introduce specific asymmetries 4 The analysis presented in this paper is based on the average share of the working age residents in 2005–2006 who had changed their region of residence during the previous year.

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and assume that two regions form a Union and that the third region represents the rest of the World. In particular, we modify three assumptions: first, the mobile factor – that we assume to be human capital, embodied in skilled workers or, equivalently, entrepreneurs (Forslid and Ottaviano, 2003) – can only migrate within the Union, composed of two regions, while there is no factor mobility between the Union and the third region; second, we assume asymmetric trade barriers across regions: the distance between the two regions forming the Union is short, while the distance between both regions of the Union and the third one is long; third, the three regions are not anymore identical: we are now exploring the impact of a change in the relative size of the third region with respect to that of the Union. The resulting new framework allows us to study the impact of both regional integration (within the Union) and international liberalisation (between the Union and the third region) on the spatial distribution of industrial activity. 3.1. Basic assumptions A 3-region economy (r ¼ 1, 2, 3) is characterised as follows: regions 1 and 2 form a Union; region 3 instead represents an outside trade partner. There are two sectors, agriculture (A) and manufacturing (M), and two types of workers. Unskilled workers are immobile (but can be reallocated across sectors), whereas skilled workers or entrepreneurs (E) can migrate but only between regions 1 and 2, i.e. within the Union. We assume that there is no factor mobility between the Union and region 3. L is the amount of unskilled labour in the overall economy and θr is the corresponding share located in region r; it follows that θr L is the endowment of unskilled workers of region r. With immobile unskilled workers and a constant and equal to one wage rate (see below), θr L can also be interpreted as the component of local demand (θr representing its share) which is not affected by skilled labour migration. When regions 1 and 2 are symmetric, we have that θ1 ¼ θ2 ¼ θ and θ3 ¼ 1  2θ. Moreover, E represents the overall number of skilled workers in the economy and E~ is the skill endowment of the Union, ~ the corresponding share. Consequently, E  E~ represents the which is mobile between regions 1 and 2. We denote by n~ ¼ E=E ~ number of immobile skilled workers located in region 3, i.e. the skill endowment of that region, and 1  n~ ¼ 1  ðE=EÞ is the ~ in relative terms) can be interpreted as the level of industrialisation of the Union corresponding share. Notice that E~ (or n, ~ in relative terms) the level of industrialisation of the outside trade partner. and, analogously, E  E~ (or 1  n, 3.2. Consumers' preferences The three regions are homogeneous in terms of tastes. Individual's (skilled or unskilled worker's) preferences are expressed by a two-tier utility function: μ

1μ

U ¼ CM CA

;

where CA is the consumption of the agricultural good and CM denotes the consumption of a composite of manufactured varieties, given by the following CES function: !σ =ðσ  1Þ CM ¼

n

ðσ  1Þ=σ

∑ ci

i¼1

;

where ci represents the quantity consumed of the variety i, with i ¼ 1; …; n; σ is the constant elasticity of substitution/taste for variety: the closer σ to 1, the greater is consumer's taste for variety, with σ 4 1; and μ and 1  μ represent the income shares devoted to the manufactured varieties and to the homogeneous agricultural good, respectively, with 0 o μ o 1. The budget constraint of an individual resident in region r is N

∑ p~ i ci þpA C A ¼ y;

ð1Þ

i¼1

where pA is the price of the homogeneous agricultural good; p~ i is the price of variety i inclusive of trade costs and y is the income of the individual agent. 3.3. Production The A sector is characterised by perfect competition and constant returns to scale. Production of 1 unit of output requires

α units of L; without loss of generality, we set α ¼1. Moreover, we assume that none of the regions has enough labour to

engage exclusively in the production of the agricultural good, that is, the so-called “non-full-specialisation condition” holds.5 The M sector is (Dixit–Stiglitz) monopolistically competitive: identical firms produce differentiated varieties with the same production technology involving a fixed component (one skilled worker), and a variable component (unskilled workers), with β units of L required for each unit of the differentiated good. Thus, compared with agriculture, manufacturing is the skill-intensive sector. 5

Further details on this condition are provided later on in footnotes 6 and 8.

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The total cost of producing the quantity qi of a variety i corresponds to CTðqi Þ ¼ π i þ wβqi ; where πi represents the fixed cost component and the remuneration of the skilled worker, with i ¼ 1; …; n. Given the standard assumptions of the FE model, the total number of firms and varieties, n, always equates the total number of skilled workers, E¼ n. Denoting by xt the share of skilled workers in the Union that are located in region 1 during the time unit t, where 0 r xt r1, the number of regional varieties produced in region r during that period can be expressed as ~ n1;t ¼ xt E~ ¼ xt nE ~ n2;t ¼ ð1  xt ÞE~ ¼ ð1  xt ÞnE ~ n3;t ¼ ð1  nÞE

3.3.1. Trade costs Typically, NEG models assume that the agricultural good is traded without costs. On the other hand, trade costs for manufacturers take an iceberg form: if one unit is shipped from region s to region r, only 1=T rs arrives at destination, where T rs Z 1 and r; s ¼ 1; 2; 3. Regions 1 and 2 (the Union) are involved in a trade agreement whereas the economic integration with region 3 (the outside region) is less deep. Therefore, the “internal distance” (trade barriers) between regions 1 and 2 is S (short); the “external distance” between 1 and 3 and 2 and 3 is the same and it is equal to L (long). Moreover, trade costs do not depend on the direction of the trade flow ðT rs ¼ T sr Þ. Trade costs between regions 1 and 2 are T 12 ¼ T S and between regions 1 and 3 and regions 2 and 3 are T 13 ¼ T 23 ¼ T L ; where T L 4T S Z 1. Finally, in order to simplify the notation, we introduce the standard transformation of trade costs into the following “trade freeness” parameters: ϕS  T 1S  σ and ϕL  T 1L  σ , where ϕL o ϕS r1. 3.4. Short-run general equilibrium The short-run general equilibrium (SRGE) in period t is defined by a given spatial allocation of skilled workers across regions, xt n~ and ð1  xt Þn~ in regions 1 and 2 and the invariant share 1  n~ in region 3. In a SRGE, which is established instantaneously in each period, supply equals demand for the agricultural commodity and each manufacturer meets the demand for its variety. With zero transport costs, the agricultural price pA is the same across regions. Since competition results in zero agricultural profits, the short-run equilibrium nominal wage w is equal to the agricultural product price and it is also equalised across regions. We use the agricultural price as numeraire, thus: w ¼ pA ¼ 1.6 Facing a wage of 1, each manufacturer has a marginal cost of β. Each maximises profit on the basis of a perceived price elasticity of  σ and sets a local (mill) price p for its variety, given by p¼

σ β: σ 1

ð2Þ

The demand facing a producer located in region r (where trade costs are also taken into account) corresponds to:7  3   3  ss;t μY σ 1 1σ ð3Þ dr;t ¼ ∑ μY s;t P s;t T rs ϕrs p  1 pσ ¼ ∑ E s¼1 s ¼ 1 Δs;t where  P r;t ¼

R

∑ n1s;t σ p1  σ T 1rs σ

s¼1

1=ð1  σ Þ

1=ð1  σ Þ 1=ð1  σ Þ

¼ Δr;t

E

p

ð4Þ

is the price index facing consumers in region r; Y s;t represents income and expenditure in region s; ss;t ¼ Y s;t =Y denotes region s's share in expenditure and s ¼ 1, 2, 3. Moreover, we have defined ~ ϕr3 : Δr;t ¼ xt n~ ϕr1 þð1  xt Þn~ ϕr2 þð1  nÞ

6 Denoting by Y the income of the overall economy, that (as confirmed below) is invariant over time, total expenditure on the agricultural product is ð1  μÞY. Assuming ð1  μÞY 4 maxð2θL; ð1 θÞLÞ all regions produce the agricultural commodity, whereas ð1 μÞY 4 maxðθL; ð1  2θÞLÞ implies that no single region is able to satisfy all the demand for the agricultural good. 7 The mathematical derivations for the equations in this section follows a standard procedure that can be found in detail in Brakman et al. (2009).

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SRGE in region r requires that each firm meets the demand for its variety. Therefore, qr;t ¼ dr;t ;

ð5Þ

where qr;t is the output of each firm located in region r. From Eq. (2), the operating profit (skilled worker remuneration) per variety is

π r;t ¼ pqr;t  βqr;t ¼

pqr;t

σ

:

ð6Þ

Since profit equals the value of sales time 1=σ and since total expenditure on manufacturers is μY, the total profit is μY=σ . Total income is Y ¼ L þ μY=σ , so that Y¼

σL : σ μ

ð7Þ

Total profit is therefore μL=ðσ  μÞ.8 Using (2) to (7), the short-run equilibrium profit in region r is determined by the spatial distribution and by the regional expenditure shares  3  1σ  R  ss;t μY σ 1 1σ p : ð8Þ π r;t ¼ ∑ μY s;t P s;t T rs ¼ ∑ ϕrs σ Δ σE s;t s¼1 s¼1 Regional incomes/expenditures are Y r;t ¼ Lr;t þnr;t π r;t : As shown in Appendix A, regional incomes ðY r;t Þ, expenditure shares ðsr;t Þ and profits ðπ r;t Þ can be expressed in terms of xt. Given that the agricultural price is 1, the indirect utility of a skilled worker in region r is

π r;t

V r;t ¼ μ ; P r;t which also depends only on xt. Notice that the real income of a skilled worker located in region 3 is affected by the distribution of the manufacturing activity between regions 1 and 2, even though no migration takes place from that region towards the other two. Finally, letting n~ ¼ 1 it is straightforward to verify (see Appendix A) that real incomes in regions 1 and 2 are not affected by the distance from region 3. Thus, interestingly enough, external trade liberalisation has no impact on the locational choices of skilled workers within the Union in the absence of a manufacturing sector in the outside region. This result follows from three features of the model set up: (i) the demand for the manufactured goods is unitary elastic: the change in trade costs, via ϕL , determines a proportional change in the price index in region 3 and a similar but inversely proportional change in the quantity demanded, so the overall change of expenditures on manufacturing in this region is zero; (ii) since region 3 does not produce manufactured varieties, a change in ϕL does not impact on price indices in regions 1 and 2; (iii) the distance between regions 1 and 3 and that between 2 and 3 are the same, so that prices of imported manufactured goods in region 3 do not depend on entrepreneurs' locational choice between regions 1 and 2. 3.5. Skilled workers migration hypothesis, full dynamical model and long-run general equilibrium Migration only involves regions 1 and 2 and it is represented by the evolution through time of the state variable xt. Provided that the share xt does not hit the obvious constraints 0 and 1, the central dynamic equation is analogous to the replicator dynamics, widely used in evolutionary game theory:     V 1;t  xt V 1;t þ ð1  xt ÞV 2;t xutþ 1 ¼ xt 1 þ γ ; ð9Þ xt V 1;t þð1  xt ÞV 2;t where γ represents the migration speed. According to (9), the unconstrained share of entrepreneurs in region 1, xutþ 1 , depends on a comparison between the indirect utility gained in that region and the weighted average of the indirect utilities in regions 1 and 2. Expression (9) can be reformulated in a more compact form as  xutþ 1 ¼ Z ðxt Þ ¼ γ 1 þ ð1 xt Þ

 Tðxt Þ ; 1 þ xt Tðxt Þ

8 Eq. (7) confirms that total income is invariant over time. From (7), ð1 μÞY 4 maxð2θL; ð1 θÞLÞ is equivalent to minð2θμ þ ð1  2θÞσ  μσ ; 2½ð1  θÞμ þ θσ  μσ Þ 4 0; and ð1  μÞY 4 maxðθL; ð1  2θÞLÞ is equivalent to minð2½θμ þ ð1  θÞσ  μσ ; ð1 2θÞμ þ 2θσ  μσ Þ 4 0. The former is a sufficient non-full-specialisation condition and the latter is a necessary one, where both are expressed in terms of the utility parameters.

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where Tðxt Þ ¼ ðV 1;t =V 2;t Þ  1. map: 8 > < 0 xt þ 1 ¼ Λðxt Þ ¼ Zðxt Þ > : 1

273

Taking into account the constraints, 0 r xt r1, the full dynamical model corresponds to the if

Zðxt Þ o 0

if

0 r Zðxt Þ r 1 :

if

Zðxt Þ 41

ð10Þ

In what follows, for convenience, we drop the subscript t. A long-run stationary equilibrium is a fixed point xn of the map (10); it is found by setting Λðxn Þ ¼ xn . Three types of fixed points can be differentiated according to the spatial distribution of manufacturing within the Union (while the share of skilled workers located in region 3 remains constant by assumption): 1. The Core-Periphery equilibria are characterised by full agglomeration in region 1 or in region 2. These are xCPð0Þ ¼ 0, corresponding to complete agglomeration in region 2; and xCPð1Þ ¼ 1, corresponding to complete agglomeration in region 1. 2. The symmetric equilibrium is characterised by an equal split of the manufacturing sector between regions 1 and 2: xn ¼ 1=2. 3. The asymmetric interior equilibria are characterised by incomplete agglomeration in one of the two regions of the Union, with some industry still present in the other region. Upon parameter configurations: no, two or four such equilibria exist. We now take a closer look at the properties of the fixed point; in particular we analyse how industrial production and employment changes within the Union, if trade is further liberalised (in both directions: regional integration, as measured by ϕS and globalisation, as measured by ϕL ); and we analyse how industrial employment changes, if the outside region is bigger than the Union or if the outside region is less endowed with skilled labour than the Union. The last question has a distinct Heckscher–Ohlin flavour: “will the Union, that is comparatively better endowed with skilled labour, specialise its employment in the manufacturing sector, that uses skilled labour?” However, it also provides useful insights in our NEG model. The sectoral employment structure can be measured by the workers' employment share in the manufacturing sector; we take into account only unskilled workers, since skilled workers are entirely employed in manufacturing, and we denote this employment share by Sr (r ¼ 1, 2, 3). In the symmetric equilibrium, regions 1 and 2 of the Union are identical and only region 1 has to be compared with the outside region 3. Thus, our central variable is the relative sectoral employment ratio defined as Semployment 1 Semployment 3

¼

Share of unskilled employment in manufacturing sector for region 1 : Share of unskilled employment in manufacturing sector for region 3

The endowment structure can be measured by the ratio between skilled and unskilled workers in a region. Again, in the symmetric equilibrium we compare region 1 (that is identical to region 2) with the outside region 3 and the relative endowment ratio is given as Sendowment 1 Sendowment 3

~ 0:5nE θ L : ¼ ~ ð1  nÞE ð1  2θÞL

In Appendix B we compare the relative sectoral employment ratio with the relative endowment ratio and show that the two are equal – and the sectoral employment structure corresponds to the skill endowment – if the following three conditions hold simultaneously: 1. if n~ ¼ 2=3; i.e. if the skilled workers are equally distributed over all three regions; ~ 2. if θ ¼ n=2 i.e. if each firm/skilled worker located in one of the regions finds the same number of unskilled workers residing in this region; taking the number of unskilled workers in each region as a measure for the (immobile) local demand, this condition holds, if no location offers advantages in terms of local market size; 3. if ϕL ¼ ϕS ; i.e. if no location offers an advantage in terms of trade costs. It can be shown that the relative employment ratio decreases with θ; it increases with n~ (note that the last result is in ~ line with a Heckscher–Ohlin intuition) and with ϕS . Concerning ϕL , only local results at θ ¼ n=2 and n~ ¼ 2=3 can be obtained and the effect is non-monotonic: starting with prohibitive costs of trade with the third, outside region, i.e. starting from ϕL ¼ 0, a slight reduction in the trade barriers fosters manufacturing in the Union (that starts specialising in manufacturing); the same specialisation occurs if trade barriers with the third region were increased starting from identical trade costs between all three regions (i.e. from ϕL ¼ ϕS ). Let us return to the stylised facts that were the starting point for our analysis; and let us assume that the EU is in a symmetric equilibrium: the reduction in trade barriers among European regions (as reflected in a reduction of ϕS ) and the deeper integration with big (as reflected in an increase of θ) and with less industrialised economies (as reflected in a

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~ outside the EU are all factors that foster manufacturing within the EU; regarding the effect of globalisation increase of n) (as reflected in a reduction of ϕL ) our results are ambiguous – and de-specialisation in manufacturing is a possible outcome for the EU. Let us now turn to the Core-Periphery equilibrium in which the mobile firms are agglomerated in region 1.9 This case turns out to be analytically more complex and only local results can be obtained. With ϕL ¼ ϕS , θ ¼ 1=3 and n~ ¼ 1=2 regions 1 and 3 are identical; and the relative employment ratio equals the relative endowment ratio. For that point, we have (local) ~ In relation to trade costs, results for the derivatives: the relative employment ratio decreases with θ, and it increases with n.

it can be shown that the relative employment ratio increases with ϕS and decreases with ϕL . The last result is more general, it holds for all 0 r ϕL r ϕS . Let us again return to the stylised facts that were the starting point for our analysis; and let us now assume that the EU is in a CP equilibrium: the local results suggest that a reduction in trade barriers among European regions (as reflected in a reduction of ϕS ), the deeper integration with big (as reflected in an increase of θ) and with less industrialised economies ~ outside the EU are all factors that foster manufacturing within the EU; globalisation (as (as reflected in a increase in n) reflected in a reduction in ϕL ) might work in the opposite direction. This analysis is centered on how the employment structure inside the 2-region Union is affected by changes in trade costs, in the relative size of the local (immobile) demand and in the relative endowment with entrepreneurs or skilled labour, always in comparison to the mirror developments in the outside region. It turned out that the Heckscher–Ohlin intuition carries over to an astonishingly large extent. However, at the heart of a NEG perspective are not so much comparative static properties as dynamic processes: under what conditions the symmetric equilibrium inside the Union is destabilised and a self-reinforcing agglomeration process sets in leading to a Core-Periphery pattern? The next section explicitly addresses the properties of the dynamic process. Note that the dynamic process is based on the mobility of firms; it thus involves only the two regions inside the Union on which the subsequent analysis focuses. 4. Local and global dynamics

This section presents analytical and numerical results related to local and global dynamics of the map Λ defined in (10). In spite of the fact that this map is one-dimensional, its complicated form allows us to obtain only a few analytical results. Therefore, we start with investigating the limiting case n~ ¼ 1 (no manufacturing sector in region 3), for which more analytic results are obtainable. The most important result is that the symmetric fixed point may lose stability not only through a subcritical pitchfork bifurcation, but also through a supercritical one. The former case is the one found in many NEG models: once the symmetric fixed point loses stability, a catastrophic agglomeration process sets in leading to full agglomeration. The supercritical case, however, is a remarkable novel feature: after the symmetric fixed point has lost stability, asymmetric, stable fixed points appear allowing a smooth transition to agglomeration. We are able to prove this result analytically for the limiting case n~ ¼ 1; however, we envisage that, by continuity, a similar result must hold also when we allow for a local manufacturing sector in region 3 – and we confirm this conjecture by numerical investigations. For the general case, we use additional numerical investigations in order to get a fuller picture of the overall bifurcation structure. In particular, we are interested in the effects of varying the size of the local market θ, the relative skill endowment n~ and the internal and external trade costs, ϕS and ϕL respectively. In doing so we show the possibility of coexistence of (cyclical or complex) attractors and we show that CP equilibria can be Milnor attractors, i.e. attractors to which not necessarily all the points from their neighborhood are actually attracted. This adds a new dimension to a central theme in NEG stories, namely that agglomeration depends sensitively upon initial conditions. 4.1. Preliminaries We begin the dynamic analysis by exploring the local stability of the fixed points of the map Λ, in particular of the two Core-Periphery fixed points, xCPð0Þ ¼ 0 and xCPð1Þ ¼ 1, and of the symmetric fixed point xn ¼ 1=2. An important property of the map Λ is related to its symmetry with respect to xn : it can be checked that Λð1 xÞ ¼ 1  ΛðxÞ. Thus, any invariant set A of the map Λ (such as fixed points, cycles, chaotic attractors, and basins of attraction) is either symmetric itself with respect to xn , or there exists one more invariant set A0 which is symmetric to A. In particular, if the map Λ has an asymmetric fixed point x ¼ xa , then the fixed point symmetric to it, x ¼ 1  xa  x0a , also exists. As already mentioned, Λ can have one or two couples of asymmetric fixed points. In Fig. 3 we show examples of the map Λ and its fixed points xCPð0Þ ; xCPð1Þ , and xn for different parameter values. Let us start with the stability properties of the symmetric fixed point xn ¼ 1=2. Its stability condition defined by 0  1 o Λ ð1=2Þ o 1 is satisfied for   γ 1 o1 ð11Þ 1 o1 þ T 0 2 4

9

The case in which the firms are agglomerated in region 2 is symmetric.

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275

Fig. 3. The map Λ for σ ¼6, μ ¼ 0.45, γ ¼ 10, θ¼ 0.25, ϕL ¼ 0:1, n~ ¼ 0:8 and ϕS ¼ 0:11; 0:2; 0:3; 0:4; 0:5 and 0.7 (the direction indicated by an arrow corresponds to increasing ϕS ).

  1 ¼ 0), or (to get this condition we have used the equality T 2   8 1 o 0:  oT0 2 γ

ð12Þ

If T0

  1 8 ¼ 2 γ

ð13Þ

0

then Λ ð1=2Þ ¼  1, so that xn undergoes a flip bifurcation, while the equality   1 ¼0 T0 2

ð14Þ

0

is related to Λ ð1=2Þ ¼ 1, so that xn undergoes a pitchfork bifurcation (due to the symmetry of the map fold or transcritical bifurcations cannot occur), which can be super- or subcritical, as we discuss in the next sections. Let us study now the stability properties of the CP fixed point xCPð1Þ (note that the same conclusions hold for xCPð0Þ due to the symmetry of the map Λ). Recall that xCPð1Þ is a border point at which the map Λ is not differentiable, thus, one can only discuss a one-side stability of xCPð1Þ . In fact, this fixed point is always one-side superstable with the related one-side 0 0 eigenvalue Λ þ ð1Þ ¼ 0. Due to the upper constraint of the map Λ we have obviously Λ  ð1Þ Z 0 (in fact, if Z 0 ð1Þ o 0 then

Λ0 ð1Þ ¼ 0), and if Λ0 ð1Þ o 1 (or 4 1) then xCPð1Þ is one-side attracting (one-side repelling, respectively). To shorten, using the

notion of stability with respect to the CP fixed points from now on we mean one-side stability. Thus, stability loss of xCPð1Þ 0 can occur if its eigenvalue passes through 1, that is, Λ  ð1Þ ¼ 1. Therefore, the stability condition of xCPð1Þ is 0

0 r Λ  ð1Þ ¼ Z 0 ð1Þ ¼ 1  γ

Tð1Þ o 1; 1 þ Tð1Þ

ð15Þ

from where we get 0 o T ð1Þ r

1

γ 1

:

Here the condition T ð1Þ ¼

1

γ 1

ð16Þ

0

corresponds to Λ  ð1Þ ¼ 0, so that xCPð1Þ is both sides superstable, and for T ð1Þ 4

1

γ 1

the flat branch ΛðxÞ ¼ 1 “enters” the interval I ¼ ½0; 1, while the condition Tð1Þ ¼ 0 0  ð1Þ ¼

ð17Þ

is related to Λ 1, that is, to the stability loss of xCPð1Þ . Given that this fixed point always exists, this bifurcation cannot be related to a (one-side) fold bifurcation; we clarify the exact nature of this bifurcation later.

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4.2. The case n~ ¼ 1 In the present subsection we assume n~ ¼ 1 in which case it is possible to get analytical results on local stability of the fixed points (see also Commendatore et al., 2012). We start with the local stability of the symmetric fixed point xn ; it requires that the condition (12) holds. Concerning the inequality on the right hand side of (12) that is related to the pitchfork   1 pf pf ¼ 0 is satisfied. Moreover, ϕS o ϕS implies bifurcation, we denote by ϕS that value of ϕS for which the condition T 0 2   1 pf o0. For 0 o n~ o1, ϕS corresponds to the positive root of a quadratic equation whose expression is quite complicated. T0 2 pf From simulations we find that for some meaningful parameter combinations, but not for all, 0 o ϕS o 1. For n~ ¼ 1, we are able to obtain the following, relatively simple, expression:

ϕpf S ¼

ðσ  μÞ½2θðσ  1Þ  μ o 1: 2θðσ  μÞðσ  1Þ þ μð3σ  2 þ μÞ

(In (27) we give the same condition solved with respect to the parameter θ). One can show: pf 1. If 1 o σ o 1 þ μ=2θ, it follows that ϕpf S o0 . Therefore the inequality ϕS o ϕS is never satisfied. pf 2. If σ 4 1 þ μ=2θ, as ϕS crosses ϕS from left to right, a pitchfork bifurcation occurs associated with two asymmetric fixed points.

Before discussing the properties of this bifurcation in detail, note a first interesting result: with n~ ¼ 1, the local stability of the symmetric fixed point does not depend on the trade distance with respect to the outside region, as measured by the parameter ϕL . This result is a direct consequence of the fact that we discussed above, namely that – in the absence of a manufacturing sector in the outside region – external trade liberalisation has no impact on the locational choices of skilled workers within the Union. The size of the (immobile) local demand as measured by θ, instead, plays a relevant role, affecting positively ϕS : pf

∂ϕS 4μðσ  μÞð2σ  1Þðσ  1Þ ¼ 4 0: ∂θ ½2θðσ  μÞðσ  1Þ þ μð3σ  2 þ μÞ2 pf

Thus, increasing the size of the local (immobile) demand has a stabilising effect on the symmetric fixed point and tends to favour dispersion. pf What happens if ϕS crosses its bifurcation value ϕS , i.e. if trade integration between regions 1 and 2 intensifies? The typical bifurcation scenario of a standard 2-region FE model is catastrophic agglomeration, with an immediate jump to a Core-Periphery equilibrium, corresponding to a subcritical pitchfork bifurcation at which stable symmetric equilibrium becomes unstable merging with two unstable interior asymmetric equilibria. However, in our framework, a smoother agglomeration process can also emerge in correspondence of a supercritical pitchfork bifurcation with the emergence of two stable interior asymmetric equilibria (see Pflüger and Südekum, 2008). In order to study in detail the properties of the pitchfork bifurcation, in the limiting case of n~ ¼ 1, we first redefine our central map to highlight the control parameter we are interested in, the trade freeness parameter ϕS . The redefined map is    TðϕS ; xÞ ð18Þ Z ϕS ; x ¼ γ 1 þ ð1  xÞ 1 þxTðϕS ; xÞ From the theory of dynamical systems (see e.g., Wiggins, 2003), in correspondence of a pitchfork bifurcation, that is, pf when ϕS ¼ ϕS and x ¼ 1=2, the following conditions must hold:     ∂Z 1 ∂T pf 1 ¼ 0 3 ¼ 0; ð19Þ ϕpf ; ϕ ; S S 2 2 ∂ϕS ∂ϕS     ∂2 Z ∂2 T pf 1 pf 1 ¼ 0 3 ¼ 0; ϕ ; ϕ ; S S 2 2 ∂x2 ∂x2

ð20Þ

    ∂2 Z 1 ∂2 T 1 a03 a 0; ϕpf ϕpf S ; S ; 2 2 ∂x∂ϕS ∂x∂ϕS

ð21Þ

    ∂3 Z 1 ∂3 T pf 1 a 0 3 3 ϕS ; a0: ϕpf S ; 3 2 2 ∂x ∂x

ð22Þ

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Fig. 4. 1D bifurcation diagram for σ¼ 2, μ¼0.45, γ ¼20, n~ ¼ 1 and θ¼ 0.25, ϕS A ½0:01; 0:1 in (a), θ¼ 0.4, ϕS A ½0:08; 0:12 in (b), θ¼ 0.32, ϕS A ½0:13; 0:19 in (c). These diagrams are related to parameter paths indicated in Fig. 5 by horizontal lines with arrows. Degenerate pitchfork bifurcation (which cannot occur in the map Λ) is shown schematically in (d). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 5. 2D bifurcation diagram in the ðϕS ; θÞ-parameter plane for σ¼ 2, μ ¼0.45, γ ¼ 20, n~ ¼ 1. 1D bifurcation diagrams related to the horizontal lines marked (a), (b) and (c) are shown in Fig. 4(a), (b) and (c), respectively. Inset presents an enlargement of the indicated window. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

The conditions (19)–(22) are verified in Appendix C. The sign of the following expression can be used to determine on pf which side of ϕS the asymmetric fixed points, at least initially, lie:     ∂3 Z 1 ∂3 T 1 ϕpf ϕpf S ; S ; 3 3 2 2 ∂x ∂x  2   4 ð o Þ0 3  2   4 ð o Þ0: ∂ Z ∂ T pf 1 pf 1 ϕS ; ϕS ; 2 2 ∂x∂ϕS ∂x∂ϕS When this expression is larger (less) than zero, the pitchfork bifurcation is supercritical (subcritical); for the particular

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case of

  ∂3 Z 1 ¼0 ϕpf S ; 3 2 ∂x

ð23Þ

the pitchfork bifurcation is critical (for example, in Fig. 5 the intersection point of the pitchfork and fold bifurcation curves θ ¼ θpf and θ ¼ θf is a codimension-2 bifurcation point at which the pitchfork bifurcation is critical: it occurs simultaneously with two fold bifurcations)10. It can be checked that the sign of (23) coincides with the sign of the following expression: K ¼ 12ðσ 1Þ2 ðσ  μÞθ þ ½2ð2σ  3Þμ2 þ 4ð3μ  σ Þðσ  1Þ2 θ  μðσ 1 þ μÞ½2ðσ  1Þ þ μ: 2

ð24Þ

This allows us to state the following proposition. 1 Proposition 1. Let us assume that σ 4 1 þ μ. Then there exists a value of the parameter θ ¼ θ with 0 o θ o such that K o0 for 2 θ o θ , K 4 0 for θ 4 θ , and K ¼0 for θ ¼ θ , where K is given in (24). For the proof of this proposition see Appendix D. In summary, the size of local (immobile) demand determines the location pattern at the bifurcation point: catastrophic agglomeration when the size of local (immobile) demand is sufficiently large; smooth transition to agglomeration involving asymmetric fixed points when the size of the local (immobile) demand is sufficiently small. Concerning the inequality on the left hand side of (12), it holds for θ o θf l where θf l is given in (26). When this condition does not hold, a flip bifurcation scenario emerges, with the possible occurrence of complex behavior, involving the global properties of dynamics. Let us now study stability properties of the CP fixed points. For n~ ¼ 1 the condition (15) related to the stability of the CP fixed point xCPð1Þ can be written as

γ 1 φoδoφ γ

ð25Þ

where

φ¼

σ ðμ  σ þ 1Þ=ðσ  1Þ and δ ¼ ϕS : 2 ðσ  μÞ½θ þ ϕS ð1  2θÞ þ½μð1  θÞ þ θσ ϕS

For 1 o σ o1 þ μ and 0 r ϕS o1, the right hand side inequality in (25) is always satisfied; for σ 41 þ μ it can be shown that the right hand side inequality in (25) is satisfied for sufficiently high values of ϕS and violated for low values. In fact, we are dealing with two monotonically decreasing functions of ϕS , the first, δðϕS Þ, tends to infinity for ϕS -0 and it is equal to 1 at ϕS ¼ 1; the second, φðϕS Þ, is positive (and larger than 1) but finite at ϕS ¼ 0 and it is equal to 1 at ϕS ¼ 1. Since at ϕS ¼ 1 the first derivative of the first function is smaller in absolute value than the derivative of the second function, the two necessarily cross at some ϕS ¼ ϕS , where 0 o ϕS o 1. It is not possible to specify the corresponding bifurcation value for the tr

tr

(internal) trade freeness parameter ϕS explicitly, however, it is possible to find an explicit expression for θ, see (28).

For σ 4 1 þ μ, as ϕS crosses ϕS from left to right, the map Λ undergoes a “one-side” transcritical bifurcation which we call tr

border-transcritical bifurcation: the CP fixed point xCPð1Þ merges with the asymmetric fixed point and it gains stability. The asymmetric fixed point can be born either due to a fold bifurcation, in which case it appears in a couple with one more asymmetric fixed point, or due to the pitchfork bifurcation of the symmetric fixed point xn that we discussed above. Symmetrically, xCPð0Þ meets the other asymmetric fixed point and it gains stability. From this we infer that the asymmetric fixed points must have always the same local stability properties in the neighborhood of the CP fixed points (as can be seen, for example, in Fig. 4). The left hand side inequality in (25) holds for a sufficiently small value of γ:

γo

1 : 1  δφ  1

When this latter condition does not hold, xCPð1Þ becomes both-side superstable. Fig. 4 displays 1D bifurcation diagrams that illustrate for specific parameter values the possible scenarios; red circles mark pitchfork bifurcations, and green circles indicate border-transcritical bifurcations. Panel (b) represents the subcritical pitchfork bifurcation that leads to catastrophic agglomeration and that is found in many standard NEG models: in that case, the pitchfork bifurcation gives rise to asymmetric fixed points that are unstable. They delimit the basin of attractions for the Core-Periphery fixed points and the symmetric fixed point, all of which are (locally) stable between the break and sustain point values for the trade freeness. Thus, fixed points coexist and the long-run pattern of regional industry location depends on parameters, as well as on the basin of attraction to which an initial condition belongs. Instead, panel (a) depicts a supercritical pitchfork bifurcation that leads to smooth agglomeration: asymmetric stable fixed points are born after the bifurcation – the model, which is entirely based on standard NEG assumptions, is thus able to 10 If the condition (23) holds not only for x ¼ 1=2, but for also in some neighborhood of x ¼ 1=2, this means so-called degenerate pitchfork bifurcation (see Sushko and Gardini, 2010) as sketched, for example, in Fig. 4(d), that cannot occur in the considered map.

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279

generate endogenously interior asymmetric outcomes (in which economic activity is neither symmetrically distributed between the two regions nor fully agglomerated in one of the regions). A last question concerns the transition between the cases depicted in panels (a) and (b): what happens at the moment when the pitchfork bifurcation of xn and the transcritical bifurcation of xCPð0Þ and xCPð1Þ occur simultaneously? One could expect a 1D bifurcation diagram like the one sketched in Fig. 4(d). However, such a diagram is impossible because it would mean that a degenerate pitchfork bifurcation occurs, with ΛðxÞ  x in the complete interval ½0; 1. In fact, the true transition is as shown in Fig. 4(c) where fold bifurcations (marked with black circles) give rise to two stable asymmetric fixed points xa , x0a and two unstable asymmetric fixed points xb , x0b . Panel (c) thus shows that also for asymmetric stable equilibria coexistence with the Core-Periphery equilibria is possible (and the related basins of attraction are delimited by additional asymmetric equilibria that are unstable). Fig. 4 displays a bifurcation pattern typically not found in standard NEG models. We have proven it for the limiting case n~ ¼ 1. However, we envisage that, by continuity, a similar result must hold also when we allow for a local manufacturing sector in region 3 and we did confirm this conjecture in numerical explorations for the general case 0 o n~ o 1. In order to conclude this section, we present in Fig. 5 a 2D bifurcation diagram that summarises the results obtained so far and allows us to gain some additional insights. This figure represents the bifurcation structure of the ðϕS ; θÞ-parameter plane for σ ¼2, μ ¼0.45, γ ¼20, n~ ¼ 1. Here different colors are related to different attracting cycles, namely the red region S to the symmetric fixed point xn ; the pink region AS to coexisting asymmetric fixed points xa and x0a ; the blue region CP to the Core-Periphery fixed points xCPð0Þ and xCPð1Þ ; the region marked as S þCP to coexisting attracting symmetric and CorePeriphery fixed points, xn , xCPð0Þ and xCPð1Þ ; the gray region shown also in an inset (bounded by the fold bifurcation curve θ ¼ θf that can be obtained only numerically) is related to coexisting attracting fixed points xCPð0Þ , xCPð1Þ and xa , x0a ; the blue region CPM to the fixed points xCPð0Þ and xCPð1Þ which are locally repelling but represent attractors in Milnor sense (we give more details about this case later); the green region to 2-cycles; the other colors correspond to cycles of periods k r 30 and white region is related either to higher periodicity or to chaotic attractors. As already mentioned, in the limiting case n~ ¼ 1 the bifurcation curves of the fixed points xn , xCPð0Þ and xCPð1Þ can be obtained analytically. In particular, the flip bifurcation boundary is defined by the condition (13) which can be written as   ðσ ð1 þ ϕS Þ  μð1  ϕS ÞÞ μ 2ð1 þ ϕS Þ μϕS þ ; ð26Þ θ ¼ θf l  þ ðσ  1Þ ð1  ϕS Þγ ðσ  μÞð1  ϕS Þ 2ðσ  μÞð1  ϕS Þ the pitchfork bifurcation boundary is defined by the condition (14) which can be written as   μ σ ð1 þ ϕS Þ  μð1  ϕS Þ þ 2ϕS ; θ ¼ θpf  ðσ 1Þ 2ðσ  μÞð1  ϕS Þ

ð27Þ

and the border-transcritical bifurcation boundary satisfies (17) that can be written as

θ ¼ θtr 

ϕS ðσ ðϕSμ=ð1  σÞ  1Þ þ μð1  ϕS ÞÞ : ðσ  μÞð1  ϕS Þ2

ð28Þ

These curves are shown in Fig. 5. In particular, one can see that the curves θ ¼ θpf and θ ¼ θtr can intersect each other, and if a ðϕS ; θÞ-parameter point moves through the intersection point according to the direction marked (c) one observes the 1D bifurcation diagram shown in Fig. 4(c) (a similar transition can be observed also for any parameter path entering the grey region), while the directions marked (a) and (b) are related to Fig. 4(a) and (b), respectively. 4.3. The case n~ o1 We are now allowing the presence of the manufacturing sector in region 3, that is, we let n~ o1. Fig. 6, left panel, represents a 2D bifurcation diagram in the ðϕS ; θÞ-parameter plane for σ ¼6, μ ¼0.45, γ ¼10, ϕL ¼ 0:1 and n~ ¼ 0:8. As it can be seen, the impact on the long-term behavior of xt – the distribution of industrial activities within the Union – of changes in ϕS and θ is qualitatively similar to the case n~ ¼ 1 (as shown in Fig. 5). The upper boundary of region S is related to the flip bifurcation of xn , while its lower boundary is the pitchfork bifurcation curve. Moreover, 1D bifurcation diagrams shown in Fig. 6, right panel , related to the paths labelled (a), (b) and (c) in Fig. 6, left panel, are quite similar to those shown in Fig. 4 and corroborate our conjecture that our previous results can be extended to the case n~ o 1. At this stage it is useful to summarise the economic interpretations that can be drawn from Fig. 6: for high trade freeness inside the Union (i.e. for a high value of ϕS ), agglomeration of economic activities in one of the two regions of the Union is a likely outcome; lowering this parameter leads to an equal distribution of economic activities (if ϕS enters in the red area S) or to an uneven distribution, but with manufacturing production in both regions (if ϕS enters in the pink area AS); these configurations lose stability for further lower values of internal trade freeness giving rise to cyclical attractors. For very low values of ϕS agglomeration of the industrial activity of the Union might again be the long-run outcome. It is interesting to note that the bifurcation lines are positively sloped, i.e. the bifurcation values for ϕS depend positively upon θ. Lower values of θ shrink the range for which the symmetric equilibrium is stable (i.e. the red area S), shifting it to the left; i.e. reducing trade distance to a country with a big local (immobile) demand (as measured by a reduction in the share of unskilled workers in the Union) favours agglomeration processes and Core-Periphery patterns inside the Union even at lower level of trade integration.

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Fig. 6. Left panel: 2D bifurcation diagram in the ðϕS ; θÞ-parameter plane for ϕL ¼ 0:1, n~ ¼ 0:8, σ ¼6, μ¼ 0.45, γ¼ 10; right panel: 1D bifurcation diagrams related to parameter paths marked (a), (b) and (c), respectively, in the 2D diagram on the left. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Interestingly enough, when considering the external trade freeness parameter ϕL (in numerical simulations not included in the paper) we found similar patterns to that shown in Fig. 6 with respect to the internal trade freeness ϕS . Next, we move our attention towards the impact of the skill endowment of the Union as measured by n~ on the long-term behaviour of industrial production in that economic area (see Fig. 7, where we show the bifurcation structure in the ~ ~ ðϕS ; nÞ-parameter plane in (a), and in the ðϕL ; nÞ-parameter plane in (b)). There exists a range for the trade freeness parameter in which the symmetric equilibrium is the long-run attractor. Outside this range, bifurcation sequences finally lead to a CorePeriphery structure. It is interesting to note that this parameter range depends on n~ in a non-monotonic way: starting from a ~ an increase in n~ shifts the range, in which the symmetric equilibrium is stable, towards higher (lower) low (high) value of n, values of trade freeness. For a highly industrialised Union (n~ close to 1) the bifurcation curves are negatively sloped, implying that the bifurcation values for both dimensions of trade freeness (internal and external) depend negatively upon its industrial share in the overall economy. Returning to the stylised facts, the EU is highly industrialised and well equipped with skilled labour (n~ is close to 1); we observe a process of internal and external integration (an increase in ϕS as well as in ϕL ) and there is a shift in the Union's trading partners towards lower industrialised regions with a lower skill endowment (n~ getting even ~ ϕS or closer to 1). If the Union was initially in a Core-Periphery position of the CPM type, an increase in one of the parameters n, ϕL alone would be sufficient to lead the economy to a symmetric equilibrium. With a simultaneous increase in n~ and ϕS (or ϕL ) the symmetric equilibrium will become the stable attractor sooner, i.e. the simultaneous increase lowers the respective bifurcation values for n~ and ϕS (or ϕL ). If the Union was initially in a symmetric equilibrium, an increase in one of the

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~ and ðϕL ; nÞ-parameter ~ Fig. 7. 2D bifurcation diagrams in ðϕS ; nÞplane for σ¼ 6, μ¼ 0.45, γ ¼ 10, θ ¼0.25 and ϕL ¼ 0:01 in (a), ϕS ¼ 0:15 in (b).

Fig. 8. In (a): 1D bifurcation diagram of the map Λ for σ¼ 6, μ¼ 0.45, γ¼ 10, θ¼ 0.25, ϕL ¼ 0:01 and n~ ¼ 0:8 related to cross-section indicated in Fig. 7 by the straight line with an arrow. In (b): an enlargement of the window indicated in (a). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

~ ϕS or ϕL alone would be sufficient to lead the economy to a Core-Periphery position of the CP type. With a parameters n, simultaneous increase in n~ and ϕS (or ϕL ) a Core-Periphery equilibrium will become the stable attractor sooner, i.e. the simultaneous increase lowers again the respective bifurcation values for n~ and ϕS (or ϕL ). Let us now comment the bifurcation scenario which is observed in the map Λ if the parameter point crosses the flip bifurcation boundary of the parameter region S. It can be done, for example, considering the 1D bifurcation diagram related to the cross-section indicated in Fig. 7(a) by the arrow. It is shown in Fig. 8 together with an enlargement. One can see in Fig. 8a that for decreasing ϕS the fixed point xn undergoes a supercritical flip bifurcation (at the point marked by a black circle) leading to an attracting 2-cycle g 2 ¼ fx0 ; x1 g, whose points are symmetric with respect to xn . Then g2 undergoes a supercritical pitchfork bifurcation (at the point marked by red circle), due to which two new attracting 2-cycles q2 and q02 are born, points of which are symmetric to each other with respect to xn . If we continue to decrease ϕS each of the 2-cycles q2 and q02 undergoes a sequence of bifurcations following well-known logistic bifurcation scenario starting with a cascade of flip bifurcations up to a homoclinic bifurcation (marked by blue points) of 2-cycle g2 (see Fig. 8b). Thus, we see that the map Λ can have coexisting attracting cycles and chaotic attractors. It is remarkable that those coexisting dynamic attractors may be asymmetric – confirming the ability of our model to generate asymmetric long-run outcome endogenously.

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Fig. 9. The map Λ in (a) at the moment and in (b) after the contact bifurcation of the chaotic attractor A ¼ ½c; c0  with its basin confined by the fixed points xCPð0Þ and xCPð1Þ . Here σ¼ 6, μ¼0.45, γ ¼ 10, θ¼ 0.25, ϕL ¼ 0:01, n~ ¼ 0:8 and ϕS ¼ 0:187626 in (a) (the related point is indicated in Fig. 8a by brown circle), ϕS ¼ 0:18 in (b). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 10. 2D bifurcation diagram in the ðϕS ; ϕL Þ-parameter plane for σ ¼ 6; μ¼0.45, γ ¼ 10, θ ¼0.25 and n~ ¼ 0:8.

Let us comment now a bifurcation marked in Fig. 8a by brown circles. It is a contact bifurcation of a one-piece chaotic attractor, bounded by the critical points11 of the map Λ denoted c and c0 , with its basin confined by the fixed point xCPð0Þ and xCPð1Þ . Such a contact occurs if a parameter point crosses the boundary of the region CPM (see Fig. 7). In Fig. 9a the map Λ is shown at the moment of such a contact defined by the condition c¼0 or c0 ¼ 1. After this bifurcation the locally repelling fixed points xCPð0Þ and xCPð1Þ become Milnor attractors. Recall that according to the most spread definition, an attractor A of a map f : I-I; I D R, is an attracting set with a dense orbit. n An attracting set A is defined as a closed invariant set for which a neighborhood U exists such that f ðUÞ  U and f ðxÞ-A as n-1 for any x A U. A Milnor attractor is defined as a closed invariant set A  I such that its stable set ρðAÞ (consisting of all points x A ρðAÞ for which the ω-limit set12 is a subset of A) has a strictly positive measure, and there is no strictly smaller closed subset A0 of A such that ρðA0 Þ coincides with ρðAÞ up to a set of measure zero (see Milnor, 1985). As we see, the basic difference of these two definitions of an attractor is related to the fact that in the first case any point from a neighborhood of A is attracted to A, while in case of a Milnor attractor not necessarily all the points from a neighborhood of A are attracted to it (in fact, the class of Milnor attractors is wider and it includes the sets which are attractors according to the first definition). For example, one can see in Fig. 9b that the points of the green intervals are mapped into xCPð1Þ and the points of the red intervals are mapped into xCPð0Þ . In fact, the interval J1 related to the left flat branch of Λ and all its preimages (a few of which are shown in Fig. 9b) constitute the stable set of the fixed point xCPð1Þ . There is a sequence of preimages of J1 accumulating to xCPð0Þ , and there is also a sequence of preimages of J 1 accumulating to xCPð1Þ . The same can be said about sequences of preimages of J2 related to the second flat branch of Λ. Thus, in any (one-side) neighborhood of xCPð0Þ or xCPð1Þ there is a

11 12

Following Mira et al. (1996), for a 1D continuous noninvertible map f : I-I, I D R, its local extrema are called critical points. A ω -limit set ωðxÞ is the set of all accumulation points under forward iterations of the orbit with initial point x.

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positive measure set of points which first escape from this neighborhood and then eventually are mapped to xCPð0Þ , as well a symmetric positive measure set of points mapped to xCPð1Þ . Clearly, not all the points of I are mapped to xCPð0Þ or xCPð1Þ : a chaotic repellor, separating the basins of the CP fixed points, remains in I, which is a Cantor set formed by all the repelling cycles and their preimages, as well as uncountably many aperiodic orbits. From an economic point of view, the possibility of coexisting cyclical or even complex attractors and of Milnor attractors emphasises that sensitivity upon initial condition – one of the main themes in NEG tales – might even be more relevant than standard NEG models actually suggest. Finally, in Fig. 10 we show the bifurcation structure of the ðϕS ; ϕL Þ-parameter plane for σ ¼6, μ ¼ 0.45, γ ¼10, θ ¼0.25 and n~ ¼ 0:8 which allows us to analyse the interaction between the two dimensions of trade freeness (internal and external): the negative slope of the bifurcation curves implies that the bifurcation value for ϕS is decreasing in ϕL and the stable range of the symmetric equilibrium (the area S) shifts to the left for higher values of ϕL . This confirms that the two types of progressive trade opening, taken separately, have the same qualitative effects and gives a clear hint that agglomeration is the most likely outcome of a simultaneous process of regional integration within the Union and external trade liberalisation. 5. Conclusions The EU has experienced a strong trend of deepening internal economic integration, including trade and factor mobility. At the same time, the ongoing strategy “Global Europe” aims at reducing trade barriers with respect to the rest of the World. Furthermore, deeper EU integration into the world economy is currently characterised by an increased importance of trade partners that are less industrialised and endowed with a lower share of skilled workers and that have a big home market (the relative size of the local demand inside the Union shrinks). Given these stylised facts, our research question was to analyse the effects of deeper regional integration and international liberalisation within one unified theoretical framework, especially shedding light on the importance of the characteristics of the external trade partners. To this end, this paper has presented a NEG model with three regions, one outside region and two closely integrated regions that form a Union: their mutual commodity trade cost is lower (their mutual trade freeness is higher) than with the outside region; and factor mobility is only possible between the regions in the Union, but not with the outside region. Given the stylised facts, we assumed that entrepreneurs or skilled labour is the mobile factor and that the outside region is less endowed with skilled labour and has the bigger local market. The model, therefore, is an asymmetric, three regions footloose entrepreneur model. We specified it in discrete time and studied the dynamic processes involved. Despite the analytical complexity typical for NEG models, we derived some results analytically; for other results, we had to use simulations as is standard in the NEG literature. We first analysed the properties for the standard NEG long-run equilibria, i.e. for the symmetric and the Core-Periphery equilibria that exist also in our model. Our analytic results suggest that a reduction in trade barriers (an increase in the trade freeness) among European regions, a reduction in the industrialisation level of the outside trading partners and an increase in the local (immobile) demand of the outside region are all factors that foster employment in the manufacturing sector within the EU; instead, a deeper integration with the outside region might work in the opposite direction. A second, most remarkable result of our analysis is that our model – although being entirely based on standard NEG assumptions – is able to endogenously generate, for significant parameter ranges, asymmetric stable equilibria, i.e. long-term positions in which the economic activity is still present in both regions of the Union even though it is not distributed symmetrically. Those asymmetric stable equilibria are born via a supercritical pitchfork bifurcation of the symmetric equilibrium if the (internal or external) trade freeness is increased above the corresponding bifurcation value. In this case – that occurs if the size of the local (immobile) demand inside the Union is sufficiently low (and the local demand in the outside region sufficiently high) – the transition to full agglomeration is smooth. Instead, if the size of the local (immobile) demand inside the Union is sufficiently high, the pitchfork bifurcation is subcritical, the asymmetric fixed points are unstable and the transition to agglomeration is catastrophic – this is the scenario found in most NEG models. In our model, the properties of the pitchfork bifurcation can be derived analytically for the special case in which the outside region is a pure rural economy without a manufacturing sector. Simulations show that it carries over to the general case, when some manufacturing production is located in the outside region. We obtained additional results on the local and global dynamics: surprisingly, the basic bifurcation scenarios are qualitatively similar in all the parameter planes which we considered. For intermediate parameter values, the symmetric fixed point is stable; for lower parameter values it loses stability via a flip bifurcation; for higher parameter values via a pitchfork bifurcation; in the latter two cases the bifurcation scenario finally leads to Core-Periphery equilibria. Note that the existence of the constraints in the model gives rise to a particular bifurcation of the CP fixed points which we call “bordertranscritical” bifurcation. This bifurcation scenario emerges by varying any of our central parameters: internal and external trade freeness, the relative endowment of the Union with skilled labour and the size of the local (immobile) demand inside the Union. In addition, we analysed how this pattern changes if two parameters are varied simultaneously and we were able to conclude: the ongoing process of deeper integration within the Union and with the rest of the World will ultimately foster a Core-Periphery pattern within the Union; the facts that the outside regions are increasingly less industrialised and have the bigger local (immobile) demand work in the same direction. In addition to the bifurcation scenarios discussed so far, bistability, i.e. coexistence of attractors, is also a characteristic feature of the model which goes well beyond the standard result in NEG models concerning the possible coexistence of the

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symmetric equilibrium and the Core-Periphery equilibria. In our model two different attractors, namely two cycles or two cyclic chaotic attractors may coexist, with quite complicated structures of the respective basins of attraction. In some cases it is even impossible to predict to which attractor the system will converge: for low values of the (internal or external) trade freeness the CP fixed points are Milnor attractors: although being locally repelling, they attract almost all the initial points (which is possible due to the flat branches of the map). Moreover, their basins are separated by such a complicated set as a chaotic repellor. Taking an initial point as close as we wish to one CP point, it cannot be stated a priori to which of the two CP fixed points the system will converge. History matters and a small perturbation may alter significantly the long-run allocation of industrial capital.

Acknowledgments This work has been prepared within the activities of the EU project COST Action IS1104 ‘‘The EU in the new complex geography of economic systems: models, tools and policy evaluation’’. The authors are grateful for financial support. Appendix A In this Appendix we show that regional incomes ðY r;t Þ, expenditure shares ðsr;t Þ, profits ðπ r;t Þ and indirect utility ðV r;t Þ depend only on xt. From Eq. (8), under our assumptions on trade costs across regions ðϕ12 ¼ ϕ21 ¼ ϕS ; ϕ13 ¼ ϕ23 ¼ ϕL Þ, we can write   s s s μY π 1;t ¼ 1;t þ 2;t ϕS þ 3;t ϕL Δ1;t Δ2;t Δ3;t σN   s s s μY π 2;t ¼ 1;t ϕS þ 2;t þ 3;t ϕL Δ1;t Δ2;t Δ3;t σN   s s s μY ð29Þ π 3;t ¼ 1;t ϕS þ 2;t ϕS þ 3;t Δ1;t Δ2;t Δ3;t σ N where ~ ϕL ; Δ1;t ¼ x1 n~ þð1  xt Þn~ ϕS þð1  nÞ ~ ϕL ; Δ2;t ¼ xt n~ ϕS þð1  xt Þn~ þ ð1  nÞ ~  ϕL Þ: Δ3;t ¼ xt n~ ϕL þð1  xt Þn~ ϕL þ 1  n~ ¼ 1  nð1 Moreover, under our assumptions on trade costs and unskilled labour endowments, regional incomes/expenditures – formulated in the main text as Y r;t ¼ Lr;t þ nr;t π r;t for r ¼ 1; 2; 3 – can be expressed as ~ π 1;t Y 1;t ¼ θL þxt nE ~ π 2;t Y 2;t ¼ θL þð1  xt ÞnE ~ π 3;t Y 3;t ¼ ð1  2θÞL þð1  nÞE

ð30Þ

Using (29) and (30) the regional income shares can be expressed in terms of xt: 2   3  ~  xt ÞϕL ϕL ϕS μnð1  σ μ θþ 6 7  ϕ Δ3;t Δ2;t Δ3;t 7 ~ t6   σ  μ θ þ μnx 6 L  7 4Δ3;t 5 1 ϕ L σ  μn~ ð1  xt Þ  Δ2;t Δ3;t Y 1;t 2 3 ¼ s1;t ¼    Y ϕL ϕS ϕL ϕ   6   S μ2 n~ 2 xt ð1 xt Þ 7 1 ϕ Δ3;t Δ2;t Δ3;t Δ1 ; t 6 7 ~ t     7 σ  μnx  L 61   5 Δ1;t Δ3;t 4 1 ϕ 1 ϕ L L ~ t σ  μn~ ð1  xt Þ  σ  μnx 

Δ2;t

2

Δ3;t

Δ1;t

Δ3;t

3

  ϕS

~ t ϕL ϕL μnx  σ μ θþ 6ϕ 7 Δ Δ Δ 7 3;t 1;t   3;t σ  μ θ þ μn~ ð1  xt Þ6 6 L  7 4Δ3;t 5 1 ϕ ~ t σ  μnx  L Δ1;t Δ3;t Y 2;t 2 3 ¼ ¼    Y ϕL ϕS ϕL ϕS   6   μ2 n~ 2 xt ð1 xt Þ 7 1 ϕ Δ3;t Δ2;t Δ3;t Δ1;t 6 7     7 σ  μn~ ð1  xt Þ  L 61   5 Δ2;t Δ3;t 4 1 ϕ 1 ϕ L L ~ ~ σ  μn ð1 xt Þ  σ  μnxt  

s2;t





Δ2;t

s3;t ¼

Y 3;t ¼ 1  s1;t  s2;t Y

Δ3;t

Δ1;t

Δ3;t

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Finally, considering that, from Eq. (4), price indexes can be written as 1=ð1  σ Þ 1=ð1  σ Þ

E

~ ϕL 1=ð1  σ Þ E1=ð1  σ Þ p p ¼ ½x1 n~ þð1  xt Þn~ ϕS þð1  nÞ

1=ð1  σ Þ 1=ð1  σ Þ

~ ϕL 1=ð1  σ Þ E1=ð1  σ Þ p p ¼ ½xt n~ ϕS þ ð1 xt Þn~ þð1  nÞ

1=ð1  σ Þ 1=ð1  σ Þ

~  ϕL Þ1=ð1  σ Þ E1=ð1  σ Þ p p ¼ ½1  nð1

P 1;t ¼ Δ1;t P 2;t ¼ Δ2;t P 3;t ¼ Δ3;t

E E

μ

Also the indirect utility of a skilled worker in region r , V r;t ¼ π r;t =P r;t , depends only on xt. Appendix B In this Appendix we show the results wrt the employment structure at the fixed point solutions, reported at the end of Section 3.5. In the symmetric equilibrium, the endowment of unskilled workers of region 1 is equal to θL (which is also the endowment of region 2); the endowment of region 3 equals ð1 2θÞL. In addition, Δ1 ¼ Δ2 ¼ 0:5n~ þ ϕL þ0:5n~ ϕS  n~ ϕL and  Δ3 ¼ 1  n~ 1  ϕL holds; as well as s1 ¼ s2 and s3 ¼ 1  2s1 . Therefore, !  s1 1 þ ϕS ð1 2s1 ÞϕL σ 1 μY n~  þ β E ~ þ ϕL þ 0:5n~ ϕS  n~ ϕL 1  n~ 1  ϕL 0:5 n βσ E 2 employment ¼ S ¼ Semployment 1 2 θL 

 2s1 ϕL 1 2s1 σ  1 μY þ βð1  n~ ÞE ~  ϕL Þ βσ E 0:5n~ þ ϕL þ 0:5n~ ϕS  n~ ϕL 1  nð1 employment ¼ S3 ð1 2θÞL The relative employment ratio is ~ 0:5nE Num1 θ L ¼ ~ Num3 ð1  nÞE Semployment 3 ð1  2θÞL Semployment 1

with ~ σ  μÞθ þ σϕL ðn~ þ 2ϕL  n~ ϕL Þ þ σϕL nð ~ ϕS  ϕL Þ Num1 ¼ 2ðϕS  2ðϕL Þ2 þ 1Þð1  nÞð and ~ σ  μÞð1 2θÞ þ2σϕL þ 2n~ σϕL ðϕL  1Þ Num3 ¼ ðϕS 2ðϕL Þ2 þ 1Þnð Note that Sendowment 1 Sendowment 2

~ 0:5nE θ L ¼ ~ ð1  nÞE ð1  2θÞL

is the relative endowment ratio. The relative employment ratio is equal to the relative endowment ratio, if ~ þ σϕL ð1  ϕL Þð3n~ 2Þ þ n~ σϕL ðϕS  ϕL Þ ¼ 0 Num1  Num3 ¼ ðϕS  2ðϕL Þ2 þ 1Þðσ  μÞð2θ  nÞ from which the three conditions in the main text follow immediately. =Semployment . The comparative statics results wrt θ, n~ and ϕS follow directly from inspecting Semployment 1 3 ~ Wrt ϕL , only local results at θ ¼ n=2 and n~ ¼ 2=3 can be obtained; for that case USR1 =USR3 has a parabolic shape in ϕL with USR1 =USR3 ¼ 1 for ϕL ¼ ϕS and ϕL ¼ 0, and with USR1 =USR3 4 1 for 0 o ϕL o ϕS . The results given in the main text follow immediately.

Appendix C Let us check the conditions (19)–(22) which have to be satisfied in case of a pitchfork bifurcation of the fixed point of the map Z given in (18). 1. Condition given in (19) is verified due to the fact that at the symmetric equilibrium V 1 ¼ V 2 and ∂V 1 =∂ϕS ¼ ∂V 2 =∂ϕS for any ϕS . 2. Condition in (20) is verified due to the symmetric properties of the map Z(x). Indeed, at the bifurcation point, this

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condition can be reduced to 0 1        ∂π 1 2 ∂π 2 2 ∂2 πPμ1 ∂2 πPμ2 B C 2 2  B ∂x ∂ V1 ∂ V2 ∂x 1 2 C  2 ¼  ¼ μ 1 þ μ B μ þ 2 π1  μ þ 2 π2 C 2 2 2 @ P A ∂x ∂x ∂x ∂x P 1

1

which holds, given that at x ¼

π1 ¼ π2;

P1 ¼ P2 ;

0

2

0 2 1 ∂2 π 1 ∂2 π 2 ∂ P1 ∂2 P 2 ∂π 1 ∂P 1 ∂π 2 ∂P 2 B C B C 2 B C B ∂x2 ∂x2 B ∂x2 C  ∂xμ þ∂x  ∂xμ C þ  2μ@ ∂xμ þ∂x A  μ@ μ þ 1 π 1  μ þ 1 π 2 A ¼ 0; 1 1 A @ Pμ P P1 P2 P1 P2 1 2 0

1 : 2

∂π 1 ∂π 2 ¼ ; ∂x ∂x

∂P 1 ∂P 2 ¼ ; ∂x ∂x

1

∂2 π 1 ∂2 π 2 ¼ ∂x2 ∂x2

and

∂2 P 1 ∂2 P 2 ¼ 2: ∂x2 ∂x

Note that both conditions 1 and 2 hold also for the case 0 o n~ o 1. 3. Concerning condition (21) we obtain the following result:   ∂2 T 1 2μð2σ  1Þ½ðσ  1Þðσ  μÞ2θ þ μðμ þ 3σ  2Þ2 ¼ ϕpf 40 S ; 2 ∂x∂ϕS ðσ  μÞðσ  1Þ2 ðμ þ 2σθÞ½ðσ 1Þðσ  μÞ2θ þ μðσ 1 þ μÞ which is always satisfied. 4. Concerning condition (22) we obtain the following result:   2 ∂3 T 32μ4 ð2σ 1Þ3 f12ðσ  1Þ2 ðσ  μÞθ þ ½2ð2σ  3Þμ2 þ 4ð3μ  σ Þðσ  1Þ2 θ  μðσ 1 þ μÞ½2ðσ 1Þ þ μg pf 1 ¼ ϕ ; S 3 2 ∂x3 ðσ  1Þ ðμ þ 2σθÞ½ðσ 1Þðσ  μÞ2θ þ μðσ 1 þ μÞ3 According to this expression condition (22) may or may not hold depending on parameter combinations.

Appendix D Proof of Proposition 1. First we consider the following second degree equation based on (24): aθ þ bθ þ c ¼ 0; 2

ð31Þ

where a ¼ 12ðσ 1Þ2 ðσ  μÞ 40;

b ¼ 2ð2σ  3Þμ2 þ4ð3μ  σ Þðσ 1Þ2 Zð oÞ0; c ¼  μðσ  1 þ μÞ½2ðσ  1Þ þ μ o0:

Therefore (31) admits one positive and one negative solution. In order to have real roots, it must be

Δ ¼ b2 4ac ¼ ð12σ  8σ 2  3Þμ4 þ 4ð10σ 2 12σ þ 3Þðσ 1Þ2 μ2 þ 4σ 2 ðσ  1Þ4 4 0:

Δ ¼0 is a quartic equation that admits four solutions of which only two at most are real (or none). Define y ¼ μ2 . Then we can write

Δ ¼ b2 4ac ¼ ð12σ  8σ 2  3Þy2 þ 4ð10σ 2 12σ þ 3Þðσ  1Þ2 yþ 4σ 2 ðσ  1Þ4 ¼ 0:

ð32Þ

This is now a second degree equation whose solutions, y1 and y2, are real since

Δ~ ¼ 144ð3σ  1Þð2σ  1Þ2 ð2σ  1Þ5 4 0:

pffiffiffi pffiffiffi and one negative for σ 4 ð3 þ 3Þ=4, i.e. Moreover, these solutions are both negative pffiffiffi for 1 o σ o ð3 þ 3Þ=4 and one positive pffiffiffi y2 o 0 o y1 . Therefore, for 1 o σ o ð3 þ 3Þ=4, Δ 4 0 always; and for σ 4ð3 þ 3Þ4, Δ 4 0 as long as 0 o x o 1 ox1 . Therefore also in this case Δ 4 0 for all relevant values of μ.13 □ Therefore, pffiffiffiffi (31) admits two real solutions, one positive and one negative. Let us call the positive solution θ , where θ ¼  ðb  ΔÞ=2a. Given that a 40, we have K o 0 for 0 o θ o θ and K 40 for θ 4 θ ; and K ¼0 when θ ¼ θ , where K is given in (24). 13

Note that x2 41 as long as the sum of the coefficients in Eq. (32) is positive, that is, 4σ 6  16σ 5 þ 64σ 4  144σ 3 þ 144σ 2  60σ þ 9 4 0

Looking at the solutions of the six degrees equation, none of them is real. It follows that this inequality is always satisfied or never satisfied. It is immediate to check, by substituting any number for σ, that the condition holds.

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Finally, notice that the condition θ o 4ðσ þ μÞðμ þ σ  1Þ½σ  ð1 þ μÞ 40:

287

1 corresponds to 2

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