The Footloose Entrepreneur Model with Three Regions
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The Footloose Entrepreneur Model with Three Regions José Maria Lopes Gaspar Master degree dissertation in Economics Supervisor: Sofia Balbina Santos Dias de Castro Gothen Co-Supervisor: João Oliveira Correia-da-Silva 2012
Biography José Maria Lopes Gaspar was born in Lisbon on 28 September of 1988. He lived in his hometown until 2006, after which he moved to Oporto to get his Bachelor (BSc) degree in Economics at the School of Economics and Management (FEP) of the University of Porto. He graduated in 2010 with a final average of fourteen (14) points out of twenty (20). During this time, he was part of the University of Porto’s team of the student organization Sharing Knowledge, where he participated in the development of projects and presentations in the field of macroeconomics. Immediately after the BSc, he joined the Master of Science (MSc) in Economics at that same institution. Upon completion of the curricular part of the MSc, his average was of eighteen (18) points out of twenty (20). Just prior to starting to work on his dissertation, he tutored students from the portuguese-speaking african countries (PALOP) in support classes of Macroeconomics I at FEP. i
Acknowledgements I would like to thank all of those who, directly or indirectly, helped and contributed for the success and completion of this work. I am very grateful to my two supervisors, Professor João Correia-da-Silva and Professor Sofia Castro, for their guidance and the countless hours spent on discussing and reviewing this work thoroughly. The benefits I get from having worked with both of them far exceed the contributions to this dissertation. I am also grateful to all the teachers of the MSc in Economics at FEP, since their lecturing is also greatly responsible for the present dissertation. Thanks to all of my friends for their support and encouragement. I would like to thank Miguel Correia, for putting up with my constant questions and for reading and commenting on this dissertation. Special thanks to all my family, specially my sister Catarina and my mother Maria Teresa. I also want to thank Isabel Figueiredo, for her patience and kindness. Dedico esta dissertação ao meu pai, Alfredo. ii
Abstract We study an analytically solvable version of Krugman’s Core-Periphery model extended to three regions. This is the 3-region Footloose Entrepreneur model based on the 2region version by Forslid and Ottaviano (2003). Solvability is achieved by employment of the skilled inter-regionally mobile work-force in the fixed costs and the unskilled immobile work-force in the variable costs of the manufacturing firms. This enables us to obtain real wages as explicit functions of the spatial distributions of skilled workers in the three regions. The main aim of this work is to study long-run equilibria of the model in terms of skilled workers’ migration, which is based on indirect utility differentials as given by differences in inter-regional real wage differences. For this purpose, we focus on the analysis of stability of three types of equilibria: concentration, total and partial dispersion. The first corresponds to full agglomeration of industry in one region, the second refers to an equalized spatial distribution of skilled workers across the three regions and the third pertains to a situation whereby skilled workers are equally dispersed in two of the three regions. We show analytical and numerical evidence in that the latter configuration is always unstable. We also compare our results concerning concentration and total dispersion with those of the 2-region model. Finally, we discuss the existence and robustness of bifurcations in our 3-region model. Keywords: new economic geography, core-periphery, footloose entrepreneur JEL Classification Numbers: R10, R12, R23 iii
Resumo Estudamos uma versão analiticamente resolúvel do modelo “core-periphery” de Krugman estendido para três regiões. Trata-se do modelo “Footloose Entrepreneur” com três regiões baseado na versão com duas regiões por Forslid e Ottaviano (2003). A solvabilidade do modelo é alcançada através da incorporação da força de trabalho qualificada e inter-regionalmente móvel nos custos fixos e da força de trabalho imóvel e não qualificada nos custos variáveis das empresas industriais. Isto permite-nos obter os salários reais como funções explícitas da distribuição espacial dos trabalhadores qualificados nas três regiões. O objetivo principal deste trabalho é o de estudar os equilíbrios de longo prazo em termos da migração de trabalhadores qualificados, que é baseada em diferenciais de utilidade indireta dados pelas diferenças inter-regionais dos salários reais. Para este fim, concentramo-nos na análise da estabilidade de três tipos de equilíbrio: concentração, dispersão total e parcial. O primeiro corresponde à aglomeração total da indústria numa região, o segundo refere-se a uma distribuição espacial dos trabalhadores qualificados equalizada pelas três regiões, e o terceiro refere-se a uma situação em que os trabalhadores qualificados estão igualmente dispersos em duas das três regiões. Apresentamos evidência analítica e numérica no sentido de que a última configuração seja sempre instável. Comparamos também os nossos resultados, relativos à concentração e à dispersão total, com as do modelo para duas regiões. Finalmente, discutimos a existência e robustez de bifurcações no modelo com três regiões. Palavras-chave: nova economia geográfica, centro-periferia, footloose entrepreneur Código JEL: R10, R12, R23 iv
Contents 1 Introduction 1 2 The model 6 2.1 Economic environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Short-run equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Long-run equilibria and stability 14 3.1 Stability of concentration . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.1 Comparing concentration in the 3-region and 2-region models . 23 3.2 Stability of total dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 Comparing total dispersion in the 3-region and 2-region models 30 3.3 Simultaneity of concentration and total dispersion . . . . . . . . . . . . 31 3.4 Stability of partial dispersion . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Bifurcation in the 3-region FE model . . . . . . . . . . . . . . . . . . . 39 4 Note on possible extensions of the FE model to n-regions 42 5 Conclusion 44 Appendix A 46 Appendix B 49 Appendix C 50 v
C.1TotalDispersion ............................... 53 C.2PartialDispersion .............................. 56 Bibliography 61 vi
List of Figures 3.1 Stability of concentration for the 2 and 3-region models. . . . . . . . . 24 3.2 Stability regions for concentration in the 2 and 3-region FE models. . . 24 3.3 Simultaneity of stability of concentration and total dispersion. . . . . . 32 3.4 Concentration and total dispersion in parameter space. . . . . . . . . . 33 3.5 Stability of partial dispersion. . . . . . . . . . . . . . . . . . . . . . . . 37 3.6 Stability of partial dispersion in parameter space. . . . . . . . . . . . . 38 3.7 Dynamics of the 3-region FE model inside the 2-simplex. . . . . . . . . 40 vii
1 Introduction The secular tendency for agglomeration of economic activity in specific industrial sectors in many countries and regions is well known and has been a matter of profound debate for a long time. However, up to recent years, the economic science had either neglected or simply failed to explain the spatial distribution of the factors of production and economic activity in general. If not due to a disregard for its evident importance and its consequences on the unequal income distributions, increasing clustering of economic activity and integration, then at least doubtlessly due to the difficulties it imposes on the analytical treatment of increasing returns, as is suggested by Fujita et al. (1999). The insistent clinging to constant returns as a simplifying assumption makes it unfeasible to explain the cumulative process of the self-perpetuating geographic concentration of economic activities. Howbeit, in the past few decades, there have been a lot of theoretical advances in this field of Economic Geography, partially thanks to new techniques developed in economic modeling, numerical methods and computation, specially in other fields such as industrial economics and economic growth. These allowed for a more rigorous treatment, based on microeconomic foundations, of the phenomena associated with the agglomeration of economic activities. Economic Geography can be understood as the study of where the different economic activities take place and the intrinsic reasons that justify it. Transport costs, increasing returns and factor mobility are among such reasons. Of course, it would be unfair to claim that such efforts in trying to explain industrial location have not been made in the somewhat distant past, like the development of models that descend from the von Thünen theory (von Thünen, 1826) on agricultural land use. Johann von Thünen was arguably a pioneer in addressing the causes that led to spatial location and how this would affect prices, before industrialization had occurred. In his famous model, different types of agriculture were placed in four concentric rings around a central city. Close proximity to the city meant low transport costs, so the farther the rings were from the city, the more agricultural products not susceptible to high transport 1
structure is such that τij = 1, if j=i, and τij =τotherwise. Inasmuch as the existing varieties of manufactures are horizontally differentiated and trade costs are the same across regions, we can establish that a consumer in region iwill be indifferent between a variety from any of the other two regions. 2.2 Short-run equilibrium We distinguish short-run from long-run equilibrium in that the former takes the spatial distribution of skilled workers as given and allows us to study the relations between variables such as skilled workers’ real wages, transport costs and share of local income spent on manufactures, whereas, in the long-run, we allow skilled workers to migrate and, hence, equilibrium corresponds to a configuration whereby skilled workers have no incentives to migrate. In this subsection we are concerned with short-run equilibrium and obtaining expressions for real wages that will allow us to study long-run equilibria in section 3.1Some of the following results are analogous to Forslid and Ottaviano (2003) and are clearly indicated. First, looking at the demand side, the representative consumer in region imaximizes utility (2.1) subject to the following budget constraint: 3 X j=1 ˆsN pji(s)dji(s)ds +pA iAi=Yi, i ={1,2,3},(2.5) where pA iis the price of the agricultural good, dji(s)is the demand by residents in location iof a variety produced in location jand pji its price. The optimization problem yields the following CES demand for dji: dji(s) = pji(s)−σ P1−σ i µYi,(2.6) 1Migration, as we shall see, depends entirely on indirect utility differentials as given by real wages. 8
with the local CES price index Piassociated with (2.2): Pi= 3 X j=1 ˆsN pji(s)1−σds 1 1−σ .(2.7) Turning now to the supply side and starting with the agricultural sector, absence of transport costs in trade of good Ameans that its price is the same everywhere, so pA 1=pA 2=pA 3.Furthermore, under perfect competition, we have marginal cost pricing so that pA i=wL iand we have zero profits for this sector in equilibrium. Consequently, the aforementioned marginal cost pricing rule implies wage equalization between regions (wL 1=wL 2=wL 3). This suggests choosing Aas numeraire, so that pA i=wL i= 1. We assume that the non-full-specialization (NFS) condition (Baldwin et al., 2003) holds, guaranteeing that agriculture is active in the three regions.2 In the industrial sector, given the fixed cost αin (2.4), skilled labour market clearing gives us the number of firms in each region in equilibrium (same as in Forslid and Ottaviano, 2003): ni=Hi α. A manufacturing firm in region ifacing the total cost in (2.4) maximizes the following profit function: Y i (s) = 3 X j=1 pij(s)dij(s)−β 3 X j=1 τijdij(s) −αwi,(2.8) where wL i= 1. Because total supply to region j6=imust include the part 1−τof the product that melts away, it is equal to τdij(s, t). The first order condition for maximization of (2.8) renders: pij(s) = τijβσ 1−σ.(2.9) 2The condition requires world expenditure on good Ato be greater than the maximum total production of Ain two regions, i.e., (1−µ)YW>2 3L, where YWis the global income.This is guaranteed if we assume that µ < σ/(3σ−2). In the 2-region model by Forslid and Ottaviano (2003), the corresponding assumption is µ < σ/(2σ−1). 9
This pricing equation is intrinsically different from that of the original CP model in that it does not depend on the wages of skilled workers, but on the wages of unskilled workers, which are equalized across regions. It is also equivalent to that of the 2-region model by Forslid and Ottaviano. This is what renders the FE model solvable. Using (2.9), the CES price index (2.7) becomes: Pi=βσ 1−σ 3 X j=1 φijni 1 1−σ ,(2.10) where φij =τ1−σ ij ∈(0,1] is what Forslid and Ottaviano have coined with the expression “freeness of trade”. It increases as the transport costs τfall, reducing the price index Pi. Absence of entrance barriers in the manufacturing industry means that there is free entry and exit, which translates in zero profits in equilibrium. This implies that operating profits must totally compensate fixed costs in terms of skilled labour and they are equal to the wages paid to entrepreneurs (i.e., skilled workers), such that: wi= 3 X j=1 pij(s)dij(s)−β 3 X j=1 τijdij(s) , which becomes, considering the prices in (2.9): wi=βxi σ−1,(2.11) where xi=Pτijdij(s)is total production by a manufacturing firm in region i. Using (2.6), (2.9) and (2.10), we can derive an expression for xithat depends on the local incomes and the number of firms of the three regions: xi=µ(σ−1) βσ 3 X j=1 φijYi P3 m=1 φmjnm .(2.12) A new expression for the nominal wage can now be derived. Replacing (2.12) in (2.11) and knowing that ni=Hi αwe have: wi=µ σ 3 X j=1 φijYj Rj ,(2.13) 10
where Rj=P3 m=1 φmjHm. By (2.3), income equals: Yi=L 3+wiHi.(2.14) The endogenous variables ni,pi,wi,xiand Yican be determined for a given allocation of skilled workers H. Recall that these results are analogous to those in Forslid and Ottaviano (2003), with the exception that we are considering a 3-region model. Of course, it is very straightforward to see that, starting from the 2-region model, building an-region FE model would practically be tantamount to building a 3-region one, as we have done so far. Until this point, that is.3The only changes required for obtaining general expressions for n-regions would be changing the superscripts from 3 to nin the aforementioned summations and changing the amount of unskilled labour in each region to Li=L n. Location decisions by skilled workers from one region to another are assumed to depend on indirect utility differentials as given by real wages. Hence, it is important to first derive the system of three linear equations in wi, for i= 1,2,3, using (2.13) and (2.14), that can be solved to obtain the equilibrium skilled (nominal) wages as explicit functions of the spatial distribution of skilled workers Hi. This is what we do in the following: Proposition 2.1. The nominal skilled wages in region iare given by: wi= µ σ L 3 3 X j=1 φ1j Rj +µ σ φ(φ−1) X k6=i Hk Y k6=i Rk +φ2−1 RiX k6=i Hk Rk +µ2 σ22φ3−3φ2+ 11 RiY k6=i Hk Rk 1−µ σ 3 X j=1 Hj Rj +µ2 σ2(1 −φ2)H1H2 R1R2 +H1H3 R1R3 +H2H3 R2R3−µ3 σ3(2φ3−3φ2+ 1) 3 Y j=1 Hj Rj .(2.15) Proof. See Appendix A. Although similar, the 2-region model cannot be derived from the present one by eliminating skilled workers from a single region. 3Some of the further results are already quite cumbersome with only three regions, let alone with four or more. 11
Remark. The 3-region model does not contain the 2-region model in Forslid and Ottaviano. Proof. If there is no third region, the unskilled labour Lhas to be divided equally between two regions, hence Li=L 2. This changes the structure of the model, as the local income Yiis now different. We would also have to exclude 1 R3from (2.13). Hence, simply setting the amount of skilled workers Hito zero would, thus, not give us the expression for the nominal skilled wage in the 2-region model (see Forslid and Ottaviano, 2003). In other words, although similar, the 2-region model cannot be derived from the present one by eliminating skilled workers from a single region. The nominal skilled wages can also be expressed as functions of the share of skilled workers in each region, hi=Hi H.Furthermore, we can conveniently omit one of the regions in our analysis, since Hk=H−Hi−Hj, with i, j, k ={1,2,3}, and define region kimplicitly as function of regions iand j. This allows us to limit our analysis of the dynamics on the 2-simplex (because hi+hj+hk= 1). Taking this into consideration, we can rewrite the numerator and denominator of (2.15), and therefore the nominal wage wi, in terms of hiand hj, leaving out region hk, as it is implicitly defined a function of hiand hj: wi(hi, hj) = Dwi(hi, hj) D(hi, hj), where Dwi=µ σ L 3 1 H(3 X m=1 φim rm +µ σφ(φ−1) hj+hk rjrk +φ2−1 rihj rj +hk rk+ +µ2 σ22φ3−3φ2+ 11 ri hjhk rjrk,(2.16) D= 1 −µ σ 3 X j=1 hj rj +µ2 σ21−φ2h1h2 r1r2 +h1h3 r1r3 +h2h3 r2r3 −µ3 σ32φ3−3φ2+ 1h1h2h3 r1r2r3 ,(2.17) 12
with ri=Ri H. Obviously, we have rk= 1 + (φ−1)(hi+hj),since hk= 1 −hi−hj4. This form of presentation is convenient for derivation purposes. Additionally, the price index Pibecomes, after (2.9): Pi(hi, hj) = βσ σ−1H αri 1 1−σ.(2.18) 4It does no harm to express Ddirectly a function of h1, h2and h3, as can be understood by reading Appendix A. 13
3 Long-run equilibria and stability Recall that skilled workers are assumed to base their location decisions on the difference between each region’s real wage and the weighted average real wage in the three regions. This assumption follows Krugman (1991b) in that it states that the skilled workers are short sighted and migrate to the location that offers them the highest indirect utility. We define long-run equilibria1as distributions of skilled workers that remain unchanged over time. An equilibrium is stable if, after occurrence of some small exogenous migration of skilled workers to any of the regions, the spatial distribution of skilled workers is pulled back to the initial one. The next paragraph contains a brief description of the dynamical system of the 2-region FE model by Forslid and Ottaviano, since it is the benchmark to our 3-region model. In the 2-region FE model, skilled worker migration follows a simple Marshallian adjustment2, whereby the rate of change of skilled workers in one region depends on the real wage differential between the two regions adjusted by some positive real parameter. If the real wage in region iis higher than in region j, skilled workers in region jwill migrate to region i. Since the shares of skilled workers in the two regions sum up to unity, the authors chose to study the dynamics through the perspective of one region only, with hbeing the share of skilled workers in that region. Naturally, the share of skilled workers in the other region is implicitly defined as 1−h, so studying the dynamics in one region renders complete information on the dynamics in general. Working with two regions pertains to studying dynamics on a 1-simplex, that is, a line segment where h goes from zero to unity. The dynamics are well defined so as to capture both interior and boundary dynamics3. As a consequence of their formulation, interior equilibria (h=1 2) always exist, whereas corner solutions (h= 0 and h= 1) are not necessarily equilibria. However, when they are equilibria, they are necessarily stable. Turning to our 3-region FE model, dynamics are described in a straightforward way, 1Hereinafter, we shall refer to long-run equilibria just as equilibria. 2See Forslid and Ottaviano (2003: 234-235). 3This refers to dynamics at points placed on the boundaries of the simplex. These correspond to h= 0 or h= 1, i.e., the boundaries of the line segment. 14
albeit different from that in the 2-region model. Workers migrate to a region if the wage in that region is higher than the weighted average real wage of the three regions, unless the former is initially absent of skilled workers, in which case it will be left empty unless there is some exogenous migration to that region. Choosing to leave the dynamics in region 3implicitly defined4, without loss of generality, the migration dynamics of skilled workers are determined by the following system: ˙ h1= ∆ω1= (ω1−¯ω)h1 ˙ h2= ∆ω2= (ω2−¯ω)h2 , h1, h2∈[0,1] ,(3.1) where ωi=wi Pµ istands for the real wage in region iand ¯ω(h1, h2) = h1ω1+h2ω2+h3ω3 is the weighted real wage average of wages in the three regions. The equations of the system are ad hoc migration equations, a standard formulation in many NEG models’ dynamics, such as in the Core Periphery model in Fujita et al. (1999)5and in the version of the FE model in Baldwin et al. (2003) based on the original FE model by Forslid and Ottaviano (2003). It clearly shows that the rate of change of the share of skilled workers is proportional to the real wage differential. We have dynamics subject to a two-dimensional simplex (an equilateral triangle) whose boundaries correspond to configurations (h1, h2)such that one of the regions is left absent of skilled workers. Such configurations pertain to either h1= 0,h2= 0, or h1+h2= 1 ⇔h3= 0. Most criticism towards the ad hoc migration equations concern the fact that there is no motion when skilled workers are fully agglomerated in one region, even if there are nonnull real wage differentials (Matsuyama, 1991) at such configurations, hence preventing us from explaining the lack of motion when this is the case. We should then just briefly clarify why the ad hoc migration equations are used in this dissertation. It would seem as if the dynamics could be explained in a more straightforward way if we did not multiply each wage differential by the share hi. However, we are dealing with both interior and boundary dynamics, which are more complex in a 3-region model 4That is, h3= 1 −h1−h2. 5They justify its use on the grounds that their CP model might be regarded as an evolutionary game. Evolutionary game theory recurrently use “replicator dynamics” in the fashion of such equations. 15
than in the 2-region model. The extra hiin each equation is necessary to guarantee that the rate of change of skilled workers in each region is null when that region is absent of skilled workers. The absence of the extra hiwould leave the dynamics ill-defined at the boundaries. For instance, imagine that both regions 2 and 3 have no skilled workers. The average real wage ¯ωis equal to the real wage in region 1, ω1, so we would have a null wage differential ∆ω1, but a differential ∆ω2whose signal would depend on which real wage was higher, ω1or ω2.Assume, hypothetically, that the real wage is higher in region 2. It follows that ˙ h1= 0 and ˙ h2>0. Furthermore, ˙ h3=−˙ h2<0.But symmetry between regions means that when both regions 2 and 3 are absent of skilled workers and their real wages are equal, then their rates of change should be the same, hence ˙ h3>0, which is a contradiction. The system would not give us enough information on the dynamics at the boundaries. On the other hand, the migration equation commonly used assures that both rates of change are equal to zero, when the regions are empty6. Furthermore, assuming the mentioned hypothetical dynamics, one could easily verify that the dynamic properties of a configuration such as (1,0,0) would be different from that of (0,0,1), so agglomerating in region 1 would be different from concentrating in region 3 as far as stability is concerned, which makes no sense since we have three regions that are identical. With the formulation implicit in the migration equation, all three regions are treated symmetrically.7Hence, the dynamics implicit in the migration equation in (3.1) are more suitable for our analysis. Next, we address the equilibria of (3.1). Based on what we have argued above, we cannot say that equilibria are strictly configurations at which there are no endogenous incentives to migrate from one region to another. This is because we have no migration at corner solutions, even if there are non-null real wage differentials. Hence, we shall use the more straightforward definition of equilibria: Configurations that satisfy ˙ h1,˙ h2= 0 are all equilibria. By direct substitution, it is then easy to see that the configurations (h1, h2, h3) = (1,0,0) ,1 3,1 3,1 3, ,0,1 2,1 28and their permutations are all equilibria of (3.1). The first one consists of 6This was already argued by Baldwin et al. (2003). 7We have ˙ h3=−˙ h1−˙ h2=. . . =h3(ω3−¯ω). 8Recall that, since h3= 1 −h1−h2, we are still working in two coordinates. For instance, the configuration (0, a, 1−a)is equivalent to (h1, h2) = (0, a). 16
full agglomeration of industry in one of the regions, leaving the other two without any skilled workers. We call this “concentration” (or full agglomeration). The second corresponds to an equilibrium for which skilled workers are equally dispersed across the three regions. This is “total dispersion”. Finally, the third configuration represents an outcome whereby skilled workers are equally dispersed across two of the three regions. This is referred to as “partial dispersion”. A relevant difference between the postulated dynamics of our 3-region model and those of the 2-region model by Forslid and Ottaviano is that corner solutions are always equilibria of the system in (3.1), whereas the corresponding corner configurations may not be equilibria in the 2-region FE model. A key ingredient in the description of the dynamics of skilled workers in NEG models is the study of stability of the following three configurations: concentration, total and partial dispersion. The equilibrium corresponding to total dispersion is fully symmetric, while the other two are partially symmetric.9The stability of each equilibrium is preserved by permutation so that the same stability conditions hold for concentration or partial dispersion in any of the regions. Equilibria are stable if, due to occurrence of some exogenous migration of skilled workers to any of the regions, the spatial distribution of skilled workers is pulled back to the initial one. We have the following result concerning configurations that are placed on a boundary of the 2-simplex: Proposition 3.1. A necessary condition for the stability of an equilibrium such that hi= 0 is: ωi<¯ω. Proof. Considering an equilibrium with hi= 0, if the real wage in region iis lower than the average real wage ¯ω, then continuity of real wages in the share of skilled workers ensures that there is a point in a neighbourhood of that equilibrium where ωi<¯ω. Thus, if there was a marginal and exogenous migration of skilled workers from any 9We can permute either the populated or unpopulated regions but not all of the three regions. 17
in the 3-region case, thus illustrating the wider area of stability of concentration in the 3-region model compared to the model with two regions. Φs,2 Φs,3 0 1 Φ SP2HΦL, SP3HΦL Stabilityof concentrationfor 2 and 3 regions Figure 3.1 – Setting the parameters µ= 0.4and σ= 5, we plot the functions SP2(φ) (dashed) and SP3(φ)(solid). Since concentration is stable to the right of the zeros, we see that concentration in the 3-region model is a sufficient condition for that of the 2-region model. 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 Φ Σ SP2HΦ,ΣL, SP3HΦ,ΣL<0; Μ=0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Φ Μ SP2HΦ,ΜL, SP3HΦ,ΜL<0; Σ=5 Figure 3.2 – Darker region represents region of stability for the 2-region model (SP2<0). The lighter one represents stability for the 3-region model (SP3<0) and clearly contains that of the 2-region model. On the left, µis fixed and SP is plotted as a function of both φand σ. On the right, σis fixed. 24
Figure 3.2 illustrates the previous proposition for a wider array of values of µand σ. The picture to the left shows the region where SP(φ, σ)<0, that is the region of stability of full agglomeration, for both the 2-region and 3-region models, for every φ and σand fixed µ= 0.4. The picture to the right has the same meaning except that we set σ= 5 and display SP (φ, µ)<0.In both pictures it is obvious that the region where concentration is a stable outcome in the 3-region model (lighter area) contains that in the 2-region model (darker area). Numerical results also indicate that the difference in the critical areas between the two regions gets smaller as σincreases and/or µdecreases. Proposition 3.8. As σapproaches infinity or µapproaches zero, the stability conditions for the concentration configuration in the 2-region and 3-region models become identical. Proof. If the difference DSP in (3.4) equals zero, then the conditions for stability in the 2 and 3-region FE models coincide. Indeed, we have: lim µ→0DSP =−φ+φ= 0 and lim σ→∞ DSP =−φ+φ= 0, concluding the proof. Intuitively, and not surprisingly, what is here patent is that a smaller weight of the manufacturing sector in the whole economy reduces the relevance of the number of regions considered in that same economy, for the purpose of determining industry location. The same can be said in the case of a manufacturing sector whose variety in its good Xis less valued by consumers. Note however that, by proposition 3.6, the limit cases correspond to a scenario where concentration is never stable in either the 2-region or the 3-region model. 25
3.2 Stability of total dispersion Stability of an outcome where skilled workers remain equally divided across the three regions, i.e. (h1, h2, h3) = 1 3,1 3,1 3, is determined by the eigenvalues of the Jacobian matrix of the system in (3.1): J= ∂∆ω11 3,1 3 ∂h1 ∂∆ω11 3,1 3 ∂h2 ∂∆ω21 3,1 3 ∂h1 ∂∆ω21 3,1 3 ∂h2 . The region under whose perspective we are going to analyze the stability of total dispersion is a matter of choice, since the regions are all identical. Invoking this symmetry between regions, if h1=h2and ω1(h1, h2) = ω2(h2, h1), then ∂ω1 ∂h11 3,1 3=∂ω2 ∂h21 3,1 3 and ∂ω1 ∂h21 3,1 3=∂ω2 ∂h11 3,1 3. Furthermore, it is obvious that ∂¯ω ∂h11 3,1 3=∂¯ω ∂h21 3,1 3. Therefore, the Jacobian Jat total dispersion corresponds to: J= α β β α . Concerning the real wage average ¯ωwe provide the following results: Proposition 3.9. Configurations of the form hi=1−hj 2, with 0< hi<1, entail ∂¯ω ∂hi= 0. Proof. See Appendix C. Proposition 3.10. The weighted real wage average ¯ωattains a critical value when skilled workers are equally dispersed across regions. Proof. By proposition 3.9, we can conclude that both ∂¯ω ∂h11 3,1 3and ∂¯ω ∂h21 3,1 3are zero, since h1=1−h2 2and also h2=1−h1 2. Because both partial derivatives are equal to zero, it means that the real wage average is at a critical value at total dispersion. We can now simplify αand βin the Jacobian: β=∂ω1 ∂h21 3,1 3and α=∂ω1 ∂h11 3,1 3. 26
Lemma 3.11. At 1 3,1 3,1 3we have ∂ωi ∂hj = 0,∀i, j 6=i. Proof. Assume, by way of contradiction, that ∂ω1 ∂h21 3,1 3>0.It must follow that, for a small ε > 0,ω11 3,1 3+ε, 1 3−ε> ω11 3,1 3,1 3.However, the real wage in one region is invariant to the permutation of the share of skilled workers in the other two regions. Therefore, ω11 3,1 3+ε, 1 3−ε=ω11 3,1 3−ε, 1 3+ε. But saying that ω11 3,1 3−ε, 1 3+ε> ω11 3,1 3,1 3is a contradiction. Hence, ∂ω1 ∂h21 3,1 3= 0.Symmetry establishes an analogous result for ∂ω2 ∂h1. On account of the previous propositions and lemma, we are able to rewrite the Jacobian matrix at 1 3,1 3as: J= α0 0α . The matrix has a double eigenvalue equal to αand total dispersion is equivalent to: α=∂ω1 ∂h11 3,1 3<0.(3.5) Stability of total dispersion depends only on whether a small increase in the share of skilled workers in region ileads to a decrease in the real wage in that same region. In accordance with the definition of stable equilibria, as explained in the beginning of this section, the previous statement is tantamount to saying that total dispersion can only be stable if, after an exogenous migration to one of the regions, say region 2, the real wage falls under the average real wage level, thus inducing skilled workers to migrate back to the regions they left (˙ h1<0) and restore the initial distribution. This is exactly what the condition in (3.5) stands for. Of course, since total dispersion is a fully symmetric equilibrium, this enables us to generalize the latter result considering any region. Thus, we need only to study what happens when there is exogenous migration to one of the regions, as it is the same as if it happened to any of the other two regions. 27
The condition for stability of total dispersion is tantamount to: ∂ω1 ∂h11 3,1 3<0⇔∂w1 ∂hi Pµ 1−∂Pµ 1 ∂hi w1<0⇔ ∂w1 ∂h1 w1 < ∂Pµ 1 ∂h1 Pµ 1 . This inequality enables us to relate stability of dispersion in terms of semi-elasticities. Skilled workers remain equally dispersed across the three regions if an increase in the percentage of skilled workers in a region induces a percentage change in the nominal wage smaller than the corresponding percentage change in the real prices. In other words, a loss in real purchase power due to an increase of the share of skilled workers hi in a region leads to an exodus of some skilled workers from that region until the initial share of skilled workers is restored, that is, until hi=1 3.13 Proposition 3.12. Total dispersion is a stable equilibrium if and only if: BP (φ) = µ2(φ−1) + (σ−1)σ(φ−1) + µ(−1+2σ)(1 + 2φ)<0.(3.6) Proof. See Appendix C.1. Here, the critical value φbsuch that BP (φb)=0is called the “break point”, following the terminology of Fujita et al. (1999): BP (φb) = 0 ⇔φb=µ+µ2−σ−2µσ +σ2 −2µ+µ2−σ+ 4µσ +σ2⇔ ⇔φb=(µ−σ)(1 + µ−σ) µ2+ (−1 + σ)σ+µ(−2+4σ).(3.7) It is also the only zero since (3.6) is clearly linear in φ. Stability of total dispersion happens for values of BP (φ)such that φ < φb. That said, 13Dividing both sides of the inequalities by h1renders a similar relation in terms of elasticities. In the present case, however, semi-elasticities fit better with intuition, insofar as h1is the percentage of skilled workers in region 1. 28
if φbis negative, total dispersion is never a stable outcome. Like Forslid and Ottaviano (2003), we rule out this possibility by finding the threshold value of σsuch that: φb≥0⇔ (µ−σ)(1 + µ−σ)≥0⇔ σ≥1 + µ. Hence, the “no-black-hole” condition (σ > 1 + µ) implies that φbis positive. A high φ, corresponding to low transport costs, naturally discourages dispersion in favour of concentration. It seems reasonable to claim that a higher percentage of local income spent on manufactures and a higher preference for variety of manufactures also favour total dispersion of skilled workers. Proposition 3.13. As σapproaches infinity or µapproaches zero, total dispersion is a stable outcome for all values of φ. Proof. Considering the value φbin (3.7) we have the following limits: lim σ→∞ φb(µ, σ) = lim σ→∞ (µ−σ)(1 + µ−σ) µ2+ (−1 + σ)σ+µ(−2+4σ)= lim σ→∞ −1−2µ+ 2σ −1+4µ+ 2σ= 1 and lim µ→0φb(µ, σ) = lim µ→0 (µ−σ)(1 + µ−σ) µ2+ (−1 + σ)σ+µ(−2+4σ)=σ(σ−1) (−1 + σ)σ= 1. Since total dispersion is a stable outcome for φ < φb= 1 and φ∈(0,1), we can say that total dispersion is always a stable outcome as µapproaches zero and σapproaches infinity. 29
3.2.1 Comparing total dispersion in the 3-region and 2-region models We compared stability conditions of full agglomeration outcomes in the 2 and 3-region models. These, although suggestive, do not extend to a comparison for the stability of total dispersion. The following propositions in this section allow us to fill this gap. Proposition 3.14. The parameter region for which concentration is stable in the 2region model contains that of the 3-region model. Proof. In the article by Forslid and Ottaviano (2003), the authors determined the breakpoint value φb,2for the model with two regions which is presented as follows: φb,2=φw σ−1−µ σ−1 + µ, where φw∈(0,1) = σ−µ σ+µ is a threshold value of φabove (below) which the region with more skilled workers offers a higher (lower) skilled worker wage wi. Subtracting the 3-region break-point in (3.7) to this one yields: DBP =φb,2−φb=(µ−σ)(1 + µ−σ) (−1 + µ+σ)(µ+σ)−(µ−σ)(1 + µ−σ) µ2+ (−1 + σ)σ+µ(−2+4σ)⇔ ⇔DBP =µ(µ−σ)(1 + µ−σ)(−1+2σ) (−1 + µ+σ)(µ+σ) (µ2+ (−1 + σ)σ+µ(−2+4σ)). The denominator is clearly positive and, provided that the “no-black-hole” condition holds, the numerator is also positive. Hence, DBP >0,∀µ, σ. This means that the critical value φbis lower in the 3-region model compared to the 2-region model. Therefore, dispersion is a more likely outcome in the model with two regions. Not surprisingly, this result is the opposite to the one proved for the concentration configuration. Considering both results, we can now conclude that the 3-region model indeed favours concentration over dispersion when compared to the 2-region model by Forslid and Ottaviano (2003). 30
Lastly, we have the following proposition concerning, yet again, the limit cases of a nonexistent manufacturing industry and when the latter is very close to perfect competition. Proposition 3.15. As µapproaches zero or as σapproaches infinity, the stability conditions for the 2 and 3-region FE models coincide. Proof. We have the following limits: lim µ→0DBP =0 [(−1 + σ)σ]2= 0 and lim σ→∞ DBP = 0 14. Inasmuch as the difference between the break-points in the two models is zero considering the limits above, the conditions for stability of total dispersion become the same. Taking propositions 3.13 and 3.15 together, we can conclude that, considering the FE model with two or three regions, total dispersion is always a stable outcome when we are either approaching an economy absent of industry or consumers give almost no value to variety in good X. This fits well with intuition. 3.3 Simultaneity of concentration and total dispersion Numerical inspection of the conditions of concentration and total dispersion in (3.2) and (3.6) suggests that, for every pair (µ, σ), there exists φ∈(0,1), for which both total dispersion and concentration are stable equilibria. In order to illustrate this in a clear way, we first set µ= 0.4and display the two dimensional regions SP(φ, σ)<0 and BP (φ, σ)<0in one picture. Next, we plot SP (φ, µ)<0and BP(φ, µ)<0by setting σ= 0.5, in the same fashion as in the previous section. Figure 3.3 illustrates this situation. The intersection between the two regions is clear in both pictures and enables us to conclude that there is indeed a region in parameter space where both full agglomeration and total dispersion are stable outcomes. 14Notice that the numerator is a polynomial function of σof degree 3 and the denominator is a polynomial function of σof degree 4 and, therefore, the limit equals zero. 31
0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 Φ Σ Stabilityof Concentration and Total Dispersion, Μ=0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Φ Μ Stabilityof Concentration and Total Dispersion, Σ=5 Figure 3.3 – Simultaneity of stability of concentration and total dispersion. One region (bottom left on the picture to the left and upper right on the picture to the right) corresponds to stability of concentration and the other corresponds to stability of total dispersion. They overlap each other in the darker region in the middle, where both equilibria are stable outcomes. If this is true, then it must hold that φs< φb, because concentration is only stable for φ > φs, while the region for stability of total dispersion implies φ < φb.If it does hold, then we have hysteresis in location (Forslid and Ottaviano, 2003), because transport costs have to rise above the corresponding break point in order for total dispersion to be unstable, even if concentration is already a stable equilibrium. If both equilibria are to be simultaneously stable, we must have SP (φb)<0,∀µ, σ. Again, this seems to be the case, however, nonlinearity of SP in φmakes it impossible to prove this analytically. Further inspection also seems to suggest that the distance between φsand φbis bigger for parameter values near the “no-black-hole” condition. Proposition 3.16. There is an open subset in parameter space (φ, σ, µ)in which both concentration and total dispersion are stable outcomes. Proof. Consider the point in the parameter space (φ, σ, µ) = 3 5,5,2 5.15 15This point was chosen by numerical inspection. 32
At this point, we have: SP 3 5,5,2 5<0and BP 3 5,5,2 5<0. Therefore, for (φ, σ, µ) = 3 5,5,2 5, both concentration and total dispersion are stable equilibria. Since SP and BP are continuous functions of (φ, σ, µ), we know the signs persist in a open neighbourhood of 3 5,5,2 5. We know for sure that concentration and total dispersion can be stable outcomes at the same time, even though we cannot define analytically the region where this happens. This also holds in the case of the 2-region model by Forslid and Ottaviano (2003). Figure 3.4 seems to suggest that it is always possible to find a value of φ, for any pair of µand σ, such that both agglomeration and total dispersion are stable outcomes. One can see the surfaces corresponding to SP (φ) = 0 and BP (φ)=0. In between the surfaces, there is simultaneity of stability and total dispersion, which is less likely for higher values of φ. Concentration and total dispersion -viewpoint I 0.0 0.5 1.0 Φ 0.0 0.5 1.0 Μ 2 4 6 8 10 Σ Concentration and total dispersion -viewpoint II 0.0 0.5 1.0 Φ 0.0 0.5 1.0 Μ 2 4 6 8 10 Σ Figure 3.4 – We have the surfaces SP = 0 and BP = 0 in the parameter space (resp. top and bottom surfaces). Concentration and total dispersion are both stable in between both surfaces, where we have SP < 0∩BP < 0. However, the region between the surfaces is very thin, and becomes thinner for a high 33
in our analysis as the expressions are too complicated. Finally, when transport costs are low enough (φ= 0.9), total dispersion is no longer stable and the only possible outcome is that of full agglomeration of skilled workers in one of the three regions. Here, we have φb< φ = 0.9. This latter case is shown in the picture to the right, where we can see convergence to either vertices of the simplex. Figure 3.7 – The dynamics of the FE 3-region model. The pictures from the left to the right depict the change in the stability of equilibria as transport costs fall. Of course, we have φs< φb, as the results obtained in section 3.3 would suggest. Another important aspect is that, throughout the three cases described in figure 3.7, partial dispersion is always unstable, which is also not surprising. All these results corroborate those in Fujita et al (1999, chap. 6). In a general way, “subcritical pitchfork” bifurcations arise in most CP models where the regions concerned are fully symmetric. It is widely acknowledged that these bifurcations disappear when there are exogenous regional differences (e.g. see Baldwin et al., 2003). In particular, the FE model for two regions is no exception, as is shown by Forslid and Ottaviano (2003). However, discussions remain on whether the core-periphery pattern implicit in such bifurcations is a sustainable one, even when the model on which it is based is symmetric. In the article concerning bifurcations in migration dynamics by Berliant and Kung (2009), the authors criticize this view, on the grounds that the existence of bifurcations in symmetric CP models rest on a strategic parametrization of the model. While many 40
authors claim that it is exogenous asymmetries alone that break core-periphery patterns based on bifurcations, Berliant and Kung argue that, while this might be one of the reasons, variations in parameters that preserve the symmetry between regions in the models may also contribute to this. Therefore, bifurcations in CP models suffer from a lack of robustness. The parametrization used in this subsection is indeed a strategic choice to facilitate the explanation of bifurcations in the 3-region FE model. However, given the numerical evidence presented in section 3.3, it seems plausible that the dynamics implicit in the model will always undergo a “subcritical pitchfork” bifurcation, as there seems to be a region in parameter space, for φ∈(0,1), where both concentration and total dispersion are stable. Nevertheless, we have shown that this region is very thin, and hardly noticeable when σand µare very high. Of course, considering propositions 3.13 and 3.6 yet again, there are no bifurcations when the industrial sector faces perfect competition or when it is non-existent. 41
4 Note on possible extensions of the FE model to n-regions All the results we have obtained so far pertain to an analysis concerning an extension of the FE model by Forslid and Ottaviano (2003) to three regions. Even with an analytically solvable version of the original CP model by Krugman (1991b), expressions such as the real wages as explicit functions of the spatial distribution of skilled workers seem hardly intelligible and most derivations concerning the study of dynamics seem to imply cumbersome calculations. If nothing else, one could at least imagine that further augmenting the dimension by increasing the number of regions considered in the analysis might transform it into an unmanageable problem. In particular, some issues that are impossible to address analytically in the model with three regions are likely to remain unaddressable in a model with four or more regions. Nonetheless, expanding the study of Economic Geography to include more regions should not be overlooked if the objective is to explain geographical concentration of industry in a world comprised of many inter-related regions. Though an extension to the n-region case is not the object of this dissertation, the analysis of the 3-region model and its comparisons with the 2-region one might give some insights and serve as a motivation in the sense of extending the assumptions implicit in the FE model to a broader context. First of all, we have proved that concentration is more likely in the 3-region model (while total dispersion is less likely) in comparison with the 2-region model. This could mean that increasing the number of regions in the analysis would result in increasing the possibility of concentration (and the opposite respective to total dispersion). Castro et al. (2012), for instance, proved that dispersion is less likely in a 4-region model compared to the 3-region model (considering Krugman’s original CP model). Another extrapolation, which may seem very straightforward, is that of considering the limit cases of µapproaching zero and σapproaching infinity on a n-region model. It would seem reasonable to claim that, in both these cases, dispersion is always stable (while concentration never is). Further results would most likely require a formal application of the FE model to a n-region context. A good next step would be to consider nregions equally spaced around a circumference, with equal trade costs between each adjacent region, and build the FE model around these assumptions. Perhaps the simplifying and uncompromising assumptions implicit in the 42
FE model by Forslid and Ottaviano (2003) will allow for a complement on the results of the Racetrack Economy (Fujita et al., 1999), which provides the aforementioned set-up. 43
5 Conclusion Building on the 2-region FE model by Forslid and Ottaviano (2003), we have obtained both analytical and numerical results from a FE model with three regions, which we have constructed along the lines of Forslid and Ottaviano. These results corroborate those already obtained in previous works on 3-region Core Periphery models. We have shown that the 3-region FE model favours concentration in comparison with the 2region one. Furthermore, we have proved analytically that both concentration and full dispersion can be simultaneously stable and provided numerical evidence in that, for every pair (µ, σ), there exists φ∈(0,1) where this is possible, though this outcome is very unlikely. This means that, like the 2-region model, the 3-region FE model exhibits a core-periphery pattern based on a “subcritical pitchfork” bifurcation. We have also concluded numerically that the dispersion of skilled workers among two regions is not sustainable in a model with three regions, where it corresponds to an outcome of partial dispersion. All of these results are tantamount to those in the CP model with three regions in Castro et al. (2012), the difference being that, additionally, we were able to obtain explicit solutions for skilled wages and obtain relations between the relevant endogenous variables and the spatial distribution of skilled workers. Additionally, we proved that, when the manufacturing sector becomes irrelevant (resp. all local income is spent on agriculture) or it approaches perfect competition, migration decisions of skilled workers are the same in both the 2-region and the 3-region FE models. This, as was previously shown, occurs because there is convergence to zero of the critical values of the transport costs, where the stability of agglomeration and full dispersion changes. However, if this happens, concentration can never be a stable outcome, insofar as transport costs cannot fall below zero, while full dispersion, on the contrary, will always be a stable equilibrium. This come as no surprise, as a significant weight of the industrial sector and heterogeneity between the goods produced by it are among some of the factors essential to sustain a self-reinforced agglomeration process. Although the FE model is able to give us closed form solutions, the assumptions it makes still do not allow enough simplification to fully assess analytically the dynamic properties when its analysis is applied to three regions. Although there is more relevant evidence in that outcomes such as partial dispersion can never be a stable outcome in a 44
Core Periphery model, nonlinearity in transport costs concerning its stability conditions still makes it impossible to analytically exclude the possibility that skilled workers might equally disperse across two regions when there are three regions available to migrate. Whereas the 2-region FE model is useful to address issues beyond the explanation capability of the original CP model by Krugman (1991b), doubts remain about whether it is suitable to tackle the more complex case of nregions, though such an analysis could be interesting, as the higher the number of regions subject to the study of CorePeriphery models, the better the insight we will get on New Economic Geography. 45
Appendix A In order to study the dynamics of the Core-Periphery model with three regions it is necessary to find an expression for wi, the reason being that skilled workers migrate to the region where they have the highest indirect utility, so naturally, a comparison between real wages is required. Proof of proposition 2.1: Proof. We have the following linear system of equations determining the nominal wages, after (2.13): w1=µ σP3 j=1 φ1jL 3+wjHj Rj w2=µ σP3 j=1 φ2jL 3+wjHj Rj w3=µ σP3 j=1 φ3jL 3+wjHj Rj , which becomes, after some manipulation: w11−µ σ H1 R1−w2µ σ φH2 R2−w3µ σ φH3 R3=µ σ L 31 R1 +φ R2 +φ R3 w1−µ σ φH1 R1+w21−µ σ H2 R2−w3µ σ φH3 R3=µ σ L 3φ R1 +1 R2 +φ R3 w1−µ σ φH1 R1−w2µ σ φH2 R2+w31−µ σ φH3 R3=µ σ L 3φ R1 +φ R2 +1 R3. This may be written in matrix form as AW =B, where Astands for the coefficients matrix, Wthe vector of nominal wages wi, while Bis the column vector of independent terms in the right-hand side of the system of equations above. Applying Cramer’s Rule, the solution to this system is of the following form: wi=Dwi D, where the denominator Dstands for the determinant of matrix Aand Dwiis the determinant of the matrix obtained by replacing the i-th column of Aby the column 46
vector B. This method is useful since we only need to solve for a specific nominal wage, e.g w1, and easily deduce the remaining solutions applying an argument of symmetry. Finding an expression for Dfirst, we have: D=1−µ σ H1 R11−µ σ H2 R21−µ σ H3 R3+ 2 −µ σφH2 R2−µ σφH3 R3−µ σφH1 R1− −1−µ σ H1 R1−φµ σ H2 R2−µ σφH3 R3−−µ σφH1 R11−µ σ H2 R2−µ σφH3 R3− −−µ σφH1 R1−µ σφH2 R21−µ σ H3 R3 = 1 −µ σH1 R1 +H2 R2 +H3 R3+µ2 σ21−φ2H1H2 R1R2 +H1H3 R1R3 +H2H3 R2R3 −µ3 σ32φ3−3φ2+ 1H1H2H3 R1R2R3 ⇔D= 1 −µ σ 3 X j=1 Hj Rj +µ2 σ21−φ2H1H2 R1R2 +H1H3 R1R3 +H2H3 R2R3− −µ3 σ32φ3−3φ2+ 1 3 Y j=1 Hj Rj ,(5.1) which is obviously invariant under any distribution of skilled workers across the regions, since it is a common denominator for every solution of the nominal wage wi. The numerator of w1,Dw1, can be calculated as follows: Dw1=µ σ L 3a1−µ σ H2 R21−µ σ H3 R3+µ σ L 3cµ σφH3 R3µ σφH2 R2+µ σ L 3bµ σφH2 R2µ σφH3 R3 −µ σ L 3c1−µ σ H2 R2−µ σφH3 R3−µ σ L 3aµ σφH2 R2µ σφH3 R3−µ σ L 3b−µ σφH2 R21−µ σ H3 R3 =µ σ L 3ah1−µ σH2 R2 +H3 R3i+µ2 σ2 L 3φbH2 R2 +cH3 R3+ +a+cφ2+bφ2−cφ −aφ2−bφµ3 σ3 L 3 H2H3 R2R3 , ⇔Dw1=µ σ L 3a+µ σhH2 R2 (φb −a) + H3 R3 (φc −a)i+a+cφ2+bφ2−cφ −aφ2−bφµ3 σ3 H2H3 R2R3 where a=1 R1 +φ R2 +φ R3,b=φ R1 +1 R2 +φ R3and c=φ R1 +φ R2 +1 R3.Simplifying 47
to eliminate a,band c, we end up with: Dw1=µ σ L 3(1 R1 +φ R2 +φ R3+µ σ"φ(φ−1) H2+H3 R2R3 +φ2−1 R1H2 R2 +H3 R3#+ +µ2 σ22φ3−3φ2+ 1H2H3 R1R2R3) ⇔Dw1=µ σ L 3 3 X j=1 φ1j Rj +µ σ"φ(φ−1) H2+H3 R2R3 +φ2−1 R1H2 R2 +H3 R3#+ +µ2 σ22φ3−3φ2+ 1H2H3 R1R2R3)(5.2) The expression for the nominal wage in region 1is obtained dividing equation (5.2) by equation (5.1): w1= µ σ L 3P3 j=1 φ1j Rj +µ σφ(φ−1) H2+H3 R2R3 +φ2−1 R1H2 R2 +H3 R3+µ2 σ22φ3−3φ2+ 1H2H3 R1R2R3 1−µ σP3 j=1 Hj Rj +µ2 σ2(1 −φ2)H1H2 R1R2 +H1H3 R1R3 +H2H3 R2R3−µ3 σ3(2φ3−3φ2+ 1) Q3 j=1 Hj Rj . One can see that, under a given distribution of H,w1(H1, H2, H3) = w1(H1, H3, H2), which is a consequence of the existing symmetry in region 2and region 3, since there is nothing to distinguish between the two regions. An analogous argument can be made concerning region 1and any of the other two regions. This means that the nominal wage in region iis invariant in the distribution of skilled workers in the other two regions. Symmetry among the regions also asserts that w1(H1, H2, H3) = w2(H2, H1, H3) = w3(H3, H1, H2). Thus, we can formulate a general expression for the numerator of the nominal wage, Dwi: Dwi=µ σ L 3 3 X j=1 φij Rj +µ σ φ(φ−1) Pk6=iHk Qk6=iRk +φ2−1 RiX k6=i Hk Rk + +µ2 σ22φ3−3φ2+ 11 RiY k6=i Hk Rk , i ={1,2,3}. It follows that the nominal wage wiis the quotient between the latter equation and the 48
determinant Din (5.1): wi= µ σ L 3 3 X j=1 φ1j Rj +µ σ φ(φ−1) X k6=i Hk Y k6=i Rk +φ2−1 RiX k6=i Hk Rk +µ2 σ22φ3−3φ2+ 11 RiY k6=i Hk Rk 1−µ σ 3 X j=1 Hj Rj +µ2 σ2(1 −φ2)H1H2 R1R2 +H1H3 R1R3 +H2H3 R2R3−µ3 σ3(2φ3−3φ2+ 1) 3 Y j=1 Hj Rj . Appendix B Proof of proposition 3.9: Proof. We have at most three different types of configurations (h1, h2,1−h1−h2)when hi=1−hj 2and 0< hi<1. First, if 0< h1=1−h2 2<1, we have: 1−h2 2, h2,1−h2 2!= (a, b, a). Second, we have 0< h2=1−h2 2<1: h1,1−h2 2,1−h2 2!= (b, a, a). Finally, if h2=h1, we end up with: 1−h2 2,1−h2 2,1−h2 2!= (a, a, a). 49
thus yielding: (φ−1) 1 2φ+ 1 (1 −φ)1 + µ2 σ2−µ σ 2+2φ+ 5φ2 2φ+ 1 +µ 1−σ1 + φ−1 2φ+ 1 2µ σ+µ2 σ2 φ−1 2φ+ 1 <0. Rewriting the inequality after placing the expression on the left-hand side under the common denominator (−1 + σ)(σ+ 2σφ)2entails: (1 −φ)(σ+µ(−1 + φ)+2σφ)µ2(−1 + φ)+(−1 + σ)σ(−1 + φ) + µ(−1+2σ)(1 + 2φ) (−1 + σ)(σ+ 2σφ)2<0⇔ (1 −φ)µ2(−1 + φ)+(−1 + σ)σ(−1 + φ) + µ(−1+2σ)(1 + 2φ)<0⇔ µ2(φ−1) + (σ−1)σ(φ−1) + µ(−1+2σ)(1 + 2φ)<0. C.2 Partial Dispersion Proof of Proposition 3.17: Proof. As our model is specified, the configuration (H1, H2, H3) = 0,H 2,H 2is the same as (h1, h2) = 0,1 2.This means region 1is the chosen to be left out of skilled workers, so the endowment His equally distributed across regions 2and 3. First of all, note the implications this has on ri.We have: r1=φand r2, r3=1 + φ 2. Stability of this equilibrium requires ξand βin to be negative. It has been shown in Section 3.3 that the stability conditions pertain to: ∂ω2 ∂h20,1 2<0, corresponding to β < 0,and: ω10,1 2< ω20,1 2, 56
which refers to ξ < 0. The first condition states that: ∂w2 ∂h20,1 2Pµ 20,1 2−∂Pµ 2 ∂h20,1 2w20,1 2<0, since price indexes have to be positive. Recalling ∂wi ∂hi, it follows: ∂w2 ∂h20,1 2= ∂Dw2 ∂h20,1 2D0,1 2−∂D ∂h20,1 2Dw20,1 2 D20,1 2. After substitution of 0,1 2in ∂D ∂h20,1 2, rendering it equal to zero, we have: ∂w2 ∂h20,1 2=∂Dw2 ∂h20,1 2D−10,1 2. Hence, we are able to rewrite the condition for stability of partial dispersion in the same fashion as we did for total dispersion: ∂Dw2 ∂h20,1 2Dw−1 20,1 2<∂Pµ 2 ∂h20,1 2P−µ 20,1 2. First, we have: ∂Dw2 ∂h20,1 2=1 H µ σ L 3(φ−1) 1−φ r2+µ σ−φ r2−2φ r2+φ−1 r2= =1 H µ σ L 3(φ−1) 41−φ (1 + φ)2−µ σ42φ+ 2 (1 + φ)2⇔ ⇔∂Dw2 ∂h20,1 2=1 H µ σ L 3 1 r2(φ−1) (1 −φ)−µ σ(2φ+ 2). 57
As for Dw2, it goes: Dw20,1 2=µ σ L 3 1 H(1 + 1 + φ r+µ σ"1 2(φ−1) 1 r+1 2 φ2−1 r2#) =µ σ L 3 1 H3 + 1 2 µ σ(φ−1) 1 r+(φ−1)(φ+ 1) r2 =µ σ L 3 1 H3 + µ σ3φ−1 φ+ 1⇔ ⇔Dw20,1 2=µ σ L H1 + µ σ φ−1 φ+ 1. Lastly, it is straightforward to see that: ∂Pµ 2 ∂h20,1 2P−µ 20,1 2=µ r 1−φ 1−σ. Thus we can finally determine the condition β < 0: β < 0⇔ 1 H µ σ L 3 1 r2(φ−1) (1 −φ)−µ σ(2φ+ 2)<µ r 1−φ 1−σ µ σ L H1 + µ σ φ−1 φ+ 1⇔ 4φ−1 (φ+ 1)2(1 −φ)−µ σ(2φ+ 2)<µ 1 + φ 1−φ 1−σ1 + µ σ φ−1 φ+ 1. Placing everything under a common denominator yields, after some manipulation: −2Lµ(−1 + φ)3µ2(−1 + φ) + 2(−1 + σ)σ(−1 + φ) + µ(−2−4φ+σ(5 + 7φ)) 3H(−1 + σ)σ2(1 + φ)2<0. The denominator is positive and so is −2Lµ(−1 + φ). Hence, we end up with: β= 3µ2(−1 + φ) + 2(−1 + σ)σ(−1 + φ) + µ(−2−4φ+σ(5 + 7φ)) <0. The second condition for stability of partial dispersions (ξ < 0) implies: w1 w20,1 2<Pµ 1 Pµ 20,1 2⇔ w1 w20,1 2< φ r! µ 1−σ . 58
where: w1 w2 =Dw1Dw−1 2DD−1⇔ w1 w2 =Dw1 Dw2 . Beginning with region 1we have: Dw10,1 2=µ σ L 3 1 H1 φ+2φ r+µ σφ(φ−1) 1 r2+φ2−1 φr +µ2 σ22φ3−3φ2+ 11 φ 1 4 1 r2. More tedious algebra renders: Dw10,1 2=Lµ(−µ+σ+ (µ+σ)φ)(σ+µ(−1 + φ)(1 + 2φ) + σφ(1 + 4φ)) 3Hσ3φ(1 + φ)2. We should rewrite Dw2: Dw20,1 2= 1 + µ σ φ−1 φ+ 1 ⇔ Dw20,1 2=Lµ(−µ+σ+ (µ+σ)φ) Hσ2(1 + φ). It is now easier to obtain the quotient Dw1 Dw2: Dw1 Dw20,1 2=Lµ(−µ+σ+ (µ+σ)φ)(σ+µ(−1 + φ)(1 + 2φ) + σφ(1 + 4φ))Hσ2(1 + φ) 3Hσ3φ(1 + φ)2Lµ(−µ+σ+ (µ+σ)φ)⇔ Dw1 Dw20,1 2=σ+µ(−1 + φ)(1 + 2φ) + σφ(1 + 4φ) 3σφ(1 + φ). Finally, the condition ξ < 0becomes: σ+µ(−1 + φ)(1 + 2φ) + σφ(1 + 4φ) 3σφ(1 + φ)<2φ 1 + φ µ 1−σ⇔ σφ(1 + φ)σ+µ(−1 + φ)(1 + 2φ) + σφ 1+4φ−32φ 1 + φ(1 + φ) <0⇔ σ+µ(−1 + φ)(1 + 2φ) + σφ "1+4φ−32φ 1 + φµ 1−σ (1 + φ)#<0. 59
Hence, the partial dispersion equilibrium is stable if and only if: ξ < 0≡σ+µ(−1 + φ)(1 + 2φ) + σφ 1+4φ−32φ 1+φµ 1−σ(1 + φ)<0. β < 0≡3µ2(−1 + φ) + 2(−1 + σ)σ(−1 + φ) + µ(−2−4φ+σ(5 + 7φ)) <0. 60
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