Economic integration and agglomeration in a customs union in the presence of an outside region

Transcription

1 Deprmen of Economics Working Pper No. 46 Economic inegrion nd gglomerion in cusoms union in he presence of n ouside region Psqule Commendore Ingrid Kubin Crmelo Pergli Iryn Sushko Ocober 0

2 ECONOMIC INTEGRATION AND AGGLOMERATION IN A CUSTOMS UN- ION IN THE PRESENCE OF AN OUTSIDE REGION Psqule Commendore (Universiy of Nples Federico II, Ily) Ingrid Kubin (Vienn Universiy of Economics nd Business Adminisrion) Crmelo Pergli (Universiy of Bsilic, Ily) Iryn Sushko (Insiue of Mhemics, Nionl Acdemy of Sciences of Ukrine) Ocober 0 ABSTRACT: New Economic Geogrphy (NEG) models do no ypiclly ccoun for he presence of regions oher hn he ones involved in he inegrion process. We explore such possibiliy in Fooloose Enrepreneur (FE) model iming sudying he sbiliy properies of long-run indusril locion equilibri. We consider world economy composed by cusoms union of wo regions (regions nd ) nd n ouside region which cn be regrded s he res of he world (region ). The effecs of economic inegrion on indusril gglomerion wihin he cusoms union re sudied under he ssumpion of consn disnce beween he cusoms union iself nd he hird region. The resuls show h higher economic inegrion does no lwys implies he sndrd resul of full gglomerion of FE models. This incomplee gglomerion oucome is due o he fc h he periphery region keeps shre of indusril civiies in order o sisfy shre of exernl demnd. Th is, he deindusrilizion process brough bou by economic inegrion in he periphery of he union is miiged by he demnd of consumers living in he res of he world. In generl, he mrke size of he hird region ffecs he number of he long-run equilibri, s well s heir sbiliy properies. In ddiion o he sndrd oucomes of FE models, we describe he exisence of wo symmeric equilibri chrcerised by unequl disribuion of firms beween regions nd, wih no full gglomerion hough. Ineresingly, hese equilibri re sble nd herefore cn be regrded s likely long-run equilibrium se of he economy. Keywords: indusril gglomerion, hree-region NEG models, fooloose enrepreneurs. JEL clssificion: C6, F, F, R.

3 . Inroducion Will furher economic inegrion increse or reduce regionl dispriies? This is one of he core quesions in Europen policy discussions. The nlysis of inegrion res is one of he clssicl opics in rde heory (for n overview see, e.g., Krishn, 008); however, in he following we pply New Economic Geogrphy (NEG) frmework, since wih is emphsis on endogenous gglomerion processes i seems o be priculrly suied o nlyse he effecs on regionl dispriies. Economic inegrion represened by reducion in rde cos influences he blnce beween cenripel nd cenrifugl forces nd depending upon which force previl he long-run he spil disribuion of economic civiy my differ. Typiclly, NEG models llow for wo sndrd long-run oucomes: n equl disribuion of he mnufcuring civiy beween he regions or, wih sufficien reducion of rde coss, full gglomerion in one of he wo regions (see, for reviews, Fuji e l., 999; nd Bldwin e l., 00). The nlyic srucure of NEG models is inrinsiclly complex, herefore mny NEG models re cully confined o he nlysis of wo regions. However, for comprehensive sudy of inegrion res his is no sufficien: one hs o differenie les beween wo regions inside he inegrion re nd one region ouside. So fr smll srnd of lierure hs developed hree-region models hough. Wihin his lierure, Pluzie (00) shows h reducion in he exernl rde cos srenghens he gglomerive forces in he home counry wih wo regions. Similr resuls re pu forwrd by Alonso-Villr (999, 00), nd Monfor nd Nicolini (000). In conrs, Krugmn nd Livs Elizondo (996) rgue h reducion in he inernionl rnspor cos my fvour dispersion of economic civiy beween he wo regions in he home counry (in heir model he domesic dispersion force is due o lnd ren nd commuing coss nd is hus exogenous nd independen of rde coss). Brülhr e l. (004) nd Croze nd Koenig-Soubeyrn (004) inroduce more geogrphicl srucure ino he nlysis, s hey ssume h one of he home regions is border region, i.e. h i hs lower rnspor cos wr he ouside region hn he oher home region. Also in his frmework, reducion of he inernionl rnspor cos fvours gglomerion in he wo-regions home counry; in priculr (bu no lwys), gglomerion in he border region. Ineresingly, Brülhr e l. (004) poin ou h he size of he hird region mers for he resuls. Tking closer look o he bove menioned conribuions revels h hese sudies only ddress one pr of he issues hnd s hey only nlyse he effecs of closer inegrion wih he res of he world; however, neglecing he effecs of closer inegrion wihin he

4 inegrion re. In our pper, we re focussing on he ler issue while deliberely neglecing he former. We explore he effecs of rde inegrion beween wo (symmeric, home) regions ( nd ) in he presence of hird region (region ), h is upon consrucion mere ouside region. As firs resul we show h our consrucion implies h rnspor cos wih he hird region do no mer. Neverheless, he hird counry is imporn s (ouside) mrke nd is size influences he blnce beween cenripel nd cenrifugl forces beween he wo counries/regions inside he inegrion re. To pu i differenly: The effecs of rde liberlision beween he wo counries inside he inegrion re will depend upon he size of he hird region, upon he impornce of he ouside links for he inegrion re. In priculr, we show h wih n increse of he size of hird region he symmeric equilibrium beween he wo regions/counries wihin he inegrion re loses sbiliy lower vlue of he rde freeness inside he inegrion re. To pu i differenly: Sronger ouside links fvour gglomerion wihin he inegrion re. In ddiion, nd his is our min resul, we show h he size of he hird region lso influences wh hppens, if he symmeric equilibrium is unsble. Wih insbiliy, fcor mobiliy ses in leding o symmeries in he fcor llocion; his symmery chnges iself he srengh of he gglomerive nd deglomerive forces nd we show nlyiclly h for smller hird region he gglomerive forces ouweigh he deglomerive forces leding o full gglomerion in one of he regions inside he inegrion re; insed, for bigger hird region, symmeries weken he gglomerive processes nd srenghen he deglomerive forces nd inerior symmeric equilibr cn be esblished. This resul is imporn, firs, becuse i shows h even if wih reducion of rde coss wih he inegrion re he symmeric equilibrium loses sbiliy (s found lso in he ppers reviewed bove), he long-run oucome need no be full gglomerion, bu my lso be pril gglomerion (hus reducion of rde coss does no led o exreme regionl disperion). Second, he resul is imporn becuse i is one of he rre exmples in he NEG lierure h produces pril gglomerion s he oucome of n endogenous process. The reminder of he pper is orgnized s follows. Secion presens he bsic frmework of he model. In secion we chrcerize he shor-run equilibrium. Secion 4 dels wih our

5 complee dynmicl model, whose locl sbiliy properies re sudied in secion 5. Secion 6 repors preliminry resuls on globl dynmics. Secion 7 concludes.. Bsic frmework The economic sysem is composed of hree regions (r =,, ) nd wo secors. The rdiionl or griculurl secor (A) is loced in ll hree regions, wheres he mnufcuring secor (M) cn be loclised mos in wo regions (s =, ). Producion involves he use of wo fcors of producion, unskilled lbour (L) nd enrepreneurs (N). In he overll economy, he moun of unskilled lbour is L: shre is eqully disribued beween he mnufcuring regions nd nd he res is loced in region (he region wihou mnufcure), i.e. = = nd =. Unskilled lbour does no migre; he N exising enrepreneurs, insed, re mobile beween regions nd. Assuming h unskilled workers nd enrepreneurs possess he sme ses, we wrie he represenive consumer s uiliy funcion s follows: U = C C () µ µ A M where C A nd C M correspond o he consumpion of he homogeneous griculurl good nd of composie of mnufcured goods: C M n i= σ σ i = d () where d i is he consumpion of good i, n is he ol number of mnufcured goods nd σ > is he consn elsiciy of subsiuion; he lower σ, he greer he consumers se for vriey. The exponens in he uiliy funcion µ nd µ indice, respecively, he invrin shres of disposble income devoed o he griculurl nd mnufcured goods, wih 0 < µ <. Only lbour is used in he producion of he homogeneous griculurl good. One uni of (unskilled) lbour is used o produce one uni of he griculurl oupu, so h consn reurns previls. Moreover, we ssume h none of he regions hs enough lbour o engge 4

6 exclusively in he producion of he griculurl good, he so-clled non-full-specilizion condiion. The mnufcuring secors involve monopolisic compeiion s modelled by Dixi-Sigliz (977). In our conex, ech firm requires fixed inpu of n enrepreneur o opere nd β unis of unskilled lbour for ech uni produced. Since one enrepreneur is needed for ech firm, he ol number of firms lwys equls he ol number of enrepreneurs. Moreover, becuse of consumers preference for vriey nd incresing reurns in producion, firm would lwys produce vriey differen from hose produced by oher firms. I follows h he number of vrieies lwys equls he number of firms. Denoing he shre of enrepreneurs loced in region in period by λ nd by N he ol number of enrepreneurs, he number of regionl vrieies produced in period in region nd re n, = λn n, = ( λ) N () where 0 λ nd where, by ssumpion, no mnufcuring civiy occurs in region. Trnsporion of he griculurl produc beween regions is cosless. Trnspor coss for mnufcures ke n iceberg form: if one uni is shipped from region s o region r only T rs rrives, where Trs, r =,, nd s =,. Region nd re involved in rde greemen wheres he economic inegrion wih region is less deep. We model his spil rrngemen s follows: he hree regions re loced on he verices of isosceles ringle 5

7 Figure The disnce (rde brriers) beween regions nd is S (shor); he disnce beween nd nd nd is he sme nd is equl o L (long). Trnspor coss beween regions nd re T = T, S nd beween regions nd nd regions nd re T = T = TL wheret > T. Finlly, in order o simplify he noion, we inroduce he following rde L S freeness prmeers: φ = φ S, φ φ φl = =, where φ nd S T σ S φ nd where L T σ L φ < φ. L S. Shor-run generl equilibrium The shor-run equilibrium in period is chrcerized by given spil llocion of enrepreneurs cross he regions, λ r,. In shor-run generl equilibrium, which is esblished insnneously in ech period, supply equls demnd for he griculurl commodiy nd 6

8 ech mnufcurer mees he demnd for is vriey. As resul of Wlrs s lw, simulneously equilibrium in he produc mrkes implies equilibrium in he regionl lbour mrkes. Wih zero rnspor coss, he griculurl price is he sme cross regions. Denoing by Y he income of he overll economy, h (s confirmed below) is invrin over ime, ol expendiure on he griculurl produc is ( µ )Y. Assuming ( µ ) Y > mx L, L ll regions produce he griculurl commodiy, wheres ( µ ) Y > mx L, ( ) L implies h no single region is ble o sisfy ll he demnd for he griculurl good. Since compeiion resuls in zero griculurl profis, he shor-run equilibrium nominl wge in period is equl o he griculurl produc price nd herefore is lwys he sme cross regions. Seing his wge/griculurl price equl o, i becomes he numerire in erms of which he oher prices re defined. Fcing wge of, ech mnufcurer hs mrginl cos of β. Ech mximizes profi on he bsis of perceived price elsiciy of σ nd ses locl (mill) price p for is vriey, given by σ p = β σ (4) The effecive price pid by consumers in region r for vriey produced in region s is The regionl mnufcuring price index fcing consumers in region r is given by pt rs. σ σ σ r, = s rs s= P np T Under our ssumpions, we cn wrie σ σ σ σ, = + ( S ) P np np T σ σ σ σ, = + ( S ) P np T np σ σ σ σ σ, = L + L ( ) P n p T n p T nd herefore 7

9 σ σ r, r, P = N p (5) where, = λ + φs( λ),, = φλ S + λ,, = φl. The demnd fcing producer loced in region s is d Y P T p s P T Yp σ σ σ σ σ σ s, = µ r, r, rs = r, r, rs µ r= r= (6) We cn wrie: ( ) d = s P + s P T + s P T µ Yp d σ σ σ σ σ σ,,,,,,, s, s, s σ, σ µ Yp = + TS + TL,,, N s, s, s, µ Yp = + φs + φl,,, N s, s, µ Yp = + φs + s,,, N = ( s P T + s P + s P T ) µ Yp σ σ σ σ σ σ,,,,,,, s, s, s σ, σ µ Yp = TS + + TL,,, N s, s, s, µ Yp = + φs + φl,,, N s, s, µ Yp = + φs + s,,, N (7) where Y r, represens income nd expendiure in region r in period, sr, Yr, Y denoes region r s shre in expendiure in period nd r =,,. Shor-run generl equilibrium in region s requires h ech firm mees he demnd for is vriey. For vriey produced in region s, x = d (8) s, s, 8

10 where x s, is he oupu of ech firm loced in region s. From equion (4), he shor-run equilibrium opering profi per vriey/enrepreneur in region s is π px s, s, = pxs, βxs, = (9) σ Since profi equls he vlue of sles imes σ nd since ol expendiure on mnufcurers is µ Y, he ol profi received by enrepreneurs is µ Y σ. Tol income is Y = L+ ( µσ) Y, so h Y σ L = σ µ (0) Tol profi is herefore µ L ( σ µ ). Equion (0) confirms h ol income is invrin over ime. From (0), ( µ ) Y > mx L, L is equivlen o ( µ + σ µσ ( ) µ + σ µσ ) > ; nd ( µ ) > mx, ( ) min ( ), 0 equivlen o ( µ σ µσ ( ) µ σ µσ ) Y L L is min + ( ), + > 0.The former is (sufficien) non-full-specilizion condiion nd he ler is necessry one, where boh re expressed in erms of he uiliy prmeers. Using (4) o (0), he shor-run equilibrium profi in region s is deermined by he spil disribuion of enrepreneurs nd he regionl expendiure shres: σ σ σ p σ σ µ Y s, = Yr, Pr, Trs = sr, Pr, rs p r= σ r= σ π µ φ Under our ssumpions on rde coss cross regions, we cn wrie σ σ σ σ µ Y, = s, P, + s, P, S + s, P, L p ( ) π φ φ σ σ σ σ µ Y, = s, P, S + s, P, + s, P, L p ( ) π φ φ σ σ 9

11 or, lernively: µ Y s, s, s, µ Y s, s, π, = + φs + φl = + φs + s, () σn,,, σn λ + φs( λ) φλ S + λ µ Y s, s, s, µ Y s, s, π, = φs + + φl = φs + + s, () σn,,, σn λ + φs( λ) φλ S + λ Regionl incomes/expendiures re Y = L+ λ Nπ,, () Y = L+ ( λ ) Nπ,, (4) Y, ( ) = L (5) Y L λ Nπ Y Y Y,, = s = + (6) Y L λ Nπ σ µ µ s s,,,, = s, = + = + λ, + φs + s, Y Y Y σ σ,, (7) Y L ( λ ) Nπ σ µ µ s s,,,, = s, ( ) = + = + λ φs + + s, Y Y Y σ σ,, (8) Y, Y ( ) L σ µ = s, = = ( ) (9) Y σ Using (7) o (9) nd king ino ccoun h s, = s, s,, he shres in s, nd s,, cn be expressed in erms of λ : 0

12 s, σ µ µφsλ σ µ φ S + ( ) σ µλ, σ, = φ S σ µλ,, (0) s, ( ) σ µ µφs λ σ µ φ S + ( ) σ µ ( λ ), σ, = φ S σ µ ( λ ),, () Given h he griculurl price is, he rel income of n enrepreneur in region s is: ω s, πs, P µ s, = () Given h he hree regions shres in ol expendiure do no depend on φ L, from () nd () king ino ccoun (9), (0) nd (), we cn derive he following proposiion: Proposiion. Profi differenils re no ffeced by he disnce of region nd region from region. Therefore, chnge of he disnce of region nd/or region from region hs no impc on opering profis. This is becuse he demnd for he mnufcured goods is uniry elsic: he chnge in rde coss, vi φ L, deermines proporionl chnge in he price index in region nd similr bu inversely proporionl chnge in he quniy demnded, so he overll chnge of expendiures on mnufcuring in his region is zero. This is lso becuse, since region does no produce mnufcured vrieies chnge in φ L does no impc on price indices in region nd. This is resul h follows from our simple se-up, he ssumpions of CES subuiliy funcion for mnufcured goods nd no mnufcuring producion in region : chnge in rnspor coss owrds he ouside region hs no effec.

13 4. The enrepreneuril migrion hypohesis nd he complee dynmicl model The cenrl dynmic equion is bsed on he replicor dynmics, widely used in evoluionry gme heory: M ( ) ( ) ( ) ( ) T ( ) ω, λω, + λ ω, T λ λ = λ + γ = λ + γ ( λ) λω, + λ ω, + λ λ () where γ represens he migrion speed nd where T ( λ ) ω, =. According o (), he ω shre of enrepreneurs in region, M ( λ ), depends on comprison beween he rel income gined in h region nd he weighed verge of he incomes in region nd. Tking ino ccoun he consrin, 0, he complee dynmicl model is represened by he following piecewise smooh one-dimensionl mp: λ, 0 if M ( λ ) < 0 λ+ = Λ ( λ) = M ( λ) if 0 M ( λ) if M ( λ ) > (4) A long-run sionry equilibrium involves * * Λ ( λ ) = λ, where poin of he mp (5). There re hree ypes of fixed poins: * λ represens so-clled fixed i) he Core-Periphery equilibri re chrcerized by full gglomerion of mnufcuring in one region. These re: CP(0) λ = 0, corresponding o complee gglomerion in region, which gives M (0) = 0 ; nd gives M () =. CP λ () =, corresponding o full gglomerion in region, which ii) he symmeric equilibrium is chrcerized by n equl spli of he mnufcuring secor * beween regions nd : λ =, h gives M = nd 0 T = ; iii) he symmeric equilibri re chrcerized by incomplee gglomerion in one of he wo regions of he cusoms union, wih some indusry sill presen in he oher region. The following cses re possible:

14 Cse : no symmeric fixed poin exiss. Cse : wo symmeric fixed poins exis which re symmeric round /: λ, λ ; Cse : four symmeric fixed poins exis h re symmeric wo by wo round /: b b λ nd λ, λ. λ, These equilibri re obined by solving M ( λ i ) = λ i i i nd M ( λ ) = λ, corresponding o i i T ( λ ) = 0 nd T ( λ ) = 0, where i =, b. 5. Exisence nd locl sbiliy of sionry equilibri In his secion, we explore he locl sbiliy nlysis of he fixed poins lised bove. Due o he symmery of he mp Λ ( λ ), generl propery is h ech equilibrium (sionry, periodic or periodic) is symmeric o iself or noher equilibrium exiss h is symmeric o such equilibrium round o ; similrly, he bsins of rcion of ech equilibrium, s well s ny oher invrin se, lso enjoy his symmeric propery. In wh follows, his symmeric rule is pplied o he fixed poins (sionry equilibri) of he mp Λ ( λ ). We find he locl sbiliy properies of he CP equilibri CP λ () = nd CP(0) λ = 0, by evluing he one-side eigenvlues of he mp M ( λ ) in correspondence of hese equilibri: ( ) ( ) T < M ( ) = γ < + T, M ( ) γt( ) 0< 0 = + 0 < 0 From which 0 < T () <, γ < T (0) < 0, γ We explore he sbiliy properies of he CP equilibrium he oher CP equilibrium by symmery. CP λ () =. The sme resuls pply o

15 We hve h < M ( ) < for γ γ µ σ+ ( ) σ ( ) κφ < φ < κφ (6) where, for convenience, we se: φs σ φ nd κφ ( ) = ( σ µ )[ + φ ( )] + [ µ ( ) + σφ ] 0 φ <. ; nd where κφ ( ) > for for < σ < + µ < nd 0 φ <, he righ hnd side inequliy in (6) is lwys sisfied; for< + µ < σ i cn be shown h he righ hnd side inequliy in (6) is sisfied for sufficienly high vlues of φ nd violed for low vlues (hin: we re deling wih wo monooniclly decresing funcions of φ, he firs ends o infiniy for φ 0 nd i is equl o φ =, he second is posiive (nd lrger hn ) bu finie φ = 0 nd i is equl o φ =, since φ = he firs derivive of he firs funcion is smller in bsolue vlue hn he derivive of he second funcion, he wo funcions necessrily cross some φ T = φ, T where 0< φ < ).I is no possible o specify he corresponding bifurcion vlue for he rde T freeness prmeer φ = φ explicily. Noe h forσ > + µ, s φ crosses T φ from lef o righ, he mp undergoes so-clled border collision bifurcion: he CP equilibrium CP λ () = mees he upper brnch of he symmeric equilibrium gining sbiliy. Symmericlly, CP(0) λ = 0 mees he lower brnch of he symmeric equilibrium gining sbiliy. From his, we infer h he symmeric equilibrium mus hve lwys he sme locl sbiliy properies in he neighborhood of he CP equilibri (see Figures nd 4). Finlly, he lef hnd side inequliy in (6) is sisfied for sufficienly smll vlue of γ: γ < µ σ+ σ φ κφ ( ) (7) 4

16 When his ler condiion does no hold, 0< φ < φ < φ <, where φ CP() λ = is sble for nd φ cn only be obined numericlly by solving (7) wih n equliy sign. [ φ nd φ correspond o wo flip bifurcion poins, which re no visible due o he consrins of he mp Λ ( λ )]. Moving on o he symmeric equilibrium, is locl sbiliy requires h he eigenvlue of he mp M ( λ ) (which coincides o h of he mp Λ ( λ ) lie wihin he inervl (,) : ) evlued his equilibrium should γ < M = + T <, 4 which implies 8 < T < 0 γ (8) Concerning he inequliy on he righ hnd side of (8), i is sisfied for ( σ µ ) ( σ ) µ ( σ )( σ µ ) + µ ( µ + σ ) P φ < φ < for < σ < + µ P, i follows h φ < 0. Therefore, his inequliy cn never be sisfied; for µ σ > +, s φ crosses undergoes so-clled pichfork bifurcion. P φ from lef o righ, he mp Λ( λ ) A firs ineresing resul we my highligh is h P φ depends posiively on : ( )( )( ) ( σ )( σ µ ) µ ( µ σ ) φ P µ σ µ σ σ = >

17 which implies h he locl sbiliy of he symmeric equilibrium depends upon he dimension of he ouside region s follows: incresing he size of he hird region (reducing ) hs desbilizing effec on his equilibrium nd ends o fvor gglomerion. In order o sudy in deil he properies of he pichfork bifurcion, we firs redefine our cenrl mp o highligh he conrol prmeer we ineresed in, rde freenessφ, nd we verify how hese my chnge when noher crucil prmeer, he size of he hird region, chnges. We could replice he sme nlysis for ny oher prmeer. The redefined mp is where Z ( φ, λ ) ( φ, λ ) T ( φ λ ) = T + λ,. M ( φ, λ ) = λ + γ ( λ ) Z( φ, λ ), From he heory of dynmicl sysems (see Wiggins, 990), in correspondence of pichforfork bifurcion, h is, whenφ = φ nd λ =, he following condiions mus P * hold: (i) (ii) (iii) (iv) M φ P, = 0; φ M P φ = λ, 0; M P φ, 0 λ φ M P φ λ, 0. Moreover, he sign of he following expression cn be used o deermine on which side of P φ he wo brnches of symmeric equilibri, les iniilly, lie: (v) M P φ λ M P, > or < 0 φ, λ φ We hve supercriicl pichfork bifurcion when his expression is lrger hn zero nd subcriicl pichfork bifurcion when i is less hn zero. 6

18 We hve h: Z condiion (i), which corresponds o φ P, = 0, is verified due o he fc h he sym- φ meric equilibrium ω, = ω, nd ω, ω, φ, = φ, φ φ for nyφ ; condiion (ii) corresponds o T T P, φ, = 0. I cn be checked (vi clcul- ion) h his equliy holds for ny φ ; P φ λ λ condiion (iii) corresponds o T P φ, 0. Afer clculion we obin he following re- λ φ sul: µ ( σ ) ( σ )( σ µ ) + µ ( µ + σ ) ( σ µ )( σ ) ( µ + σ) ( σ )( σ µ ) + µ ( µ + σ ) > 0 which is lwys sisfied; condiion (iv) corresponds o, 0. Afer clculion we obin he following resul: T P φ λ { } ( ) ( ) ( ) + ( ) + ( ) ( ) + ( + ) + ( ) 4 µ σ σ µ σ σ µ σ σ µ µ µ σ µ σ ( σ ) ( µ + σ) ( σ )( σ µ ) + µ ( µ + σ ) This expression could be negive, posiive or zero depending on prmeer vlues. If i is differen from zero, given h 0< µ < < σ, he sign of (v) corresponds o h of he following expression: ( σ ) ( µ σ) + ( σ ) µ + ( σ ) ( σ µ ) + µ ( µ + σ ) µ + ( σ ) (9) This llows us o se he following proposiion : 7

19 Proposiion : I is possible o show h i exiss = such h (9) is posiive for < nd i is negive for >, wih 0< < forσ > + µ. Proof. Firs we rewrie (9) s A + B + C where: ( ) ( ) A σ µ σ < 0, ( ) ( ) ( ) B σ µ + σ σ µ ( < )0 ( ) ( ) C µ µ + σ µ + σ > 0. Therefore (9) dmis one posiive nd one negive soluion. In order o hve rel roos, i mus be: ( ) ( )( ) ( ) 4 4 B 4AC = σ 8σ µ + 4 0σ σ + σ µ + 4σ σ > 0 = 0 is quric equion h dmis 4 soluions of which only wo mos re rel (or none). Define x µ. We hve h: ( σ σ ) x ( σ σ )( σ ) x σ ( σ ) 4 = = 0 This is now second degree equion whose soluions, x nd x b, re rel since ( σ )( σ ) ( σ ) = 44 5 > 0 8

20 Moreover, hese soluions re boh negive for + < σ <.8 nd hey re one posi- 4 + ive x nd one negive x b (wih he posiive lrger hn he negive) forσ >.8. 4 Therefore: + for < σ <.8 > 0 lwys. 4 + For σ >.8 > 0 for 0< x< x >. Therefore lso in his cse > 0 for ll relevn vlues of µ 4. Therefore A B C + + = 0 dmis wo rel soluions one posiive nd one negive. Le s cll B + he posiive soluion, where A. Given h A < 0, he expression (9) is posiive for 0 < nd i is negive for >. Finlly, noice h he condiion < corresponds o ( σ ) ( σ µ )( µ σ )( σ µ ) + + < 0 Th cn be furher reduced o σ > + µ. Q.E.D. A his sge i could be ineresing wo consider wo simpler cses: Firs cse: =. By seing =, we re bck o he sndrd FC model. Expression (9) becomes: ( )( ) µ + σ µ σ + ( σ + µ ) which is negive forσ > + µ. Th is, for his cse, only subcriicl pichfork bifurcion cn occur. As shown in Fig. (), he curve of symmeric equilibri lies on he lef of P φ. 9

21 () σ = µ = 0.45 = Size of hird region: = 0 0. φ 0.5 (b) σ = µ = 0.45 = Size of hird region: = 0. φ 0.4 (c) σ = µ = 0.45 = Size of hird region: = 0. φ 0.5 Figure 0

22 The wo exising symmeric equilibri λ nd λ re unsble. Th is, for ny such equi- librium M ( λ ) > nd M ( λ ) > h correspond o T ( λ ) > 0 nd T ( λ ) > 0. This cn be verified only numericlly s he Figure (so-clled wiggle digrm) below shows: Fig. : Wiggle digrm, showing he sbiliy of equilibri for =, µ = 0.45 φ = 0. nd σ = As shown in Fig., ploing T ( λ ) wih respec o λ, for he given prmeer configurion, he symmeric equilibrium is loclly sble sble since T < 0. A he sme ime, lso he CP equilibri re rcing given he boundry condiions in (4), which give Λ (0) =Λ () = 0 (h is, due o he presence of borders he CP equilibri re supersble, i.e. he firs derivive of he mp evlued hose equilibri is equl o zero). Insed, he wo symmeric equilibri re unsble, given h he slope of T ( λ) in correspondence of hese equilibri is posiive. λ nd λ sepre symmericlly wihin he uniry inervl (0, ), he bsins of rcion of he symmeric equilibrium, ( λ, λ ), of he CP equilib- CP(0) CP() rium λ = 0, 0, λ ), nd of he CP equilibrium λ =, ( λ,. Second cse: =. This cse corresponds o n equl disribuion of he griculurl secor mong he hree regions. If =, expression (9) cn be rewrien s µ + + ( ) ( ) σ 4 5µ σ µ µ (0)

23 Solving for σ, we obin: 5 µ ± µ (49µ + 6) σ i = + i =, 4 wih 0< σ < < σ. We cn disregrd σ nd conclude h (0) is lrger hn zero for < σ < σ nd i is less hn zero forσ > σ >. In Figure (b), we se σ > σ >, herefore (0) is negive. As φ crossesφ P subcriicl bifurcion emerges. The curve of symmeric equilibri lies enirely on he lef of See Figure 4. P φ. The wo exising symmeric equilibri re unsble. Fig. 4: Wiggle digrm showing he sbiliy of equilibri for = /, µ = 0.45 φ = 0. nd σ = P In Figure (c), we se < σ < σ, herefore (0) is posiive. As φ crosses φ from lef o righ, supercriicl pichfork bifurcion emerges. The curve of symmeric equilibri lies, les P iniilly, on he righ ofφ. Four symmeric equilibri my exis. The wo exernl b b equilibri, λ nd λ, re unsble. This is due o he fc h due o he border collision bifurhe CP equilibri gin sbiliy (see bove). Insed, he wo inerior symmeric equilibri, λ nd λ, re sble. This is due o he fc h in he neighborhood of he symmeric equilibrium he pichfork bifurcion mus be supercriicl. The disppernce of he four symmeri equilibri occurs vi so-clled fold bifurcion. Typiclly ccording o such ype of bifurcion by vrying prmeer (in our cse by reducing φ ) wo equilibri

24 emerge (one sble nd one unsble). In our cse, due o symmery, his occurs boh below nd bove λ * =. To check furher on he sbiliy properies of he symmeric equilibri see Figure 5, where, for < σ < σ, we hve ploed T ( λ ) for differen vlues of he rde freeness prmeer. For φ = 0.09 nd φ = 0., only he exernl symmeric equilibri coexis wih he symmeric nd he CP equilibrium; he exernl symmeric equilibri delimi he bsins of rcion of he sble equilibri: siuion nlogous o Fig. 4. By incresing slighly φ, in he exmple up o φ = 0., he wo sble inerior symmeric equilibri emerge, for which M ( λ ) < nd M ( λ ) <, h correspond o T ( λ ) < 0 nd T ( λ ) < 0. b The bsins of rcion re now given by 0, λ ) for he CP equilibrium for he inerior symmeric equilibrium b λ, ( 0.5, λ ) CP(0) λ = 0, ( λ ) b,0.5 for oher sble symmeric equilibrium b λ nd finlly, ( λ, for he CP equilibrium CP() λ =. Noice h he symmeric equilibrium, which is unsble fer crossing he bifurcion vlue rcion of he wo inerior symmeric equilibri. P φ, sepres he bsins of Fig. 5: Wiggle digrm showing he sbiliy of equilibri for = /, µ = 0.45 nd σ = nd for differen vlues of he rde freeness prmeer: φ = 0.09 φ = 0. φ = 0. nd φ = 0.

25 Finlly, in Fig. 6 we presen he generl cse 0< <. These simulions h he negive impc of he size of he hird region on P φ is significn. Finlly, concerning he inequliy on he lef hnd side of (8), i holds for sufficienly smll vlue of γ or for sufficienly high vlue of (see lso below). 4

26 σ = µ = 0.45 = Size of hird region: = / σ = µ = 0.45 =.09 Size of hird region: = 0.7 σ = µ = 0.45 =. Size of hird region: = 0.67 Figure 6 6. Preliminry resuls on globl dynmics As i is sed in he previous secions, he mp Λ hs wo CP-fixed poins λ=0 nd λ =, he symmeric fixed poin λ * =/, nd i cn lso hve four more, symmeric, fixed poins, λ, - λ nd λ b, - λ b (which re symmeric wih respec o λ * by pirs). Le us wrie down he expressions of he bifurcion curves of he fixed poin λ * =/, reled o is eigenvlue ±. The eigenvlue he mp Λ evlued λ * cn be wrien s 5

27 γ ( φ ) µ µφ ( σ µ )( φ) M '( λ*) = η + +. ( + φ) ( σ ) σ( + φ ) µ ( φ) The vlue η = corresponds o he pichfork bifurcion. This bifurcion holds if µ σ( + φ ) µ ( φ) = pf φ. ( σ µ )( φ) ( σ ) The vlue η = - is reled o he flip bifurcion. The flip bifurcion occurs if ( σ ( + φ ) µ ( φ ) µ ( + φ ) µφ = fl + +. ( σ µ )( φ) ( σ ) γ( φ) ( σ µ )( φ) To ge n ide bou he globl dynmics of he mp Λ le us fix μ=0.45, γ=0 nd consider he (f, )-prmeer plne for differen vlues of σ. Firs, le σ=. In Fig.7 we presen he Dim bifurcion digrm in he (f, )-prmeer plne where differen colours correspond o rcing cycles of differen periods (up o he period equls ). The wo curves = pf nd = fl reled o he pichfork nd, respecively, he flip bifurcion of he fixed poin λ * re ploed using he reled equions. To ge his Dim digrm only one iniil condiion ws used, so, mulisbiliy cnno be observed in such cse. In order o clrify he dynmics le us consider Dim bifurcion digrms. Figure 7 Dim bifurcion digrm in he (f, )-prmeer plne σ=, μ=0.45, γ=0. 6

28 The Dim digrms reled o he srigh lines wih rrows re shown in Fig. 8 (horizonl line) nd Fig. 9 (vericl line). Firs, le us fix he vlue = / nd vry he prmeer f in he rnge (0, 0.5) (he reled prmeer pss is shown in Fig.7 by horizonl line wih n rrow). The corresponding Dim bifurcion digrm is shown in Fig.8 () ogeher wih is wo enlrgemens, in (b) nd (c). Le us commen on he bifurcion sequence sring from he rcing fixed poin λ * (e.g., f = 0.055) nd will decrese he vlue of f (see Fig.8 (b)). A φ = φ supercriicl perioddoubling bifurcion occurs leding o he rcing -cycle denoed g₂; hen φ = φ his cycle undergoes supercriicl pichfork bifurcion resuling in wo more (coexising) rcing -cycles. Ech of hese cycles undergoes cscde of period-doubling bifurcions following he 'logisic' scenrio up o he pirwise merging of wo coexising -cyclic rcors in one -cyclic rcor due o he homoclinic bifurcion of he -cycle g₂. A f=f₁ he choic inervl hs conc wih is bsin of rcion, bounded by he wo repelling CP fixed poins, fer which hese wo fixed poins become sble. Now consider he enlrgemen of Fig.8 () shown in (c). A f=f₄ border collision bifurcion leds o he sbilision of he CP fixed poins nd o he ppernce of wo more repelling fixed poins λ, - λ. A f=f₅ he fixed poin λ * undergoes supercriicl pichfork bifurcion leding o wo more fixed poins, λ b nd - λ b. Then f=f₆ we observe reverse fold bifurcion due o which he four fixed poins λ, - λ nd λ b, - λ b dispper. merge by pirs nd 7

29 Figure 8 In (): Dim bifurcion digrm μ=0.45, σ=, γ=0, = /, f (0, 0.5); In (b) nd (c): enlrgemens of wo windows indiced by recngles mrked I nd II in (). Le us come bck o he Dim digrm in Fig. 7, fix f = 0.0 nd will vry he vlue of he prmeer. The Dim bifurcion digrm for f= 0.0 nd (0., 0.65) is shown in Fig. 9. We observe h = he fixed poin λ * =/ undergoes he supercriicl pichfork bifurcion (for decresing ) leding o wo fixed poins λ * ₁=λ nd λ * ₂=-λ. If we coninue o decrese he vlue of = ech of hese fixed poins undergoes supercriicl period- 8

30 doubling bifurcion nd hen i follows he 'logisic' bifurcion scenrio. For he vlues of less hn cerin vlue his scenrio is observed in he reverse order up o he period-doubling bifurcion = leding o he rcing fixed poin. Thus, for exmple, = 0. we re bck o he wo rcing fixed poins λ * ₁ nd λ * ₂. Figure 9 Dim bifurcion digrm μ=0.45, σ=, γ=0, f=0.0, (0., 0.65). Figure 0 An enlrgemen of he window indiced in Fig. 9. 9

31 Sring gin from he rcing fixed poin λ * =/ (see Fig. 9) nd incresing he vlue of one cn see h = 5 he fixed poins λ * undergoes subcriicl period-doubling bifurcion h cn be seen in Fig. 0 which shows n enlrgemen of he Fig. 9. Here he dshed colour lines re reled o he poins of repelling cycle of period denoed g₂ born 4 = due o he fold bifurcion ogeher wih n rcing cycle g₂ of period. The prmeer rnge ( 4, 5) corresponds o coexising rcing fixed poin λ * nd -cycle g₂. Then, he subcriicl period-doubling bifurcion he poins of g₂ merge wih he fixed poin λ * nd fer his fixed poin becomes repelling so h he only rcor is he -cycle g₂. If we coninue o increse he vlue of, = 6 he -cycle g₂ undergoes supercriicl pichfork bifurcion leding o wo new -cycles g ₂ nd g b ₂. Ech of hese cycles undergoes 'logisic' sequence of bifurcions (for ( 6, 7) we gin hve coexising rcors) up o he momen of homoclinic bifurcion of he -cycle g₂ leding o merging of he rcors. Afer his bifurcion he rcor is unique. I exiss up o he conc wih is bsin of rcion bounded by he repelling CP fixed poins. As resul hese fixed poins re sbilised. Noe h he period-doubling bifurcion of λ * observed in Fig. 8 (b) is supercriicl nd i is subcriicl in Fig. 0, while he pichfork bifurcion of λ * in Fig. 8 (b) is subcriicl nd i is supercriicl in Fig. 9. To compre he bifurcion srucure shown in Fig. 7 wih he one for lrger vlue of σ, we show in Fig. he Dim bifurcion digrm in he ( φ, ) -prmeer plne σ=8, μ=0.45, γ=0. The bsic sequence of bifurcions is similr, bu we leve is complee chrcerizion, s well s more deiled invesigion of he dynmics of he mp, for fuure work. 0

32 Figure Dim bifurcion digrm in he ( φ, ) -prmeer plne σ=8, μ=0.45, γ=0. 7. Finl remrks The (scn) NEG lierure on hree-region models hs del so fr wih he impc of rde policy wih hird region on indusril gglomerion in wo-region home counry. In conrs, he FE-NEG model presened in his pper hs delivered resuls on how economic inegrion beween wo regions ( nd ) priciping in cusoms union impcs on he disribuion of indusril firms wihin he union, holding consn he disnce beween he union iself nd he res of he world (region ). We hve shown h: ) becuse our simple se up, chnge in rde policy wih he res of he world does no impc on profi differenils beween regions nd, hus leving unffeced he disribuion of indusril firms wihin he union; ) i is he size (of he mrke) of he hird region h mers for he blnce beween cenripel nd cenrifugl forces beween he wo counries/regions inside he inegrion re. This leds o he generl conclusion h he effecs of rde liberlision beween he wo counries inside he inegrion re sricly depends upon he size of he mrke exernl o he union. Such resul holds looking boh sble nd unsble symmeric equilibri s summrized in he following wo poins; ) n increse of he exernl demnd coming from he hird region, leds he symmeric equilibrium beween region nd o lose sbiliy lower vlue of he rde free-

33 ness inside he inegrion re. Th is, s shown in Brülhr e l. (004), sronger ouside links fvour gglomerion wihin he inegrion re; 4) in ddiion, nd his is our min resul, he size of he hird region lso influences wh hppens, if he symmeric equilibrium is unsble. For smller hird region he gglomerive forces ouweigh he deglomerive forces leding o full gglomerion in one of he regions inside he inegrion re. Insed, for bigger hird region, symmeries weken he gglomerive processes nd srenghen he deglomerive forces nd inerior symmeric equilibr cn be esblished. This ler resuls is imporn for wo resons: ) becuse i shows h even if wih reducion of rde coss wih he inegrion re he symmeric equilibrium loses sbiliy (s found lso in he ppers reviewed bove), he long-run oucome need no be full gglomerion, bu my lso be pril gglomerion (hus reducion of rde coss does no led o exreme regionl dispersion); b) becuse i is one of he rre exmples in he NEG lierure h produces pril gglomerion s he oucome of n endogenous process.

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