Federal Reserve Ban of Mnneapols Research Deparmen Saff Repor 378 Ocober 8 (Frs verson: Sepember 6) Trade, Growh, and Convergence n a Dynamc Hecscher-Ohln Model* Clausre Bajona Ryerson Unversy Tmohy J. Kehoe Unversy of Mnnesoa, Federal Reserve Ban of Mnneapols, and Naonal Bureau of Economc Research ABSTRACT In models n whch convergence n ncome levels across closed counres s drven by faser accumulaon of a producve facor n he poorer counres, openng hese counres o rade can sop convergence and even cause dvergence. We mae hs pon usng a dynamc Hecscher- Ohln model a combnaon of a sac wo-good, wo-facor Hecscher-Ohln rade model and a wo-secor growh model wh nfnely lved consumers where nernaonal borrowng and lendng are no permed. We oban wo man resuls: Frs, counres ha dffer only n her nal endowmens of capal per worer may converge or dverge n ncome levels over me, dependng on he elascy of subsuon beween raded goods. Dvergence can occur for parameer values ha would mply convergence n a world of closed economes and vce versa. Second, facor prce equalzaon n a gven perod does no mply facor prce equalzaon n fuure perods. *A prelmnary verson of hs paper wh he le On Dynamc Hecscher-Ohln Models II: Infnely-Lved Consumers was crculaed n January 3. We are graeful for helpful commens from parcpans n conferences and semnars a he 3 Amercan Economc Assocaon Meeng, he Unversdad Carlos III de Madrd, he Unversy of Pennsylvana, he Inernaonal Moneary Fund, he Cenre de Recerca en Economa del Benesar, he Insuo Auónomo Tecnológco de Méxco, he 4 CNB/CERGE-EI Macro Worshop n Prague, he Unversa Pompeu Fabra, Sanford Unversy, he Unversy of Texas a Ausn, he Banco de Porugal, and he Unversy of Mchgan. We also han Joann Bangs and Jaume Venura for helpful dscussons. Kehoe hans he Naonal Scence Foundaon for suppor. The vews expressed heren are hose of he auhors and no necessarly hose of he Federal Reserve Ban of Mnneapols or he Federal Reserve Sysem.
. Inroducon In 7, GDP per capa n he Uned Saes was roughly 46, U.S. dollars. Usng exchange raes o conver pesos o dollars, we calculae GDP per capa n Mexco n 7 o be roughly 9,6 U.S. dollars. In 94, he Uned Saes had ncome per capa of abou 9,4 7 U.S. dollars. To predc wha wll happen n he Mexcan economy over he nex 7 years, should we sudy wha happened o he U.S. economy snce 94? Or should we ae no accoun ha, n 94, he Uned Saes was he counry wh he hghes ncome per capa n he world, whle, n 7, Mexco had a very large rade relaon wh he Uned Saes a counry wh a level of ncome per capa approxmaely fve mes larger? In oher words, does he ably o rade nernaonally radcally change a counry s growh pah? In parcular, does nernaonal rade accelerae growh n poor counres and, hus, lead o faser ncome convergence, or does cause poor counres o grow a a slower rae han rch counres, leadng o ncome dvergence? Much of he dscusson of convergence of ncome levels n radonal growh heory reles on models of closed economes. (See, for example, Barro and Sala--Marn 3.) In hs paper we as: Do he convergence resuls obaned n closed economy growh models change when we nroduce rade? We answer hs queson usng a dynamc Hecscher-Ohln model a combnaon of a sac wo-good, wo-facor Hecscher-Ohln rade model and a wo-secor growh model wh nfnely lved consumers where borrowng and lendng are no permed. We fnd ha nroducng rade no he growh model radcally changes he convergence resuls: In many envronmens where ncome levels converge over me f he counres are closed, for example, hey dverge f he counres are open. Ths s because favorable changes n he erms of rade for poor counres reduce her ncenves o accumulae capal. In models le he neoclasscal model n whch convergence n ncome levels across closed counres s drven by faser accumulaon of a producve facor n he poorer counres, openng hese counres o rade can sop convergence and even cause dvergence of ncome levels. The model ha we use s a specal case of he general dynamc Hecscher-Ohln model suded by Bajona and Kehoe (6). There are n counres ha dffer only n her populaon szes and her nal endowmens of capal. There are wo raded goods ha are produced usng capal and labor; one of he goods s more capal nensve han he oher. Tme s dscree, and here s a nonraded nvesmen good ha s produced usng he wo raded goods. Consumers have uly
funcons ha are homohec and dencal across counres, and derve uly from he consumpon of he wo raded goods. The model ha we sudy s boh a classc Hecscher-Ohln model and a classc growh model n he sense ha he wo facors of producon are denfed as labor and physcal capal. A counry ha s capal abundan n he ermnology of Hecscher-Ohln heory s rch n he ermnology of growh heory. I would be sraghforward o redo he analyss for a model n whch he wo facors of producon were labor and human capal. I would be more complcaed o exend he analyss o a model wh more han wo facors of producon. Neverheless, even n models wh more han wo facors, we would expec he cenral message of hs paper o carry over: Consder a model of closed economes n whch counres become rcher because hey accumulae a facor, or facors, of producon. Suppose ha convergence n ncome levels s drven by reurns o a facor beng hgher n counres ha are poorer because hey have less of he facor. Openng he economes n hs model o nernaonal rade wll reduce he reurns o he facor, hereby reducng ncenves o accumulae he facor and reducng he endency owards convergence. There s a large leraure ha s a leas parally relaed o he opc suded here. Bardhan (965) and On and Uzawa (965) sudy he paerns of specalzaon and rade n a Hecscher- Ohln model n whch consumers have fxed savngs raes. Deardorff and Hanson (978) consder a model n whch hese fxed savngs raes dffer across counres and show ha he counry wh he hgher savngs rae wll expor he capal nensve good n he seady sae. Sglz (97) also consders models wh fxed savngs behavor, n hs case a Marxan specfcaon where all labor ncome s consumed and all capal ncome s saved. In addon, he consders a model n whch here are nfnely lved, uly-maxmzng consumers wh dfferen dscoun raes n each counry. Sglz sudes he paern of rade and specalzaon n he seady sae of hs model and sudes dynamc equlbrum pahs n a small open economy verson of he model. Chen (99) sudes he long-run equlbra of wo-counry, dynamc Hecscher-Ohln models wh uly-maxmzng agens and dencal preferences n boh counres under he assumpon ha boh counres produce boh goods. He fnds ha here s a connuum of seady saes n such models and ha, unless nal capal-labor raos are equal, here s rade n he seady sae. Chen also shows ha cycles are possble n such models when one good s he consumpon good and he oher s he nvesmen good f he consumpon good s capal
nensve. Baxer (99) consders a model smlar o Chen s bu n whch ax raes dffer across counres. She shows ha he paern of rade and specalzaon n he seady sae s deermned by hese axes. Brecher, Chen, and Choudhr () consder a model wh dfferences n echnologes across counres. Nshmura and Shmomura (), Bond, Tras, and Wang (3), Do, Nshmura, and Shmomura (), and Ono and Shbaa (5) sudy dynamc Hecscher- Ohln models wh endogenous growh or exernales. A number of researchers have suded dynamc Hecscher-Ohln models usng he small open economy assumpon: Fndlay (97), Mussa (978), Smh (984), Aeson and Kehoe (), Chaerjee and Shuayev (4), and Obols-Homs (5). Aeson and Kehoe and Chaerjee and Shuayev are of parcular relevance o our paper. Aeson and Kehoe sudy a model n whch he res of he world s n s seady sae and he small open economy sars wh eher a lower or a hgher capal-labor rao. They show ha, f he small open economy s ousde he res of he world s cone of dversfcaon, hen he counry converges o he boundary of hs cone. If he small counry sars nsde he cone of dversfcaon, hen oo s n seady sae and says here. Ths resul s n sharp conras o our resul ha, for ceran parameer values and nal condons, even f all counres sar n he cone of dversfcaon, some necessarly leave. Conrasng our resuls wh hose of Aeson and Kehoe shows how srong are her assumpons ha he res of he world s n s seady sae and ha here are no general equlbrum prce effecs. Chaerjee and Shuayev consder a model smlar o ha of Aeson and Kehoe n whch here are sochasc producvy shocs and show ha he comparave advanage conferred by dfferen nal endowmens can dsappear over me. The paper mos closely relaed o ours s Venura (997), who sudes rade and growh n a dynamc Hecscher-Ohln model wh uly-maxmzng consumers and dencal preferences across counres. He assumes ha here are wo raded goods one capal nensve and one labor nensve ha are used n consumpon and nvesmen. Venura absracs away from sudyng he paerns of specalzaon by assumng ha each good uses only one of he facors n s producon process. Under hs assumpon, facor prces are equalzed across counres ndependenly of her relave facor endowmens. Venura sudes he evoluon of capal socs over me. Our paper dffers from hs n ha () we use dscree me raher han connuous me because maes easer o oban analycal resuls, alhough we show how our resuls can be exended o a connuous-me verson of he model, () we sudy he evoluon of ncome levels as 3
well as of capal socs, (3) n models wh more general producon srucures, we oban condons under whch facor prce equalzaon n a gven perod ensures facor prce equalzaon n fuure perods, and (4) we complee Venura s (997) analyss by consderng he possbly of corner soluons n whch one or more counres have zero nvesmen n some perods and showng ha convergence resuls may brea down n such cases. (Venura analyzes only neror soluons.) I s also worh menonng he wor of Cuña and Maffezzol (4), who presen numercal expermens usng a hree-good, wo-facor verson of he Venura model. In hs paper, we sudy he paerns of rade, capal accumulaon, and ncome growh over me as a funcon of he counres nal relave endowmens of capal and labor. We fnd, as does Venura (997), ha, f boh counres dversfy over he enre equlbrum pah, he elascy of subsuon beween raded goods s crucal n deermnng convergence behavor. Ths s no longer rue when one of he counres specalzes n producon n some perod. For a gven elascy of subsuon, wheher counres converge or dverge depends on he paern of specalzaon over me. We presen an example n whch counres ncome levels converge n equlbra whou facor prce equalzaon for an elascy of subsuon ha mples dvergence n ncome for equlbra wh facor prce equalzaon along he equlbrum pah. We also presen an example n whch corner soluons n nvesmen cause our convergence resuls o brea down.. The general model There are n counres, =,..., n. Each has a connuum of measure L of dencal, nfnely lved consumer-worers, each of whom s endowed wh uns of capal n perod and one un of labor n every perod, =,,.... There are hree goods: an nvesmen good, x, whch s no raded, and wo raded goods, y, j =,, whch can be consumed or used n he j producon of he nvesmen good. Uly funcons and echnologes are he same across counres. Each raded good j, j =,, s produced wh a connuously dfferenable, consan reurns o scale funcon ha uses capal,, and labor, : y = φ (, ). () j j j j 4
We assume ha good s relavely capal nensve and ha he producon funcons are such ha here are no facor nensy reversals. Producers mnmze coss ang prces as gven and earn zero profs. The frs-order condons from he producers problems are r p φ (, ), = f > () j jk j j j w p φ (, ), = f > (3) j jl j j for each j, j =,, where r s he renal rae, w s he wage, and p j s he prce of good j, j =,. Addonal subscrps φ (, ), φ (, ) denoe paral dervaves. jk j j jl j j The nvesmen good s produced accordng o he consan-reurns producon funcon j x = f( x, x ). (4) Leng q be he prce of he nvesmen good, he frs-order condons for prof maxmzaon are p qf( x, x), = f x > (5) p qf( x, x), = f x >. (6) In each perod, consumers decde how much of each raded good o consume, c, c, and how much capal o accumulae for he nex perod, +. We assume ha here s no nernaonal borrowng or lendng. Bajona and Kehoe (6) prove ha nernaonal borrowng and lendng ensures facor prce equalzaon bu resuls n ndeermnacy of producon and rade. The represenave consumer n counry solves he uly maxmzaon problem max β uc (, c ) = s.. p c + p c + q x w + r (7) ( + δ ) x c, x j, where β, < β < s he common dscoun facor and δ, δ, s he deprecaon rae. The feasbly condon for good j, j =,, s 5
. (8) n n L( c ) j + x j = Ly = = j Labor and capal are perfecly moble across secors whn a counry, bu no across counres. Therefore, he feasbly condons n each counry, =,..., n, are + (9). () + Lewse, he nvesmen good s nonraded and he feasbly condon n each counry s x = f( x, x ). () I s easy o show ha allowng for rade of he nvesmen good would only generae ndeermnacy of rade n hs model, whou oherwse changng he se of equlbra. Before analyzng he properes of he model, we ls he man assumpons: A.. There are n counres, whch are populaed by nfnely lved consumers. Counres dffer only n her populaon szes, L >, and her nal endowmens of capal, >. A.. There are wo raded goods, whch can be consumed or used n he producon of he nvesmen good. The producon funcons of he raded goods, φ (, ), are ncreasng, concave, connuously dfferenable, and homogeneous of degree one. A.3. Traded good s relavely capal nensve, and here are no facor nensy reversals: For all / >, K K j φl( /,) φl( /,) <. () φ ( /,) φ ( /,) A.4. Labor and capal are perfecly moble across secors bu are no moble across counres. A.5. There s an nvesmen good n each counry, whch s no raded. The producon funcon for he nvesmen good, f ( x, x ), s ncreasng, concave, connuously dfferenable, and homogeneous of degree one. 6
A.6. The perod uly funcon uc (, c ) s homohec, srcly ncreasng, srcly concave, and wce connuously dfferenable, and sasfes lm u ( c, c) =. Defnon : An equlbrum of he world economy s sequences of prces, {,,,, } consumpons, nvesmens, and capal socs { c,,, c x }, producon plans for he raded goods, { y,, j j j}, and producon plans for he nvesmen goods{ x,, x x }, such ha:. Gven prces { p,,,, p q w r }, he consumpons and capal socs {,, } consumers problem (7). cj j p p q w r, c c solve he. Gven prces { p,,,, p q w r }, he producon plans { y,, j j j} and {,, } cos mnmzaon and zero prof condons (), (3), (5), and (6). 3. The consumpon, capal soc, { c,, c }, and producon plans, {,, j j j} {,, } x x x, sasfy he feasbly condons (), (4), (8), (9), (), and (). x x x sasfy he y and Noce ha, snce rade equalzes he prces of he raded goods across counres, he prces of he nvesmen good are also equal, q = q. Snce he cos mnmzaon problems are he same across counres, hs s rue even f some counry does no produce he nvesmen good n perod. The homogeney of he budge consrans n (7) and he cos mnmzaon and zero prof condons (), (3), (5), and (6) n curren perod prces allow us o mpose a numerare n each perod. We se q =, =,,... (3) I s worh nong ha he assumpon of no nernaonal borrowng and lendng mples ha rade balance holds: p ( y c x ) + p ( y c x ) =. (4) Ths condon can be derved from he budge consan n he consumer s problem (7) and he cos mnmzaon and zero prof condons (), (3), (5), and (6). 7
Defnon. A seady sae of he world economy s consumpon levels, an nvesmen level, and ˆ, ˆ, ˆ, ˆ ˆ, ˆ, ˆ y l, a capal soc, { c c x }, facors of producon and oupu for each raded ndusry, { j j j} j =,, facors of producon and oupu for he nvesmen secor { xˆ, xˆ ˆ, x }, and prces { ˆ, ˆ ˆ ˆ,, } p p w r, for =,..., n, ha sasfy he condons of a compeve equlbrum for approprae nal endowmens of capal, ˆ =. Here we se ν ˆ = ν for all, where ν represens a generc varable. We say ha a seady sae s a nonrval seady sae f a leas one of he counres has a posve level of capal n ha seady sae: ˆ > for some =,..., n. The seady sae resuls for general dynamc Hecscher-Ohln models wh nfnely lved consumers derved n Bajona and Kehoe (6) apply o he Venura model. The followng proposons sae hem whou proof. Proposon : Under assumpons A.-A.6, n any nonrval seady sae facor prces are equalzed. Proposon : Under assumpons A.-A.6, f here exss a nonrval seady sae, hen here exss a connuum of hem, whch have he same prces and world capal-labor rao, ˆ. These seady saes are ndexed by he dsrbuon of capal-labor raos across counres, Furhermore, nernaonal rade occurs n every seady sae n whch ˆ 3. The negraed economy ˆ,..., ˆ n. ˆ for some. The characerzaon and compuaon of equlbrum of he model descrbed n he prevous secon s dffcul n general, snce nvolves deermnng he paern of specalzaon n producon over an nfne horzon. (See Bajona and Kehoe 6 for some resuls on he equlbrum of he general model.) Numercal mehods are usually needed o compue equlbrum. The characerzaon and compuaon of equlbrum becomes much easer, however, when he model specfcaon s such ha we can solve for he equlbrum by dsaggregang he equlbrum of he negraed economy a closed economy wh nal facor endowmens equal o he world endowmens whch s equvalen o a wo-secor growh model. (See Dx and Norman 98 for a descrpon of he mehodology.) In hs case, he equlbrum prces and aggregae 8
consumpon, producon, and nvesmen of our economy concde wh he equlbrum prces, consumpon, producon, and nvesmen of he negraed economy. Consder he socal planner s problem max β uc (, c ) = s.. c + x y = φ (, ) (5) c + x y = φ (, ) ( δ ) x = + f( x, x) + + c, x j j, where L / L n n =. Noce ha assumpon A. mples ha = = >. Proposon 3: Suppose ha he allocaon { c, c, }, { y,, }, { y,, }, { x, x, x } solves he socal planner s problem (5). Then hs allocaon, ogeher wh he prces {,,,, } p p q w r, s an equlbrum of he negraed economy where q =, p = f( x, x), p = f( x, x), r = p φk(, ), and w = p φl(, ). Conversely, suppose ha { p, p, q, w, r }, { c, c, }, { y,, }, { y,, }, {,, } x x x s an equlbrum of he negraed economy. Then he equlbrum allocaon solves he socal planner s problem (5). Furhermore, f he socal planner s problem has a soluon, hen s he unque equlbrum allocaon of he negraed economy. Proof: The frs clam s jus he second heorem of welfare economcs, and he second clam s he frs heorem. In our seng, s sraghforward o prove hese clams by showng ha he frsorder condons and ransversaly condon for he socal planner s problem are equvalen o he equlbrum condons n he defnon of equlbrum where here s only one counry, n =. If he uly funcon s bounded on he consran se of he socal planner s problem, here exss a soluon o hs problem. Snce he funcon u s srcly concave and he funcons φ, φ, and f 9
are concave, he soluon o he planner s problem s unque, whch mples ha here s a unque equlbrum o he negraed economy. Once we have he equlbrum of he negraed economy, o compue an equlbrum of he world economy, we need o dsaggregae he consumpon, nvesmen, and producon decsons across counres. Wheher an equlbrum can be solved hs way s a guess-and-verfy approach. Frs consder he dsaggregaon of producon decsons. If capal-labor raos are very dfferen across counres, assgnng nonnegave producon plans for boh goods o all counres s no conssen wh her havng he same facor prces, and solvng for equlbrum usng he negraed approach s no possble. Fgure, nown as he Lerner dagram, shows he endowmens of capal and labor ha are conssen wh usng he negraed economy approach o solve for equlbrum for a sac Hecscher-Ohln model. Le p, p be he equlbrum prces of he raded goods n he negraed economy. The rays / and / represen he capal-labor raos used n he producon of each good n he equlbrum of he negraed economy. The area beween boh rays s called he cone of dversfcaon. If all counres have endowmens of capal and labor n he cone of dversfcaon, he equlbrum prces of he negraed economy are conssen wh nonnegave producon plans for boh goods n all counres. To fnd he cone of dversfcaon, we solve he problem max pφ (, ) + p φ (, ) s.. + (6) +,. j If p and p are he equlbrum prces of he negraed economy, hen, snce c and c are boh srcly posve by assumpon A.6, he soluon o hs problem s such ha φ (, ) > and φ (, ) >. Assumpon A. mples ha / > /. The cone of dversfcaon s specfed by hese secor-specfc capal-labor raos, whch depend only on relave prces, ( p / p ) κ and κ ( p / p ). I s he se of counry specfc capal-labor raos j such ha κ ( p / p ) κ ( p / p ). (7)
In our dynamc economy, he cone of dversfcaon changes over me snce he capallabor rao and, consequenly, he equlbrum prces of he negraed economy, change over me. Therefore, o solve for an equlbrum usng he negraed economy approach, we need o dsaggregae he nvesmen decsons n such a way ha counres say n he correspondng cone of dversfcaon for all me perods. Gven ha he perod uly funcon s dencal and homohec across counres, facor prce equalzaon mples ha we can use he negraed economy approach o solve for equlbrum n a sac model. In our dynamc economy, here s an addonal possble complcaon: If one of he counres has a corner soluon n whch chooses zero nvesmen n some perod whle anoher counry chooses posve nvesmen, hen we canno dsaggregae he consumpon and nvesmen decsons of he negraed economy. Laer, we wll show how hs possbly maes dffcul o characerze equlbra. In he res of he paper, we assume ha consumers combne he wo raded goods n consumpon n he same way ha producers of he nvesmen good combne hese wo goods n producon: uc (, c) v( f( c, c) ) =,where v s a srcly concave, srcly ncreasng funcon. Ths assumpon smplfes he dynamcs of he model, snce maes he negraed economy equvalen o a one-secor growh model and, herefore, cycles and chaos are ruled ou as possble equlbrum behavor of he negraed economy. Neverheless, he dynamcs of he smplfed model are rch enough for he purpose of he paper: o llusrae he dfferen convergence behavor beween rade and closed economes. To furher smplfy he analyss we assume, as does Venura (997), ha he funcon v s logarhmc. A.7. The perod uly funcon u aes he form uc (, c) log ( f( c, c) ) =. Consder he producon funcon defned by solvng F (, ) = max f( y, y) s.. y φ (, ) (8) y φ (, ) + +
,. j Assumpons A.5 mples ha f s srcly quas-concave, whch, ogeher wh he concavy of φ and φ, mples ha for any (, ) here s a unque soluon o hs problem. I s sraghforward o prove ha F s ncreasng, concave, connuously dfferenable, and homogeneous of degree one. Le f, F s srcly quas-concave. Assumpon A.7 s useful because allows us o solve he wo-secor socal planner s problem (5) by solvng he relaed one-secor socal planner s problem max = j β log c s.. c + x F(,) (9) ( δ ) + x c, x. We sae he followng proposon whou gvng a proof because, frs, he proof s jus a sraghforward applcaon of he maxmum heorem, and, second, we wll no employ he proposon n s general form, bu raher wll only consder producon funcons for whch we can analycally solve problem (8). Proposon 4. Le y (, ), y (, ), (, ), (, ), (, ), (, ) denoe he soluon o (8). If { c, c, }, { y,, }, { y,, }, {,, } problem (5), hen {,, } x x x solves he wo-secor socal planner s c x solves he one-secor socal planner s problem (9) where c = f( c, c). Conversely, f {,, } { c, c, }, { y,, }, { y,, }, {,, } c x solves he one-secor socal planner s problem (9), hen x x x solves he wo-secor socal planner s problem (5) where y = y (,), = (,), = (,) j j [ ] x = x /( c + x ) y (,). j j j j, [ ] j j c = c /( c + x ) y (,), and j j We frs consder a verson of he model n whch he producon funcon φ j for each raded good uses only one facor of producon. Under hs assumpon, facor prces equalze
along he equlbrum pah ndependenly of nal condons. Snce hs s he assumpon made by Venura (997), we call hs verson of he model he Venura model. By dsaggregang he equlbrum of he negraed economy, we derve resuls on he evoluon of he world dsrbuons of ncome and of capal n he Venura model. We also show by means of an example ha, even hough facor prce equalzaon holds n every equlbrum of he Venura model, here may be equlbra n whch here s zero nvesmen n some counres and n whch our convergence resuls do no hold. I s worh ponng ou ha Venura (997) consders a connuous-me verson of our model. For compleeness, we sech ou our resuls for he connuous-me model n appendx A. We hen consder a verson of he model n whch he more general producon funcons φ j have he same consan elascy of subsuon as does he producon funcon for he nvesmen good f. In such models, facor prces need no equalze along he equlbrum pah, bu, f hey do, he equlbra have he same properes as hose of he Venura model. We refer o hs verson of he model as he generalzed Venura model. For hs model, we derve he cone of dversfcaon analycally, and gve condons under whch, f counres are n he cone of dversfcaon, hey say here. We also derve condons under whch, even f counres sar n he cone of dversfcaon, hey leave n a fne number of perods. 4. Venura model Followng Venura (997), we assume ha he producon funcon for each of he raded goods uses only one facor of producon: y y = φ (, ) = () = φ (, ) =. () Ths assumpon mples ha he cone of dversfcaon s he enre nonnegave quadran, ndependenly of he prces p and p, and ha facor prces equalze along any equlbrum pah: r = r = p and w = w = p. Noce ha, n hs case, F (,) = f (,). Furhermore, we assume ha he producon funcon of he nvesmen good has a consan elascy of subsuon beween he npus of he wo raded goods: 3
b b ( ) / f( x, x ) = d a x + a x () b f b, and f s f ( x, x ) = dx x (3) a a n he lm where b =. Here a > and a+ a =. The elascy of subsuon s σ = /( b). In wha follows, we can easly ranslae saemens nvolvng b no saemens nvolvng σ. Suppose ha we fnd he equlbrum of he negraed economy by solvng he one-secor socal planner s problem (9). To dsaggregae consumpon and nvesmen, we solve he uly maxmzaon of he represenave consumer : max = β log c s.. c + x w + r (4) ( δ ) + x c, x j. If we solve (7), we can oban a soluon o (4) by seng c = f( c, c ), and, f we solve (4), we can oban a soluon o (7) by seng c = c /( w + r ), c = c /( w + r ), x = x /( w + r ), x = x /( w + r ). The necessary and suffcen condons for a sequence of consumpon levels and capal socs o solve (4) are ha ( r ) + β δ + +, wh equaly f x > (5) c c + and ha he ransversaly condon c + ( δ ) = w + r, (6) + lm β + c = (7) 4
holds. If x > for all and all, hen we are jusfed n usng he negraed economy approach. We solve for he negraed economy equlbrum n he Venura model by solvng for he equlbrum of a one-secor growh model. Noce, however, ha he wo secors maer a lo for dsaggregang he equlbrum. In parcular, we canno solve for he equlbrum values of he varables for one of he counres by solvng an opmal growh problem for ha counry n solaon. Insead, he equlbrum pah of a specfc counry s capal soc and s seady sae value depend no only on he counry s nal endowmen of capal bu also hrough he neres rae r and wage w on he equlbrum pah of he world s capal soc, and s seady sae value. If here s posve nvesmen n every perod, hen he equlbrum pah for he negraed economy s deermned by he dfference equaons n he nal condon ( ) L / L ( δ ( )) c = β + r c (8) + + c + + ( δ ) = f(,), (9) n = =, and he ransversaly condon = = lm β c =. (3) + Here r ( ) s he renal rae of capal, b b ( b)/ b a d( a + a) f b r ( ) = a ad f b=. (3) Sandard resuls for one-secor models (for example, Rebelo 99) say ha he equlbrum of he negraed economy has susaned growh for some values of he parameers. The exsence of seady sae depends upon wheher he renal rae of capal as a funcon of nal endowmens, r ( ), can ae he value / β + δ for some >. If r ( ) < / β + δ for all, hen converges o. If, however, r ( ) > / β + δ for all, hen grows whou bound. Consder an economy whou labor, where feasble allocaons sasfy / b ( δ ) c + = da +. (3) + 5
Ths economy has a susaned growh pah n whch ( )( / b β δ) / b ( δ) = + β da +. c = da + and Defnon 3. We say ha an equlbrum converges o he susaned growh pah of he correspondng economy whou labor f / b ( β )( δ ) lm c / = da + (33) / b ( δ ) lm + / = β da +. (34) Sandard resuls from, for example, Soey, Lucas, and Presco (989) provde he followng characerzaon of he equlbrum of he negraed economy. Lemma : The behavor of he equlbrum of he negraed economy of he Venura model depends on parameer values:. If b < and / β + δ > da, he rval seady sae s he unque seady sae, and he unque / b equlbrum of he negraed economy converges o.. If b =, f b < and / b / β + δ da, or f b > and / β + δ > da, here s a unque nonrval sable seady sae characerzed by he soluon of he equaon and he unque equlbrum of he negraed economy converges o. 3. If b > and / b / b r ( ˆ) = / β + δ, / β + δ da, here s no nonrval seady sae, and he unque equlbrum of he negraed economy converges o he susaned growh pah. In he case where b = and δ =, here s an analycal soluon o he one-secor socal planner s problem (9) for he negraed economy: a = x = + β ad (35) c = ( βa ) d (36) a For oher parameer values, we need o use numercal mehods o solve for he equlbrum. Neverheless, we can derve analyc resuls on he evoluon of he dsrbuon of ncome levels over me ha depend on he values of varables n he negraed economy equlbrum. 6
Qualavely characerzng he negraed economy equlbrum hen allows us o qualavely characerze he evoluon of ncome levels. In parcular, we can fnd condons under whch relave ncome levels converge and condons under whch hey dverge. The nex proposon derves a formula ha compares he level of ncome per capa n a gven counry measured n curren prces, y = w + r, o he world s average a any gven perod, y = w + r, o he same relave ncome poson n he prevous perod. Ths formula s ey o our analyss, snce a decrease n he rao ( y y )/ y over me ndcaes convergence n ncome levels (counres become closer o he mean) and an ncrease n he rao ndcaes dvergence n ncome levels. Proposon 5. In he Venura model, f x > for all and all, he ncome level of counry relave o he world s ncome level evolves accordng o he rule y y s y y s y y = =, (37) y s y s y where s = rc / y,,,... =, and s rc /[ β( r δ) y ] deprecaon, δ =, s = c / y, =,,.... = +. When here s complee Proof: Subracng he frs-order condon for he consumer s problem n he open economy from he same condon for he negraed economy, we oban: c+ c+ c c =. (38) c c + I s here ha he assumpon of no corner soluons n nvesmen s essenal, allowng us o mpose he frs-order condons (5) and (8) as equales. Manpulang he frs-order condons (5) and he budge consrans (4), we oban he famlar demand funcon for logarhmc uly maxmzaon: c s ( β) ( ) s s r w r δ = = + + τ =+ + τ δ Noce ha, snce we have facor prce equalzaon, 7. (39) c c = ( β)( + r δ)( ). (4)
Usng (4) a wo dfferen daes and (8), we derve he expresson: c = ( ). (4) + + c The dfference beween a counry s ncome per worer and he world s ncome per worer s y y = r ( ). (4) + + + + + Usng he expresson for + + n (4), we oban y+ y r c / y + + + y y =. (43) y+ rc / y y We can use he frs-order condon (8) o rewre hs expresson as where s = rc / y for,,... becomes s = c / y. y y s y y s y y = =, (44) y s y s y = and s rc /[ β( r δ) y ] = +. When δ =, c / c = βr and s The proof of hs proposon reles on facor prce equalzaon occurrng n every perod and on here never beng a corner soluon n nvesmen. If facor prces are no equal n some perod n he fuure, he demand funcons (39) for each ndvdual counry and for he negraed economy would have dfferen prces n ha perod and, herefore, equaon (4) would no hold. Lewse, f a corner soluon n nvesmen occurs, equaon (38) need no hold. We laer provde examples n whch lac of facor prce equalzaon and he lac of neror soluons for nvesmen cause he characerzaon of behavor of relave ncome n proposon 5 o fal. Equaon (37) n he prevous proposon compares a counry s ncome relave o he world average. Wheher counres converge or dverge n her ncome levels depends on wheher he rao rc / ydecreases or ncreases over me. If he rao ncreases, counres ncomes move furher away from he world s average ncome and, hus, here s dvergence n ncome levels. If he rao decreases, counres ncome levels become closer o he world s average ncome level, and here s convergence n ncome levels. If he rao s consan, counres manan her nal ncome dfferences and, herefore, he dsrbuon of ncome says consan over me. We should 8
sress ha here convergence means ha counres ncome levels become more smlar over me. I does no mean ha hey converge o he same level of ncome: Alhough he absolue value of ( y y )/ y can be srcly decreasng over me, n general converges o a consan dfferen from. Usng proposon 5, we can reduce he characerzaon of he convergence properes of equlbra n he case wh complee deprecaon o a characerzaon of he behavor of s = c / y n he soluon o he one-secor socal planner s problem (9). Lemma. In he unque equlbrum of he negraed economy of he Venura model wh complee deprecaon, he behavor of s = c / y depends on parameer values:. If b < and / β > da / b, hen s s a srcly decreasng sequence ha converges o β. / b. If b < and / β da, hen s converges o [ f ( ˆ,) ˆ]/ f( ˆ,) where ˆ s he unque nonrval sable seady sae. If ˆ <, s s a srcly ncreasng sequence; f ˆ >, s s a srcly decreasng sequence. 3. If b =, hen s = βa s consan. 4. If b > and / β > da / b, hen s converges o [ f ( ˆ,) ˆ]/ f( ˆ,). If ˆ <, s s a srcly decreasng sequence; f ˆ >, s s a srcly ncreasng sequence. 5. If b > and / β da / b, hen s s a srcly decreasng sequence ha converges o β. Proof: Snce he resul for he case 3, where b =, follows rvally from equaon (36), we concern ourselves wh he oher cases, where b < or b >. Mulplyng and dvdng he Euler equaon, (8), by / y and usng he feasbly condon, (9), we oban s s where s = c / y and h = ( r ) / y. We defne he funcon = β h, (45) 9
r ( ) a h ( ) = = f (,) a a b b +. (46) Noce ha h'( ) < f b < and ha h'( ) > f b >. In he lm where b =, h ( ) = a and h'( ) =. We use he monooncy of he sequence n any soluon o he one-secor socal planner s problem (9) o esablsh monooncy properes for he sequence h. The heorem s hen esablshed by showng ha he monooncy properes for he sequence h mply he desred monooncy properes for he sequence s. b < and Consder he dfferen cases enumeraed n he saemen of he heorem. In case, where / β > da / b, s a srcly decreasng sequence ha converges o, whch mples ha h s a srcly ncreasng sequence ha converges o. In case, where b < and and n case 4, where b > and / β > da / b / β da, / b, s a srcly ncreasng sequence ha converges o ˆ f ˆ < and a srcly decreasng sequence ha converges o ˆ f ˆ >. In case, hs mples ha h s a srcly decreasng sequence f ˆ < and a srcly ncreasng sequence f ˆ >. In case 4, however, h s a srcly ncreasng sequence f ˆ < and a decreasng sequence f ˆ >. In boh cases, h converges o ˆb ˆb a /( a + a) no maer wha he nal value of. In case 5, where b > and mples ha / β da / b, s a srcly ncreasng sequence ha grows whou bound, whch h s an ncreasng sequence ha converges o. We now argue ha, f h s srcly ncreasng along a soluon pah o (9), hen s s srcly decreasng and, f h s srcly decreasng, hen s s srcly ncreasng. We begn wh he case where h s srcly ncreasng. Suppose, o he conrary, ha, alhough h s srcly ncreasng, s s no srcly decreasng, ha s, here exss T such ha st st. Snce h s srcly ncreasng, equaon (45) mples ha Snce st st, hs mples ha s > s. (47) T+ T st st
s T+ > st, (48) st st whch mples ha st+ > st. Ierang, we fnd ha, for all > T, he sequence s s srcly ncreasng. Usng equaon (48), for all > T, we oban: h s s s = > = β s β s β +. (49) In he lm, he sequences h and ( s ) / β boh converge o he same lm, ( sˆ ) / β. Equaon (49) mples ha h sˆ >, (5) β whch conradcs our assumpon ha h s srcly ncreasng. We prove ha, when h s srcly decreasng, s s srcly ncreasng, usng he same argumen and jus reversng he nequales. The nex proposon provdes our man resuls on convergence n ncome levels for he Venura model. I follows mmedaely from proposon 5 and lemma. Proposon 6. (Convergence n relave ncome levels) In he Venura model wh complee deprecaon, f x > for all and all :. If b < and. If b < and / β > da / / b, hen here s convergence n relave ncome levels. / b β da, hen here s dvergence n relave ncome levels f convergence n relave ncome levels f ˆ >. 3. If b =, relave ncome levels say consan. 4. If b > and / b < ˆ and / β > da, hen here s convergence n relave ncome levels f ˆ < and dvergence n relave ncome levels f ˆ >. 5. If b > and / β da / b, hen here s convergence n relave ncome levels.
We have analyzed all of he cases enumeraed n he saemen of proposon 6 for he sae of compleeness. Case and cases and 4 where ˆ > are less neresng han he ohers. The conras of he remanng resuls wh he analogous resuls for a world of closed economes s srng: In cases, 3, and 4, f he counres are closed o rade, we now ha relave ncome levels converge over me because all counres have equlbra ha converge o he seady sae of he negraed economy. If we open he counres o rade, however, relave ncome levels dverge f b < and say fxed f b =. Noce ha, f b >, relave ncome levels converge, bu no o he same level as hey do n a world of closed economes. In case 5, f he counres are closed o rade, we now ha relave ncome levels dverge over me because growh acceleraes over me and counres ha sar wh lower ncome levels because hey have lower nal capal socs grow more slowly. If we open hese economes o rade, however, ncome levels converge. The nuon for he resuls n proposon 6, a leas for he cases where b < and / β da and whereb =, s obvous: In a world of closed economes, poor counres ha s, / b counres wh lower nal capal socs grow faser han rch counres because lower capal socs lead o hgher reurns on nvesmen. Trade equalzes he reurn on capal n poor and rch counres, elmnang he ncenve for hgher nvesmen n poor counres. The resuls n proposon 6 were derved under he assumpon ha x > for all =,..., n and =,.... The nex proposon derves necessary and suffcen condons on parameer values and nal capal endowmens under whch here exss an equlbrum wh posve nvesmen for all counres and all perods. Roughly speang, he proposon saes ha he only problemac cases are he ones where proposon 6 would mply ncome dvergence and one of he counres nal endowmen of capal s low enough ha s opmal for he counry o evenually ea up s capal soc and specalze n he producon of he labor nensve good. Even hough he cases wh zero nvesmen may be emprcally rrelevan, hey are heorecally valuable, snce hey sgnal ha he resuls n proposon 6 may no longer be vald f one of he counres reaches he border of he cone of dversfcaon. Our analyss of he general verson of he Venura model n secon 5 corroboraes hs suspcon. Proposon 7. In he Venura model wh complee deprecaon, () for he cases enumeraed n he saemen of lemma where he sequence s = c / y n he equlbrum of he negraed
economy s consan or srcly decreasng, here exss an equlbrum where x > for all and all. () For he cases where s s srcly ncreasng, le z / c =, z c / ( β r ) =, and c zˆ lm =. (5) Ths lm s well defned. Le mn be he counry wh he lowes nal endowmen of capal per worer, mn, =,..., n. If zˆ z mn, (5) hen here exss an equlbrum where x > for all and all. Oherwse, here s no equlbrum where x > for all and all. When here exss an equlbrum wh no corner soluons n nvesmen, s he unque such equlbrum. Proof: Noce ha, snce z s =, (53) s he sequence z has he same monooncy properes as he sequence s. () In he cases where s converges o β z converges o ( β ) / β. () In he cases where s converges o, f ( ˆ,) ˆ / f( ˆ,) z converges o f ( ˆ,) ˆ / ˆ. Equaon (4) mples ha, z z = =. (54) z z Snce assumpon A. mples ha >, we now ha ( )/ >. If z s wealy ncreasng, or f z s srcly decreasng bu condon (5) holds, hen x = > for all and all. If, on he oher hand, z s srcly decreasng and condon (5) does no hold, hen he nvesmen decsons n he negraed economy equlbrum canno be dsaggregaed as n proposon 5 o assgn nonnegave nvesmen o each counry n every perod. 3
Unqueness of he dsaggregaon of he negraed economy equlbrum, f exss, follows from he unqueness of he soluon o he one-secor socal planner s problem (9). Unqueness of he dsaggregaon of he negraed economy equlbrum s easy o esablsh because hs equlbrum solves a socal planner s problem. I s more dffcul o say anyhng abou unqueness of equlbra, f hey exs, ha nvolve corner soluons n nvesmen or, n he more general model n he nex secon, ha nvolve lac of facor prce equalzaon. In such equlbra, Pareo mprovemens are possble f we allow nernaonal borrowng and lendng. 5. Generalzed Venura model Consder a generalzaon of he Venura model n whch he producon funcons f, φ, and φ are general consan reurns o scale producon funcons. Proposons 3 and 4 ndcae ha we can fnd he negraed economy of he generalzed Venura model by solvng he onesecor growh socal planner s problem (9). In hs generalzed Venura model, facor prce equalzaon need no occur a any gven perod of me. Counres can specalze n he producon of one of he raded goods, facor prces can dffer across counres and, herefore, n general, he equlbrum canno be solved usng he negraed approach and he resuls n proposon 6 no longer apply. In wha follows, we consder a specfc verson of he generalzed Venura model for whch he cone of dversfcaon can be solved analycally and derve condons under whch facor prce equalzaon n a gven perod mples facor prce equalzaon n every subsequen perod. In such cases, he resuls of he Venura model on he evoluon of he dsrbuon of ncome apply o he generalzed Venura model. For suaons where facor prces do no equalze afer a fne number of perods, he analyss done n he Venura model s no longer vald. In fac, usng a numercal example, we show ha he predcons of he Venura model can acually be reversed. Our example llusraes he resrcveness of he facor prce equalzaon assumpon. 5.. The C.E.S. model Le us consder a model, whch we refer o as he C.E.S. model, where all producon funcons are C.E.S. In parcular, assume ha he producon funcons have he form: 4
b b ( ) / y = φ (, ) = θ α + ( α ) for j =, (55) j j j j j j j j j b b ( ) / b b f( y, y ) = d a y + a y, (56) where b, and a = a. Noce ha, snce we have assumed no facor nensy reversals n he producon of he raded goods, he producon funcons φ j, j =,, need o have he same consan elascy of subsuon σ = / ( b). Seng hs elascy equal o ha of he producon funcon for nvesmen good f allows us o analycally solve for he funcon F : where b b ( ) / b F (, ) = D A + A (57) A b b ( a αθ b ) ( aαθ b + ) = b b b b ( aαθ ) + ( aαθ ) + ( a( α) θ ) + ( a( α) θ ) b b b b b b b =, (58), A A and b b b b b b b b b b b D= d ( aαθ ) + ( aαθ ) + ( a( α) θ ) + ( a( α) θ ) f b, and n he Cobb-Douglas case ( b = ): F (, ) =, (59) A A D (6) A = aα + aα, A = A (6) α α a α α a ( ) ( ) d θ aα α θ a α α D =. (6) A A A A To deermne when facor prce equalzaon occurs and when does no, we need o characerze he cone of dversfcaon n he negraed economy and how changes wh he world capal-labor rao. Solvng he maxmzaon problem ha defnes F, (8), we fnd ha κ ( ( )/ ( )) p p = κ, j =,, (63) j j 5
where he consans κ, κ ha deermne he cone of dversfcaon have he form, for b, b b b b ( a( α) θ b ) + ( a( α) θ ) b b ( aαθ ) ( aαθ ) α κ j = α + b b, j =,. (64) For he Cobb-Douglas case he consans are α j A κ j = α j A, j =,. (65) The nex proposons esablsh condons under whch, for he C.E.S. model, facor prce equalzaon n a gven perod mples facor prce equalzaon n all subsequen perods. Proposon 8: In he C.E.S. model wh complee deprecaon, suppose ha he sequence s = c / y n he equlbrum of he negraed economy s wealy decreasng. Suppose ha facor prce equalzaon occurs n perod T. Then here exss an equlbrum n whch facor prce equalzaon occurs a all T. Furhermore, hs equlbrum s he only such equlbrum. Proof: Assume ha all counres are n he cone of dversfcaon a perod T. Defne usng he formula, > T, z T T =, (66) zt T where z / = c and z = c/( β r ) are defned as n he saemen of proposon 7. We need o show ha dsaggregang capal hs way eeps counres n he cone of dversfcaon and ha solves he equlbrum of our model economy. To prove ha he counres reman n he cone, we need o show ha, for all Tha s, for all T, T, κ κ. (67) κ. (68) κ 6
Snce we have assumed ha sequence s s wealy decreasng, we now from he proof of proposon 7 ha he sequence ( )/ s wealy decreasng, mplyng ha he nequales above are sasfed. To prove ha hese sequences of capals, ogeher wh he equlbrum prces of he negraed economy, are a soluon o he model economy, we defne where c s ( β ) s s r w r = = + τ =+ + τ δ, (69) w s and r s are equlbrum prces of he negraed economy, and show ha consumpons and capal socs defned hs way solve he equlbrum of our model economy. For he Cobb-Douglas model ( b = ), f facor prce equalzaon occurs along he equlbrum pah, we can acually solve for he equlbrum analycally. The nex proposon descrbes how he capal sequence of he negraed equlbrum s spl across counres n order o deermne he rade equlbrum. The complee analycal soluon s gven n appendx B. Proposon 9: In he Cobb-Douglas model wh complee deprecaon, suppose ha facor prce equalzaon occurs n perod T. Then facor prce equalzaon occurs a all T. Furhermore, = γ, (7) where γ = T / T and + = β AD for T. A Proof: The proof of hs proposon s a specal case of he proof of proposon 8. Proposon : In he C.E.S. model wh complee deprecaon, suppose ha he sequence s s srcly ncreasng. Agan le z / = c, z = c/( β r ), and c zˆ lm =. (7) Le mn be he counry wh he lowes nal endowmen of capal per worer, and le max be he mn max counry wh he hghes,, =,..., n. If 7
max mn zˆ zˆ κ κ, (7) z z hen here exss an equlbrum wh facor prce equalzaon n every perod. When such an equlbrum exss, s unque. If, however, eher of he condons (7) or Error! Reference source no found. s volaed, here s no equlbrum wh facor prce equalzaon n every perod.. Proof: Ths proof s an obvous generalzaon of he proof of proposon 7 usng he defnons n proposon 8. Even hough s proof s rval gven he machnery ha we have developed, proposon 9 s a powerful resul. Under some general condons, even f facor prce equalzaon occurs a a gven perod, a some pon n he fuure facor prces wll dffer across counres. In he case where b < and / β DA / b, for example, he unque equlbrum of he negraed economy converges o he nonrval seady sae, bu, f nal endowmens of capal per worer are suffcenly dfferen n he sense ha eher of he condons (7) s volaed, hen here s no dsaggregaed equlbrum ha corresponds o. Even f he world economy sars wh all counres dversfyng n producon and facor prces equalzed, a some pon a leas one counry necessarly has s capal-labor rao leave he cone of dversfcaon. Absracng away from he paerns of specalzaon, as Venura (997), Chen (99), and many ohers do, msses ou on some mporan dynamcs precsely n he neresng cases, he cases n whch here s poenally dvergence of ncome levels. In such cases, we canno use he negraed economy approach o solve for he equlbrum, and none of he analyss n proposons 3, 4, and 5 apples. Insead, we need o use numercal mehods o compue he equlbrum. In wha follows we presen numercal examples ha volae he assumpons of proposon 6 or of condon (7) or n proposon. In all hese examples, he convergence resuls drascally dffer from he convergence predcons under facor prce equalzaon n proposon 6. We explan how o compue equlbrum for he generalzed Venura model n secon 6. We presen hree numercal examples ha llusrae dfferen equlbrum paerns for he Venura model when here are corner soluons n nvesmen and for he generalzed Venura model when counres are no n he cone of dversfcaon along he enre equlbrum pah. In 8
each of he examples, we consder a wo-counry economy. We se β =.95, δ =, and L = L =. Example (Venura model): Ths example shows ha gnorng he possbly of corner soluons n he Venura model can lead o he wrong predcons on convergence n ncome levels. Suppose ha and ha = 6 and.5.5 ( ) f( x, x ) =.5x +.5x, (73) =. The equlbrum pahs for are depced by he sold lne n fgure. Noce ha counry has x = = sarng n perod 3. Fgure 3 depcs he relave ncome of counry. Symmery mples ha ( y y)/ y ( y y)/ y =. Proposon 5 gves us no ndcaon of wha happens o he dsrbuon of ncome across counres: ( y y)/ y ncreases from.363 n perod o.3 n perod and hen seadly declnes, unl equals.977 n he seady sae. Noce ha relave ncomes converge beween perods and even hough nvesmen s srcly posve n boh counres n boh perods. For he sae of comparson, we also plo n fgures and 3 he equlbrum pahs of capal socs and ncome dsrbuon for an economy wh nal endowmens = 5 and = (n dashed lnes). In hs case, x > for all 3 and and proposon 6 apples. In he economy wh more smlar nal capal socs, boh counres accumulae capal over me, and ncome levels dverge over me, wh ncreasng from.53 n perod o.77 n he seady sae. ( y y)/ y Example (Cobb-Douglas model): In hs example, he nal endowmens are dfferen enough so ha one of he counres (counry ) specalzes n producon along he equlbrum pah. Suppose ha φ (, ) =.6.4 (74) φ (, ) =.4.6 (75) f ( x, x ) = x x, (76).5.5 9
and ha he nal endowmens are = 4 and socs n each counry, =.. Fgure 4 shows he evoluon of capal (depced n sold lnes), and he cone of dversfcaon, κ, κ (depced n dashed lnes). The labor abundan counry, counry, specalzes n he producon of he labor nensve good, and he capal abundan counry, counry, dversfes. Ths paern of producon s mananed along he equlbrum pah, wh asympocally convergng o he boundary of he cone of dversfcaon. Fgure 4 also plos he cone of dversfcaon of he negraed economy assocaed o our model economy (depced n doed lnes), κ, κ. Noce ha he cone of dversfcaon of he negraed economy dffers from he cone of dversfcaon of he rade economy. Therefore, he paern of specalzaon canno be deermned by usng he former. Fgure 4 llusraes hs clearly: n perods,, and, 3 s nsde he cone of dversfcaon (counry dversfes), even hough s ousde he cone of dversfcaon of he negraed economy. The evoluon of relave ncome levels s presened n fgure 5. Income levels n our economy converge even hough proposon 6 says ha relave ncomes would say consan f boh counres were o dversfy. Example 3 (C.E.S. model wh b < ): Ths example complemens proposon by presenng an economy where one of he counres exs he cone of dversfcaon afer a fne number of perods. Suppose ha and ha = 5 and (.5.5 ) φ (, ) =.8 +. (77) (.5.5 ) φ (, ) =. +.8 (78).5.5 ( ) f( x, x ) =.5x +.5x, (79) =. Fgure 6 shows he evoluon of equlbrum capal socs over me (sold lnes). The cone of dversfcaon s depced n dashed lnes. Noce ha counry dversfes along he equlbrum pah. The evoluon of s depced n more deal n fgure 7, a blowup of he deal n fgure 6. As seen n he fgure, counry, he labor abundan counry, produces boh goods n perods and. In perod, jumps ousde he cone of dversfcaon and counry specalzes n he producon of he labor nensve good. (We could say ha he