Agglomeration and Growth with and without Capital Mobility

Transcription

1 Agglomeratio ad Growth with ad without Capital Mobility Richard Baldwi Rikard Forslid Philippe Marti Giamarco Ottaviao Frédéric Robert-Nicoud HWWA DISCUSSION PAPER 130 Hamburgisches Welt-Wirtschafts-Archiv (HWWA) Hamburg Istitute of Iteratioal Ecoomics 001 ISSN

2 The HWWA is a member of: Wisseschaftsgemeischaft Gottfried Wilhelm Leibiz (WGL) Arbeitsgemeischaft deutscher wirtschaftswisseschaftlicher Forschugsistitute (ARGE) Associatio d Istituts Europées de Cojocture Ecoomique (AIECE)

3 Agglomeratio ad Growth with ad without Capital Mobility Richard Baldwi a Rikard Forslid b Philippe Marti c Giamarco Ottaviao d Frédéric Robert-Nicoud e a Graduate Istitute of Iteratioal Studies (Geeva) ad CEPR (Lodo) b Uiversity of Stockholm ad CEPR (Lodo) c Uiversity of Lille-1, CERAS-ENPC (Paris) ad CEPR (Lodo) d Uiversità di Bologa ad CEPR (Lodo) e Lodo School of Ecoomics This paper was preseted by Philippe Marti at the HWWA workshop Ecoomic Geography ad Regioal Growth Theory ad Evidece, Hamburg, March 3/ It is the draft of a chapter for a book o Public Policies ad Ecoomic Geography. The subject of this paper is assiged to the HWWA s research programme Europea Itegratio ad Spatial Developmet.

4 HWWA DISCUSSION PAPER Edited by the Departmet EUROPEAN INTEGRATION Head: Dr. orad Lammers Hamburgisches Welt-Wirtschafts-Archiv (HWWA) Hamburg Istitute of Iteratioal Ecoomics Öffetlichkeitsarbeit Neuer Jugferstieg Hamburg Telefo: 040/ Telefax: 040/ Iteret: Richard Baldwi Graduate Istitute of Iteratioal Studies (Geeva), CEPR (Lodo) Rikard Forslid Uiversity of Stockholm, CEPR (Lodo) Philippe Marti Uiversity of Lille-1, CERAS-ENPC (Paris), CEPR (Lodo) Giamarco Ottaviao Uiversità di Bologa, CEPR (Lodo) Frederic Robert-Nicoud Lodo School of Ecoomics

5 Abstract This paper presets a simple framework i which the locatio ad the growth rate of ecoomic activities are edogeous ad iteract. We show that the ature of the equilibrium ad of the relatio betwee growth ad locatio depeds fudametally o whether capital is assumed to be mobile (i which case we iterpret it as physical capital) or immobile (huma capital). I the first case, with costat returs to scale, growth ad locatio are idepedet ad o divergece or covergece process takes place. We show that ewly created firms ca relocate to the poor regio, eve though there is always a higher share of firms i the rich regio, if the idustry is competitive ad if the retur to capital is low. With immobile capital, a process of covergece betwee regios takes place whe trasactio costs o goods are sufficietly high but a process of catastrophic agglomeratio occurs whe these costs are sufficietly high ad regioal iequality is ot affected betwee regios. With localized techological spillovers, higher spatial cocetratio of ecoomic activities spurs growth, whether capital is mobile or ot. This implies that lowerig trasactio costs o goods ca spur growth but icrease regioal iequality. Lowerig trasactio costs o trade i techologies betwee regios may icrease both regioal equality ad growth.

6 Cotets Page I INTRODUCTION 7 II THE CASE WITHOUT LOCALIZED SPILLOVERS: GROWTH MATTERS FOR GEOGRAPHY 9 1 The basic framework of growth ad agglomeratio 9 Perfect capital mobility 14 3 No capital mobility Stability of the symmetric equilibrium The Core-Periphery equilibrium III THE CASE WITH LOCALIZED SPILLOVERS: GEOGRA- PHY MATTERS FOR GROWTH (AND VICE VERSA) 7 1 The case of perfect capital mobility 8 The case without mobility: the possibility take-off ad agglomeratio 34 Appedix I 41 Appedix II 41 Refereces 4 Graph 1: The orther shares of expediture ad capital: the stable case 19 Graph : The orther shares of expediture ad capital: the ustable case 0 Graph 3: Stability properties of equilibria 3 Graph 4: Equilibrium growth, agglomeratio ad icome iequality 31 Graph 5: Fallig trasactio costs o goods: Stability properties of equilibria i the presece of localized spillovers 37 Graph 6: Fallig trasactio costs o ideas: Stability properties of equilibria i the presece of localized spillovers 40

7 I INTRODUCTION Spatial agglomeratio of ecoomic activities o the oe had ad ecoomic growth o the other had are processes difficult to separate. Ideed, the emergece ad domiace of spatial cocetratio of ecoomic activities is oe of the facts that uzets (1966) associated with moder ecoomic growth. This strog positive correlatio betwee growth ad geographic agglomeratio of ecoomic activities has bee documeted by ecoomic historias (Hoheberg ad Lees, 1985 for example), i particular i relatio to the idustrial revolutio i Europe durig the ieteeth cetury. I this case, as the growth rate i Europe as a whole sharply icreased, agglomeratio materialized itself i a icrease of the urbaizatio rate but also i the formatio of idustrial clusters i the core of Europe that have bee by ad large sustaied util ow. The role of cities i ecoomic growth ad techological progress has bee emphasized by urba ecoomists (Hederso, 1988, Fujita ad Thisse, 1996), developmet ecoomists (Williamso, 1988) as well as by ecoomists of growth (Lucas, 1988). At the other had of the spectrum, as emphasized by Baldwi, Marti ad Ottaviao (001), the growth takeoff of Europe took place aroud the same time (ed of eighteeth cetury) as the sharp divergece betwee what is ow called the North ad the South. Hece, growth sharply accelerated (for the first time i huma ecoomic history) at the same time as a dramatic ad sudde process of agglomeratio took place at the world level. Less dramatically ad closer to us, Quah s results (1996) suggest also a positive relatio betwee growth ad agglomeratio. He fids that amog the Cohesio group of coutries (Greece, Spai, Portugal ad Irelad, though there are o Irish regioal data), the two coutries that have achieved a high rate of growth ad coverged i per capita icome terms towards the rest of Europe (Spai ad Portugal) have also experieced the most marked regioal divergece, Portugal beig the coutry to have exhibited the sharpest icrease i regioal iequalities. By cotrast Greece, which has a low growth rate ad has ot beefited from a tedecy to coverge with the rest of Europe, has ot experieced a rise i regioal iequalities. A recet study by INSEE (1998) shows also that the coutries with a per capita GDP level above the Europea Uio average also experiece above-average regioal disparities. These studies are cosistet with the results of De la Fuete ad Vives (1995), for istace, buildig o the work of Esteba (1994), ad Marti (1998) who suggest that coutries have coverged i Europe but that this process of covergece betwee coutries took place at the same time as regios iside coutries either failed to coverge or eve diverged. 7

8 Hece, these empirical results poit to the iterest of studyig growth ad the spatial distributio of ecoomic activities i a itegrated framework. From a theoretical poit of view, the iterest should also be clear. There is a strog similarity betwee models of edogeous growth ad models of the ew ecoomic geography. They ask questios that are related: oe of the objectives of the first field is to aalyze how ew ecoomic activities emerge through techological iovatio; the secod field aalyzes how these ecoomic activities choose to locate ad why they are so spatially cocetrated. Hece, the process of creatio of ew firms/ecoomic activities ad the process of locatio should be thought as joit processes. From a methodological poit of view, the two fields are quite close as they both assume (i some versios) similar idustrial structures amely, models of moopolistic competitio. I this chapter, we will attempt to clarify some of the liks betwee growth ad agglomeratio. Partly, the iterest of such coectio will be to explai the ature of the observed correlatio betwee growth ad agglomeratio. We will aalyze how growth alters the process of delocatio. I particular, ad cotrary to the fudametally static models of the ew ecoomic geography, we will see how spatial cocetratio of ecoomic activities may be cosistet with a process of delocatio of firms towards the poor regios. We will also show that the growth process ca be at the origi of a catastrophic agglomeratio process. Oe of the surprisig features of the rugma (1991) model, was that the itroductio of partial labor mobility i a stadard ew trade model could lead to catastrophic agglomeratio. We will show that the itroductio of edogeous growth i the same type of stadard ew trade model could lead to the same result. The advatage is that all the results are derived aalytically i the edogeous growth versio. I this first part, the causality lik will be from growth to locatio: growth may be at the origi of a agglomeratio process. The relatio betwee growth ad agglomeratio depeds crucially o capital mobility. Without capital mobility betwee regios, the icetive for capital accumulatio ad therefore growth itself is at the heart of the possibility of spatial agglomeratio with catastrophy. I the absece of capital mobility, some results are i fact familiar to the New Ecoomic Geography (Fujita, rugma ad Veables, 1999): a gradual lowerig of trasactio costs betwee two idetical regios first has o effect o ecoomic geography but at some critical level iduce catastrophic agglomeratio. I the model preseted i this chapter, i the absece of migratio, catastrophic agglomeratio meas that agets i the south have o more private i- 8

9 cetive to accumulate capital ad iovate. We show that capital mobility elimiates the possibility of catastrophic agglomeratio ad is therefore stabilizig i this sese. This is i sharp cotrast with labor mobility which we kow to be destabilizig. However, capital mobility also makes the iitial distributio of capital betwee the two regios a permaet pheomeo so that both the symmetric ad the Core-Periphery equilibria are always stable. Oe iterestig fidig i this chapter is that a commo very simple threshold level of trasactio costs determies i) whe the symmetric equilibrium looses stability ad whe the Core-Periphery gais stability i the absece of capital mobility or with imperfect capital mobility: ii) the directio of capital relocatio betwee the poor ad the rich coutry with capital mobility. I a secod sectio of this chapter, we will cocetrate o the opposite causality ruig from spatial cocetratio to growth. For this, we will itroduce localized spillovers which will imply that the spatial distributio of firms will have a impact o the cost of iovatio ad the growth rate. This chapter uses modified versios of Baldwi (1999), Baldwi, Marti ad Ottaviao (000) ad Marti ad Ottaviao (1999). The first two papers aalyze models of growth ad agglomeratio without capital mobility. I cotrast to the first paper which uses a exogeous growth model, this chapter aalyses edogeous growth. I cotrast to the secod paper, we restrict our attetio to the case of global techology spillovers. The last paper presets a model of growth ad agglomeratio with perfect capital mobility. II THE CASE WITHOUT LOCALIZED SPILLOVERS: GROWTH MATTERS FOR GEOGRAPHY 1 The basic framework of growth ad agglomeratio Cosider a world ecoomy with two regios (orth ad south) each with two factors (labor L ad capital ) ad three sectors: maufactures M, traditioal goods T, ad a capital-producig sector I. Regios are symmetric i terms of prefereces, techology, trade costs ad labor edowmets. The Dixit-Stiglitz M-sector (maufactures) cosists of differetiated goods where productio of each variety etails a fixed cost (oe uit of ) ad a variable cost (a M uits of labor per uit of output). Its cost fuctio, therefore, is π +w a x M i, where π is 's retal rate, w is the wage rate, ad x i is total output of a typical firm. Traditioal goods, which are assumed to be homogeous, are produced by 9

10 the T-sector uder coditios of perfect competitio ad costat returs. By choice of uits, oe uit of T is made with oe uit of L. Regioal labor stocks are fixed ad immobile, so that we elimiate oe possible source of agglomeratio. Each regio's is produced by its I-sector. I is a memoic for iovatio whe iterpretig as kowledge capital, for istructio whe iterpretig as huma capital, ad for ivestmet-goods whe iterpretig as physical capital. Oe possible iterpretatio of the differece betwee the situatio of capital mobility ad oe of capital immobility is that i the first case we view as physical capital (mobility the meas the delocatio of plats) or as kowledge capital that ca be marketable ad tradable through patets. The secod case, capital immobility, would be more cosistet with the iterpretatio of huma capital. I this case, labor immobility implies capital immobility. The I-sector produces oe uit of with a I uits of L, so that the margial cost of the I sector, F, is w a I. Note that this uit of capital i equilibrium is also the fixed cost of the maufacturig sector. To idividual I-firms, a I is a parameter, however followig Romer (1990) ad Grossma ad Helpma (1991), we assume a sector-wide learig curve. That is, the margial cost of producig ew capital declies (i.e., a I falls) as the sector's cumulative output rises. May justificatios of this learig are possible. Romer (1990), for istace, ratioalizes it by referrig to the o-rival ature of kowledge. We ca summarize these assumptios by the followig: L I W W * = ; F = wai ; ai = 1/ ; = + a (1) I where ad * are the orther ad souther cumulative I-sector productio levels. Note that we assume that spillovers are global: the North lears as much from a iovatio made i the south tha i the orth 1. I sectio II of this chapter, we will itroduce localized techological spillovers. Followig Romer (1990) ad Grossma ad Helpma (1991), depreciatio of kowledge capital is igored. Fially, the regioal 's represet three quatities: regio-specific capital stocks, regio-specific cumulative I- sector productio, ad regio-specific umbers of varieties (recall that there is oe uit of per variety). The growth rate of the umber of varieties, o which we will focus, is therefore: / = g. 1 The ext sectio aalyzes the case of localized spillovers. 10

11 We assume a ifiitely-lived represetative cosumer (i each coutry) with prefereces: 1 1 1/ σ * + ρt 1 α α 1 1/ σ U = l ; ; C e Qdt Q = CY CM M = ci di () t= 0 i= 0 where ρ is the rate of time preferece, ad the other parameters have the usual meaig. Utility optimizatio implies that a costat fractio of total orther cosumptio expediture E falls o M-varieties with the rest spet o Y. Norther optimizatio also yields uitary elastic demad for T ad the CES demad fuctios for M varieties. The optimal orther cosumptio path also satisfies the Euler equatio which requires E / E = r ρ (r is the orth's rate of retur o ivestmet) ad a trasversality coditio. Souther optimizatio coditios are isomorphic. O the supply side, free trade i Y equalizes omial wage rates as log as both regios produce some T (i.e. if α is ot too large). Takig home labor as umeraire ad assumig Y is freely traded, we have w=w*=1. As for the M-sector, we choose uits such that a M =1-1/σ so that we get the usual pricig rules. With moopolistic competitio, equilibrium operatig profit is the value of sales divided by σ. Usig the goods market equilibrium ad the optimal pricig rules, the operatig profits are give by: w αe π = B w σ * * αe π = B σ w w ; B s ; B * s se + φ(1 s φse + φ(1 s φ(1 se ) + ) φs + 1 s 1 se + ) φ s + 1 s (3) Where s E E/ E w is orth s share of world expediture E w. s is the share of firms which are located i the orth. Whe capital is immobile, this share is the share of capital owed by the Norther regio: s. φ is the usual trasformatio of trasactio costs. Also, B is a memoic for the 'bias' i orther M-sector sales sice B measures the extet to which the value of sales of a orther variety exceeds average sales per variety worldwide (amely, αe w / w ). There are may ways to determie optimal ivestmet i a geeral equilibrium model. Tobi's q-approach (Tobi, 1969) is a powerful, ituitive, ad well-kow method for doig just that. Baldwi ad Forslid (000) have show how to use Tobi s q i the 11

12 cotext of ope ecoomy edogeous growh models. The essece of Tobi's approach is to assert that the equilibrium level of ivestmet is characterized by the equality of the stock market value of a uit of capital which we deote with the symbol v ad the replacemet cost of capital, F. Tobi takes the ratio of these, so what trade ecoomists would aturally call the M-sector free-etry coditio (amely v=f) becomes Tobi's famous coditio q =v/f=1. Calculatig the umerator of Tobi's q (the preset value of itroducig a ew variety) requires a discout rate. I steady state, E / E = 0 i both atios, so the Euler equatios imply that r=r * = ρ. Moreover, the preset value of a ew variety also depeds upo the rate at which ew varieties are created. I steady state, the growth rate of the capital stock (or of the umber of varieties) will be costat ad will either be the commo g=g* (i the iterior case), or orth's g (i the core-periphery case). I either case, the steady-state values of ivestig i ew uits of are: π v = ρ + g ; v * * π = ρ + g (4) It ca be checked that the equality, v=f, is equivalet to the arbitrage coditio preset i edogeous growth models such as Grossma ad Helpma (1991). The coditio of v π o arbitrage opportuity betwee capital ad a asset with retur r implies: r = +. v v O a ivestmet i capital of value v, the retur is equal to the operatig profits plus the chage i the value of capital. This coditio ca also be derived by statig that the equilibrium value of a uit of capital is the discouted sum of future profits of the firm with a perpetual moopoly o the productio of the related variety. The free etry coditio i the iovatio sector esures that the growth rate of the value v of capital is equal to growth rate of the margial cost of a iovatio, F, which due to itertemporal spillovers is g. With r = ρ, we get the regioal q's: q = F π ; q * = * π ( ρ + g) F( ρ + g) (5) w w To see this, use the world labor market equilibrium: + ( α)e + g which 1 σ 1 L = αe 1 σ says that world labor supply ca be used either i the maufacturig sector, the traditioal sector or the iovatio sector. It implies that a steady state with costat growth oly exists if E w itself is costat.

13 Usig the defiitio of F, the margial cost of iovatio, Tobi s qs are: q = π w ; q ( ρ + g) ( ρ + g) * * π = w (6) Note that i the case of global spillovers, the commo growth rate is easy to fid as it does ot deped o geography. For this, we ca use the world labor market equilibrium: w σ 1 w L = αe + ( 1 α)e + g, which states that labor ca be used either i the σ maufacturig sector (remember that the uit labor requiremet i this sector is ormalized to (σ-1)/σ), i the Y sector or i the iovatio sector ( is the productio of w the sector per uit of time ad F=1/ w is the labor requiremet i the iovatio sector). The world level of expediture is simply give by: E w = L + ρ which states that, with uit itertemporal elasticity of substitutio, world expediture is equal to world labor icome plus ρ times steady-state world wealth, F w =1. To fid the growth rate, we therefore do ot eed to kow aythig about the locatio of firms or the distributio of capital. Usig these equatios, the growth rate of the umber of varieties ad of the world capital stock is give by: α σ α g = L ρ (7) σ σ Usig equatios (6) ad (7) as well as the defiitio of world icome, it is easy to check that q=b ad q*=b*. Fially, a simple equilibrium relatio exists betwee s E ad s, the orther share of expeditures ad the orther share of capital. It ca be show that optimizig cosumers set expediture at the permaet icome hypothesis level i steady state. That is, they cosume labor icome plus ρ times their steady-state wealth, F = s, ad, F*= (1- s ) i the orth ad i the south respectively. Hece, E = L+ρ s, ad E* = L+ρ(1-s ). Thus, we get: 13

14 s E E E w = L + ρ s L + ρ ρ(s = 1/ + 1) ( L + ρ) (8) This relatio betwee s E ad s, ca be thought as the optimal savigs/expediture fuctio sice it is derived from itertemporal utility maximizatio. The ituitio is simply that a icrease i the orther share of capital icreases the permaet icome i the orth ad leads therefore to a icrease i the orther share of expeditures. From ow o two roads are ope: 1) we ca let capital owers decide where to locate productio. Capital is mobile eve though capital owers are ot, so that profits are repatriated i the regio where capital is owed. I this case, s, the share of firms located i the orth ad s, the share of capital owed by the orth, may be differet. s is the edogeous ad determied by a arbitrage coditio that says that locatio of firms is i equilibrium whe profits are equalized i the two regios. Because of capital mobility, the decisio to accumulate capital will be idetical i both regios so that the iitial share of capital owed by the orth, s, is permaet ad etirely determied the iitial distributio of capital owership betwee the two regios. ) a secod solutio is to assume that capital is immobile. Presumably, this would be the case if we focus o the iterpretatio of capital beig huma (coupled with immobile agets). I this case, the locatio of productio, s, is pied dow by capital owership: s = s. Because the case of capital mobility elimiates the possibility of a catastrophe similar to the ew ecoomic geography model ad from that poit of view is simpler, we start with it. Perfect capital mobility With perfect capital mobility, operatig profits have to be the same i both regios which also implies that the value of capital has to be the same i both regios. Hece, π =π* ad q = q* =1. This, together with the assumptio of costat returs to scale, ad the assumptio of global spillovers (implyig that the cost of iovatio is the same i both regios) meas that the two regios will accumulate capital at the same costat rate so that ay iitial distributio of capital is stable ad o catastrophic sceario ca ufold (see Marti ad Ottaviao, 1999). The reaso is that the retur to capital accu- 14

15 mulatio is the same i both regios ad therefore the icetive to accumulate are idetical i the two regios whe capital is perfectly mobile. With capital mobility, a obvious questio arises: where does capital locate? Capital owed i oe regio ca be located elsewhere. We have that +* = +*, but (*) does ot eed to be equal to (*). Agai, the arbitrage coditio, which implies that profits across regios eed to be equal for firms to be idifferet betwee the two locatios, pis dow the equilibrium locatio of firms. Usig equatio (3), we get that there is o more icetive for relocatio whe the followig relatio betwee s ad s E is verified: (1 + φ)(se 1) s = 1/ + (9) (1 φ) This is a example of the home market effect: firms locate i the large market (the market with the highest share of expediture) because of icreasig returs i the moopolistic competitio sector. Usig equatio (8), we get the equilibrium relatio betwee the share of firms located i the orth (s ) ad the share of capital owed by the orth (s ): s ρ(1 + φ)(s 1) 1/ + 0 s 1 1 = ( φ )( L + ρ ) (10) Note also that if the iitial distributio of capital i the orth is such that s > ½, the more firms will be located i the orth tha i the south: s > ½. A icrease i the share of capital i the orth, s, iduces relocatio to the orth as it icreases expediture ad market size there. Note also that lower trasactio costs (higher φ) will reiforce the home market effect, implyig that a uequal distributio of capital owership will traslate i a eve more uequal distributio of firms. Remember that, because of free capital mobility, the growth rate of capital is the same i both regios so that s is etirely determied by the iitial exogeous distributio of capital ad is costat through time. It also implies that the share of icome ad expeditures i the orth does ot deped o the locatio of firms. This elimiates a potetial likage that will prove crucial whe we relax the assumptio of perfect capital mobility. Hece, the equilibrium described by (10) is always stable. I particular, the symmetric equilibrium where s = s = 1/, is always stable for ay level of trasactio costs o trade i goods. To see this, oe ca aalyze the effect of a exogeous icrease i s, by a small amout ad 15

16 check the impact of this perturbatio o the ratio of profits i the orth to profits i the south. That is, ask the questio whether a icrease i geographic cocetratio i the orth decreases or icreases the icetive to relocate i the orth. The symmetric equilibrium is stable, if ad oly if (π/π*)/ s is egative. Ideed this is the case for all positive levels of trasactio costs sice, evaluated at the equilibrium geography: ( π ) ( 1 φ ) π * = s ( 1+ φ ) s E 1 (1 s E < 0 ) Evaluated at the equilibrium give by (10), a exogeous icrease i the share of firms located i the orth always decreases relative profits there, so that it leads firms to go back to the south. The locatio equilibrium determied i (10) is always stable. The reaso is that whe more firms locate i the orth, this icreases competitio there (ad decreases it i the south). We will come back to this local competitio effect. There are several iterestig questios that we ca aalyze i this framework. First, i this model with growth meaig with costat creatio of ew firms, do we have relocatio of firms towards the orth or towards the south? I ecoomic geography models without growth, idustrial cocetratio implies that firms are destroyed i the south ad relocated i the orth. Here, the relocatio story is richer. To see what is the directio of relocatio we eed to look at the differece betwee the share of capital owed by the orth ad the share of firms located i orth. This is give by: s s = [ L(1 φ) ρφ ][ s 1] ( 1 φ )( L + ρ) (11) I the symmetric equilibrium, where both regios are edowed origially with the same amout of capital there is o relocatio of course. If the iitial distributio of capital is such that s > 1/, so that the orth is richer tha the south, the the directio of the capital flows is ambiguous ad depeds o the sig of the followig expressio: L(1-φ)- ρφ. If this expressio is positive, the s > s so that some of the capital owed by the orth relocates to the south. The reaso of the ambiguity of the directio of locatio is that two opposite effects are preset: a local competitio effect that makes the poor capital regio attractive because firms (each usig oe uit of capital) istalled there face less competitio; a capital icome effect that makes the rich regio attractive because due to its high level of icome ad expediture the rich regio represets a larger 16

17 market. The first effect domiates whe trasactio costs are high (φ is low) because the, the local competitio effect is strog as the souther market is protected from orther competitio. Also if the umber of workers is high, the share of icome that comes from capital is low relative to labor icome so that the capital icome effect is small. O the cotrary, whe the rate of time preferece is high, the retur to capital is high also which makes the capital rich regio more attractive. There is a threshold level of trasactio costs that determies the directio of capital flows. It is give by: φ CP L = L + ρ (1) Whe trasactio costs are below this level, relocatio takes place towards the south ad vice-versa. The reaso why we attach CP (for Core-Periphery) to this threshold will become clear later whe we aalyze a versio of the Core-Periphery model, as we will see that this threshold value comes back agai ad agai. A iterestig feature here is that cocetratio i the orth (s ad s > ½), is compatible with relocatio of firms from orth to south (s < s ) whe φ < φ CP. This comes from the itroductio of growth ad the fact that a larger umber of ewly created firms are created ad owed by the orth tha by the south; the competitio effect the kicks i ad teds to drive s below s. A secod iterestig questio we ca ask is the followig: whe is that that whe all capital is owed by the orth, all firms are also located i the orth ad o relocatio occurs towards the south? That is, whe is it that whe s = 1, the s = 1? We ca already thik of this situatio as a Core-Periphery oe. Usig equatio (10), it is easy to see that this will be the case whe φ > φ CP as defied i equatio (1). Hece, with capital mobility, whe trasactio costs are low eough the Core-Periphery is a stable equilibrium i the sese that if all the capital is owed by the orth, all firms are also located i the orth. 17

18 3 No capital mobility Restrictig capital mobility (together with the assumptios of labor mobility) has two implicatios. First, the umber of firms ad the umber of uits of capital owed i a regio are idetical: s = s. Secod, because the arbitrage coditio of the previous sectio does ot hold, profits may be differet i the two regios. This i tur implies that, cotrary to the previous sectio, the two regios may ot have the same icetive to accumulate capital so that the iitial owership of capital does ot eed to be permaet. This meas that the aalysis will be quite differet from the previous sectio. We will ask the followig questios which are the usual oes i the ew ecoomic geography models. Startig from a equal distributio of capital, the symmetric equilibrium, we will determie uder which circumstaces it remais a stable equilibrium. The we will look at the Core-Periphery equilibrium ad agai ask whe this equilibrium is stable. 3.1 Stability of the symmetric equilibrium We first cosider iterior steady states where both atios are ivestig, so q =1 ad q * =1. Usig (3) ad (6) i (7), q = q * =1 ad imposig s = s we get: s (1 + φ)(se 1) = 1/ + (1 φ) (13) Note that this is the same relatios as the oe i (9) except that it ow determies the locatio of capital owership ad ot oly the locatio of productio. Together with equatio (8), this defies a secod positive relatio betwee s E ad s. The ituitio is that a icrease i the orther share of expediture raises demad for locally produced maufactured goods more tha for goods produced i the south. This relative icrease i orther demad icreases profits i the orth ad therefore the margial value of a extra uit of capital. I other words, the umerator of Tobi s q icreases i the orth. Hece, this raises the icetive to iovate there ad the orth ideed icreases its share of capital s. The ituitio is therefore very close to the home market effect except that it iflueces here the locatio of capital accumulatio. Together with the optimal savig relatio of (8), it is easy to check that the symmetric solutio s E = s = ½ is always a equilibrium, i particular it is a equilibrium for all levels of trasactio costs. The symmetric equilibrium is the uique equilibrium for which both regios accumulate 18

19 capital (q = q* =1). However, the fact that there are two positive equilibrium relatios betwee s E ad s, the share of expeditures ad the share of capital i the orth, should war us that the symmetric equilibrium may ot be stable. Ideed, i this model a 'circular causality' specific to the presece of growth ad capital immobility teds to destabilize the symmetric equilibrium. It ca be related to the well-kow demad-liked cycle i which productio shiftig leads to expediture shiftig ad vice versa. The particular variat preset here is based o the mechaism first itroduced by Baldwi (1999). There are several ways to study the symmetric equilibrium's stability. We ca first graph the two equilibrium relatios betwee s E ad s, the Permaet Icome relatio (call it PI) give by equatio (8) ad the Optimal Ivestmet relatio (call it OI) give by equatio (13). I the case where the slope of the PI relatio is less tha the OI relatio we get graph 1. At the right of the permaet icome relatio, s E, the share of expeditures i the orth, is too low give the high share of capital owed by the orth (agets do ot cosume eough). The opposite is true at the left of the PI relatio. At the right of the optimal ivestmet relatio, s, the share of capital i the orth, is too high give the low level of s E, the share of expeditures i the orth (agets ivest too much). The opposite is true is at the left of the OI relatio. This graphical aalysis suggests that i this case the symmetric equilibrium is stable. Graph 1: s E The orther shares of expediture ad capital: the stable case OI PI 1/ 1/ s 19

20 I the case where the slope of the PI relatio is steeper tha the OI, the the same reasoig leads to graph. This suggests that i this case, the symmetric equilibrium is ustable. Graph : The orther shares of expediture ad capital: the ustable case PI s E 0I 1/ 1/ s Accordig to this graphical aalysis, the trasactio cost below which the symmetric equilibrium becomes ustable is exactly the oe for which the slope of the PI curve equals the slope of OI curve. This turs to be the threshold level φ CP give by equatio (1). To gai more ituitio o this result, we ca also study the symmetric equilibrium's stability i a differet ad more rigorous way. We ca aalyze the effect of a exogeous icrease s, by a small amout ad check the impact of this perturbatio o Tobi s q, allowig expediture shares to adjust accordig to (8). The symmetric equilibrium is stable, if ad oly if q/ s is egative: i this case, a icrease i the share of orther capital lowers Tobi's q i the orth (ad therefore the icetive to iovate) ad raises it i the south (by symmetry q/ s ad q * / s have opposite sigs). Thus whe q/ s < 0, the perturbatio geerates self-correctig forces i the sese that the 0

21 icetive to accumulate more capital i the orth falls ad icreases i the south. If the derivative is positive, the icrease i the share of capital i the orth reiforces the icetive to accumulate more capital i the orth: the symmetric equilibrium is ustable i this case. Differetiatig the defiitio of q with respect to s, we have: q / q s 1 φ s = 1 φ + s s = 1/ s = 1/ E ( 1 φ ) ( 1+ φ ) (14) This expressio illustrates the two forces affectig stability. The first term is positive, so it represets the destabilizig force, amely the demad-liked oe, which ca also be iterpreted as a capital icome effect as a icrease i the capital share of the orth icreases its capital icome ad its expediture share. This effect was abset i the case of capital mobility. The egative secod term reflects the stabilizig local-competitio effect which was the oly preset i the capital mobility case. Clearly, reducig trade costs (a icrease i φ) erodes the stabilizig force more quickly tha it erodes the destabilizig demad-likage. Usig (8) to fid s E / s = ρ/[l+ρ], the critical level of φ at which the symmetric equilibrium becomes ustable is defied by the poit where (14) switches sig. It is easy to check that agai this critical level is give by φ CP of equatio (1). The appedix uses stadard stability tests ivolvig eigevalues ad derives the same result. Whe trade costs are high the symmetric equilibrium is stable ad gradually reducig trade costs produces stadard, static effects more trade, lower prices for imported goods, ad higher welfare. There is, however, o impact o idustrial locatio, so durig a iitial phase, the global distributio of idustry appears uaffected. As trade freeess moves beyod φ CP, however, the equilibrium eters a qualitatively distict phase. The symmetric distributio of idustry becomes ustable, ad orther ad souther idustrial structures begi to diverge; to be cocrete, assume idustry agglomerates i the orth. Sice s caot jump, crossig φ CP triggers trasitioal dyamics i which orther idustrial output ad ivestmet rise ad souther idustrial output ad ivestmet fall. Moreover, i a very well defied sese, the south would appear to be i the midst of a 'vicious' cycle. The demad likages would have souther firms lowerig employmet ad abstaiig from ivestmet, because souther wealth is fallig, ad souther wealth is fallig sice souther firms are failig to ivest. By the same logic, the orth would appear to be i the midst of a 'virtuous' cycle. 1

22 3. The Core-Periphery equilibrium I additio to the symmetric equilibrium, a core-periphery outcome (s =0 or 1, but we will focus oly o the secod oe where the orth gets the core) ca also exist. For s =1 to be a equilibrium, it must be that q =1 ad q*<1 for this distributio of capital owership. Cotiuous accumulatio is profitable i the orth sice v=f, but v*<f* so o souther aget would choose to setup a ew firm. Defiig the Core-Periphery equilibrium this way, it implies that it is stable wheever it exists. Usig (3), (6) ad (7), (8), q* with s =1 simplifies to: q * (1 + φ ) L + φ ρ = (L + ρ) φ (15) If q* is less tha 1 whe s =1, the the Core-Periphery equilibrium exists ad is stable as there is o icetive for the south to iovate i this case. The threshold φ that solves q*=1 defies the startig poit of the core-periphery set. Agai, this threshold is φ CP of equatio (1). This implies that at the level of the trasactio costs for which the symmetric equilibrium becomes ustable, the Core-Periphery becomes a stable equilibrium. Whe trasactio costs are high eough, the Core-Periphery equilibrium is ot a stable equilibrium: i this case the south would have a icetive to iovate because the profits i the south are high eough. This is because eve though the souther market is small i this case (it has o capital icome i the Core-Periphery equilibrium), it is protected from orther competitio thaks to high trasactio costs. Whe trasactio costs are low eough, this protectio dimiishes ad the fact that the market i the south is small becomes more importat: i this case, above the threshold φ CP, it becomes o profitable to operate a firm i the south. Usig s =1, the remaiig aspects of the core-periphery steady state are simple to calculate. I particular, sice s =1, q=1, ad q*<1, we have that o labor is used i the iovatio or maufacturig sectors i the south ad all iovatio is made i the orth. Note that the core-periphery outcome (s =1) is reached oly asymptotically. This is because the stock of capital i the south does ot depreciate ad oce the level of φ CP is crossed, stays costat, whereas the stock of capital i the orth keeps growig at rate g.

23 Graph 3 summarizes the model s stability properties i a diagram with φ ad s o the axes: Graph 3: Stability properties of equilibria s Symmetric (stable) Core-Periphery (stable) 1/ 0 (o trade) φ CP Symmetric (ustable) 1 (free trade) φ Followig the traditio of the ew ecoomic geography we have aalyzed here the existece ad stability coditios of the symmetric ad Core-Periphery equilibria. I this simple model we ca go further ad aalyze what would happe if we started from a situatio i which the orth had more capital tha the south (1/ <s <1). It ca be checked, usig equatios (3), (6) ad (7) that i this case q <1 (ad q*>1) if: ( 1 s )( s 1/ )( 1 φ )[ L( 1 φ ) + ρ φ] < 0 that is if φ < φ CP. Hece, i this case, the orth would ot iovate (the large stock of capital implies a high degree of competitio) ad the south would iovate. Hece, if we start from such a iterior asymmetric equilibrium the oe would coverge back to the symmetric equilibrium as log as trasactio costs are high eough. If φ < φ CP, the the ecoomy coverges to the core-periphery equilibrium. 3

24 Cocludig remarks: If we compare the case of perfect capital mobility ad o capital mobility, we ca get the followig coclusios: - whe φ < φ CP, the absece of capital mobility leads to covergece betwee the two regios: if oe regio starts with more capital tha the other the, the two regios coverge to the symmetric equilibrium. O the cotrary, with capital mobility, ay iitial distributio of capital owership becomes permaet. However, some of the firms owed by the orth will relocate ad produce i the south. This will produce some sort of covergece i terms of GDP but ot i terms of GNP. - whe φ > φ CP, the absece of capital mobility leads to divergece betwee the two regios: asymptotically, whatever the iitial distributio of capital (with some advatage to the orth though), all the capital is accumulated ad owed by the orth. With capital mobility, as log as all the capital is ot etirely i the orth, some firms will still produce i the south. However, some of the souther capital will be delocated i the orth. Hece, i the case of mobile capital (physical or tradable iovatios such as patets), the key parameter for regioal icome distributio is the exogeous iitial distributio of capital. I the case of immobile (huma) capital, the key parameter is the level of trasactio costs. The regioal distributio of capital affects the log term regioal icome distributio oly to the extet that it determies which regio becomes the core, through a small iitial advatage i capital edowmets for example. To simplify matters we have used a model where oly oe type of capital exists. To make it more realistic, i particular for the Europea case, it would be iterestig to exted it ad take ito accout the differet atures of capital so that part of the capital is mobile ad part is ot. We have aalyzed the polar cases of o capital mobility ad full capital mobility ad have show that they are very differet. A atural questio is whether our aalysis ca be exteded to the itermediate case where some fiite trasactio costs exist o capital movemets. I short, the aswer is that with small trasactio costs the aalysis is idetical to the case of o capital mobility. To see this, remember that the iovatio sector is perfectly competitive so that i equilibrium if a regio has a iovatio sector the 4

25 margial cost of a iovatio is equal to its margial value, that is the value of the discouted profits of a moopolistic firm: there are o profits to be made i equilibrium i the iovatio sector. If a firm relocates its productio from oe regio to aother it will ecessarily make a loss: the reaso is that the value of discouted profits i the regio where it may relocate is pied dow to F, the commo margial cost, due to free etry i that regio. Hece, payig the trasactio, however small, will imply a loss. Hece, if we restrict ourselves to the case of positive trasactio costs o capital movemets, the equilibrium with imperfect capital mobility is essetially idetical to the case of o capital mobility. Does this imply that the case of perfect capital mobility, which is the oly oe i which actual capital movemets ca take place i this framework, is too special to be relevat? Remember that the reaso why trasactio costs o capital movemets elimiate ay trade i capital is that capital productio is perfectly competitive ad that capital is a homogeous good. Trade i capital could appear to be destabilizig despite the presece of such trasactio costs if these assumptios are relaxed, for example alog the lies of the model of Marti ad Rey (1999). 5

26 6

27 III THE CASE WITH LOCALIZED SPILLOVERS: GEOGRA- PHY MATTERS FOR GROWTH (AND VICE VERSA) I the previous sectio, we showed that growth could dramatically alter ecoomic geography i the sese that the process of accumulatio of capital could lead to catastrophic agglomeratio. However, geography had o impact o growth. This was due to the fact that we assumed global spillovers: the learig curve, which as i ay edogeous growth model, was at the origi of sustaied growth, was global i the sese that the orth ad the south would lear equally from a iovatio made i ay regio. I this sectio, we aalyze how localized spillovers give a role i growth to the geography of productio ad iovatio activities. The presece of localized spillovers has bee well documeted i the empirical literature. Studies by Jacobs (1969) ad more recetly by Jaffe et al. (1993), Coe ad Helpma (1995 ad 1997), Ciccoe ad Hall (1996) provide strog evidece that techology spillovers are either global or etirely localized. The diffusio of kowledge across regios ad coutries does exist but dimiishes strogly with physical distace which cofirms the role that social iteractios betwee idividuals, depedet o spatial proximity, have i such diffusio. Itroducig localized techological spillovers requires a mior modificatio to oe of the assumptios made i the previous sectio. Equatio (1) that described the iovatio sector assumed global spillovers i the sese that the margial cost of a iovatio, idetical i both regios, was: F= w a I = 1/ W, so that it was decreasig i the total stock of existig capital. Hece, this was similar to the assumptio of Grossma ad Helpma (1991) that past ivetios (the stock of ) had a positive effect o the productivity of R&D. Now suppose that these spillovers are localized i the sese that the cost of R&D i oe regio also depeds o the locatio of firms (capital). Hece, the orther cost of iovatio depeds more o the umber of firms located i the orth tha i the south so that equatio (1) becomes (takig ito accout that the wage rate is equal to (1): F = a I ; a I 1 W A ; A s + λ ( 1 s ) (16) where λ measures the degree of localizatio of techology spillovers. The lower λ the more localized these spillovers are or put it differetly, the higher the trasactio costs 7

28 o the mobility of ideas, techologies, iovatios If λ=1, we go back to the case of global spillovers. The cost fuctio of the iovatio sector i the south is isomorphic. Agai the case of perfect capital mobility is easier tha the case without capital mobility. Hece, we will start with the former followig some of the aalysis of Marti ad Ottaviao (1999) ad the describe the model without capital mobility followig Baldwi, Marti ad Ottaviao (000). 1 The case of perfect capital mobility There are three edogeous variables that we are iterested i: g, the growth rate which is commo to both regios; s, the share of firms that are producig i the orth; ad s E, the share of expediture i the orth which also ca be thought as a measure of icome iequality betwee orth ad south. Remember that with perfect capital mobility, s the share of capital i the orth is give by the iitial distributio of capital as the stocks of capital i both regios are growig at the same rate. We wat to fid the differet equilibrium relatios betwee these three edogeous variables. Due to localized spillovers, it is less costly to iovate i the regio with the highest umber of firms (which represet also capital or iovatios). This implies that, because of perfect capital mobility, all the iovatio will take place i the regio with a higher umber of firms. Remember that due to perfect competitio the value of a iovatio is equal to its margial cost. The shares of firms are perfectly tradable across regios (perfect capital mobility) so the value of capital (or firms) caot differ from oe regio to aother ad o iovatio will take place i the south. But the south will be able to simply buy (without trasactio costs) iovatios or capital produced i the orth. Hece, i the case whe s > 1/, that is whe the iitial stock of capital is higher i the orth tha i the south, we kow from the previous sectio that this will imply that more firms will be located i the orth (s > 1/) so that all iovatio will take place i the orth. I this case the world labor market equilibrium will be give by: g σ 1 W L = + α E + (1 ) E s + λ (1 s ) σ α W (17) Remember also that world expediture is give by: E W = L+ρF W. The value ad margial cost of capital is give by F i (16). The reaso agai is that the R&D sector is 8

29 perfectly competitive so that i equilibrium the margial cost of a ivetio (F) must be equal to its value v. Usig this ad equatio (17), we get the growth rate of capital g as a fuctio of s, our first equilibrium relatio: g αl σ σ α σ [ s + λ(1 s )] ρ 1/ < s 1 = (18) Compared to the growth rate derived i the previous sectio, this oe differs because of the presece of localized spillovers: spatial cocetratio of firms (a higher s ) implies a lower cost of iovatio ad therefore a higher growth rate. Note also that for a give geography of productio (a give s ), less localized spillovers (a higher λ) also implies a lower cost of iovatio i the orth (as the iovatio sector i the orth beefits more from spillovers of firms producig i the south) ad a higher growth rate. The arbitrage coditio cosistet with the assumptio of perfect capital mobility requires profits to be equalized i the two locatios so that π =π*=αe W / (σ W ). This gives the same relatio betwee s ad s E as i the previous sectio (equatio 9), which we called the home market effect ad which is the secod equilibrium relatio we will use: s (1 + φ)(se 1) = 1/ + (1 φ) (19) To fid the third equilibrium relatio, oe betwee s E ad g, remember that due to itertemporal optimizatio, E= L+ρv where v is the value of capital which itself is equal to the discouted value of future profits. Usig these relatios, it is easy to get the last equilibrium relatio: s E αρ(s 1) = 1/ + σ ( g + ρ) (0) Note that as log as s > 1/, that is, as log as the orth is iitially better edowed i capital tha the south, the s E > 1/, that is, icome per capita is higher i the orth tha i the south. Note also that icome iequality is decreasig i the growth rate. The reaso is that the value of capital is lower with higher growth because of more future competitio due to faster etry of ew firms. If the orth is relatively rich i capital 9

30 (s > 1/), the level of capital icome declies more i the orth tha i the south, leadig to decreasig icome iequality. The equilibrium characterized by these three relatios is stable for the same reasos as i the case of perfect capital mobility of the previous sectio. Capital mobility allows southerers to save ad ivest buyig capital accumulated i the orth (i the form of patets or shares). Hece, the lack of a iovatio sector does ot prevet the south from accumulatig capital: the iitial iequality i wealth does ot lead to selfsustaiig divergece. No circular causatio mechaism which would lead to a coreperiphery patter, as i the ew geography models of the type of rugma (1991), will occur. Usig equatios (17), (18) ad (19), the equilibrium is the solutio to a quadratic equatio give i appedix I. The equilibrium growth rate follows from equatio (17). Oe ca fid the trasactio cost such that relocatio goes from orth to south i the case where s > 1/ (which implies also that s > ½). s > s if: λl(1 s ) + Ls φ < λl(1 s ) + Ls + ρ Note that whe all the capital is owed by the orth (s =1), the the threshold level of trasactio cost is agai φ CP give i the previous sectio. Note also that i the less extreme case where s <1, less localized spillovers imply, everythig else costat, that relocatio will take place towards the south. The reaso is that less localized spillovers imply a lower cost of iovatio i the orth, ad therefore a lower value of capital of which the orth is better edowed with. Hece, less localized spillovers geerate, for a give distributio of capital, a more equal distributio of icomes ad expeditures ad therefore attract firms i the south. Aother way to say it is that less localized spillovers weake the capital icome effect which was described i the previous sectio. Oe could aalyze the properties of this equilibrium by aalyzig the equilibrium locatio s give i appedix. However, it is more revealig to use a graphical aalysis. Equatio (19) provides a positive relatio betwee s ad s E, the well kow home market effect. O graph 4, this relatio is give by the curve s (s E ) i the NE quadrat. Equatio (18) provides a positive relatio betwee g ad s. This is the localized spill- 30