NBER WORKING PAPER SERIES TRADE, GROWTH, AND CONVERGENCE IN A DYNAMIC HECKSCHER-OHLIN MODEL. Claustre Bajona Timothy J. Kehoe

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1 NBER WORKING PAPER SERIES TRADE, GROWTH, AND CONVERGENCE IN A DYNAMIC HECKSCHER-OHLIN MODEL Clausre Bajona Tmohy J. Kehoe Worng Paper 567 hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 5 Massachuses Avenue Cambrdge, MA 38 Ocober 6 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 Auonomo Tecnologco de Mexco, he 4 CNB/CERGE-EI Macro Worshop n Prague, he Unversa Pompeu Fabra, Sanford Unversy, he Unversy of Texas a Ausn, and he Banco de Porugal. 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. The vews expressed heren are hose of he auhor(s) and do no necessarly reflec he vews of he Naonal Bureau of Economc Research. 6 by Clausre Bajona and Tmohy J. Kehoe. All rghs reserved. Shor secons of ex, no o exceed wo paragraphs, may be quoed whou explc permsson provded ha full cred, ncludng noce, s gven o he source.

2 Trade, Growh, and Convergence n a Dynamc Hecscher-Ohln Model Clausre Bajona and Tmohy J. Kehoe NBER Worng Paper No. 567 Ocober 6 JEL No. F,F43,O5,O4 ABSTRACT Ths paper sudes he properes of 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, even f facor prces are equalzed, 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. Clausre Bajona Deparmen of Economcs 55 Unversy Drve Coral Gables, FL cbajona@exchange.sba.mam.edu Tmohy J. Kehoe Unversy of Mnnesoa Deparmen of Economcs 7 9h Avenue Souh Mnneapols, MN and NBER and Federal Reserve Ban of Mnneapols ehoe@econ.umn.edu

3 . Inroducon In 4, GDP per capa n he Uned Saes was roughly 4, U.S. dollars. Usng exchange raes o conver pesos o dollars, we calculae GDP per capa n Mexco n 4 o be roughly 6,5 U.S. dollars. In 935, he Uned Saes had ncome per capa of abou 6,6 4 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 935? Or should we ae no accoun ha, n 935, he Uned Saes was he counry wh he hghes ncome n he world, whle, n 4, Mexco had a very large rade relaon wh he Uned Saes a counry wh a level of ncome per capa approxmaely sx mes larger? If we use a purchasng power comparson mehod o calculae Mexcan GDP per capa n 4, we come up wh 9,8 U.S. dollars, whch was roughly he U.S. level n 94, bu he qualave naure of our queson remans he same. 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? Specfcally, we consder 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. 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. They combne he wo raded goods o oban uly n he same manner as frms combne hese goods o oban he nvesmen good n he sense ha he perod uly funcon has he form uc (, c) = log f( c, c), where f s he producon funcon of he nvesmen good. As we wll see, hs assumpon allows us o

4 reduce he calculaon of equlbra n whch all counres produce boh raded goods n every perod o he calculaon of an equlbrum of an appropraely specfed one-secor model. 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

5 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 her assumpons are 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, over me, 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, all counres produce boh goods 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 well as of 3

6 capal socs, (3) we oban condons under whch counres reman n he cone of dversfcaon and under whch hey leave n models wh more general producon srucures, and (4) we sudy he possbly of equlbra n whch one or more counres have zero nvesmen n some perods, a possbly ha s presen n Venura s (997) model, bu whch s gnored. 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 n he economy: an nvesmen good, x, whch s no raded, and wo raded goods, y, j =,, whch can be j consumed or used n he producon of he nvesmen good. Each raded good j, j =,, s produced wh a consan reurns o scale echnology ha uses capal,, and labor, : y = φ (, ). () j j j j We assume ha good s relavely capal nensve and ha he echnologes 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 4

7 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) argue ha allowng nernaonal borrowng and lendng ensures facor prce equalzaon bu resuls n ndeermnacy of producon and rade n equlbrum. The represenave consumer n counry solves he 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. (8) n n L( c ) j + x j = Ly = = j 5

8 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 descrbed above, we ls he man assumpons of he model: 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

9 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

10 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 8

11 for a descrpon of he mehodology.) In hs case, he equlbrum prces and aggregae 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 9

12 soluon o hs problem. Snce he funcon u s srcly concave and he funcons φ, φ, and f 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, o fnd, for example, c, =,..., n, such ha. (6) n n Lc / L = c = = 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.. + (7) +,. 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 j

13 specfed by hese secor-specfc capal-labor raos, whch depend only on he relave prce p / p, p p κ ( / ) and κ ( p / p). I s he se of counry specfc capal-labor raos such ha κ ( p / p ) κ ( p / p ). (8) 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 fnd a way o dsaggregae he nvesmen decsons such 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), (9) 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. To smplfy he analyss we furher 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)

14 s.. y φ (, ) () y φ (, ) + +,. j Assumpons A.6 and A.7 mply 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(,) () ( + 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 (). Proposon 4. Le y (, ), y (, ), (, ), (, ), (, ), (, ) denoe he soluon o (). 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 () where c = f( c, c). Conversely, f {,, } { c, c, }, { y,, }, { y,, }, {,, } c x solves he one-secor socal planner s problem (), hen x x x solves he wo-secor socal planner s problem

15 (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 resuls for he negraed economy do no hold. 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. In 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. Fnally, for he specal case of Cobb- Douglas producon funcons, we analycally solve he model when here s facor prce equalzaon. 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 (, ) =. (3) 3

16 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: f b, and f s b b ( ) / f( x, x ) = d a x + a x (4) b f ( x, x ) = dx x (5) 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. I s worh ponng ou ha Venura (997) consders a connuous-me verson of hs model. For compleeness, we laer sech ou our resuls for he connuous-me model. Suppose ha we fnd he equlbrum of he negraed economy by solvng he one-secor socal planner s problem (). To dsaggregae consumpon and nvesmen, we solve he uly maxmzaon of he represenave consumer, (7): max = β log c s.. c + x w + r (6) ( δ ) + x c, x j. If we solve (7), we can oban a soluon o (6) by seng c = f( c, c ), and, f we solve (6), 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 (6) are ha 4

17 ( r ) + β δ + +, f = x > (7) c c + and ha he ransversaly condon c + ( δ ) = w + r, (8) + lm β c = (9) + 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 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 he nal condon L / L n ( δ ( )) c = β + r c (3) + + c + ( δ ) = f(,), (3) + 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=. (33) 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 5

18 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 + (34) + Ths economy has a susaned growh pah n whch ( )( β δ) / b β( da + δ) 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 + (35) 6 / b ( δ ) lm + / = β da + (36) 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 () for he negraed economy:

19 a = x = + β ad (37) c = ( βa ) d (38) 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. 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. 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 = =. (39) 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 =. (4) c c + I s here ha he assumpon of no corner soluons n nvesmen s essenal, allowng us o mpose he frs-order condons (7) and (3) as equales. Manpulang he frs-order condons (7) and he budge consrans (6), we oban he famlar demand funcon for logarhmc uly maxmzaon: 7

20 c s ( β) ( ) s s r w r δ = = + + τ =+ + τ δ Noce ha, snce we have facor prce equalzaon, The budge consran (6) mples ha Combnng (3), (4), and (43), we oban. (4) c c = ( β)( + r δ)( ). (4) c c = ( + r δ )( ). (43) c = ( ). (44) + + c The dfference beween a counry s ncome per worer and he world s ncome per worer s y y = r ( ). (45) Usng he expresson for + + n (44), we oban y+ y r c / y y y =. (46) y+ rc / y y We can use he frs-order condon (3) o rewre hs expresson as where s = rc / y for,,... y y s y y s y y = =, (47) y s y s y = and s rc /[ β( r δ) y ] = +. When δ =, c / c βr mples ha we can facor β ou of he numeraor and he denomnaor of (47) and se s = c / y. = 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 (4) 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 (4) need no hold. We laer provde 8

21 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 (39) 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 r+ c / y+ decreases or ncreases over me. If he rao ncreases, counres ncomes move furher away from he ncome of he negraed economy and, hus, here s dvergence n ncome levels. If he rao decreases, counres ncome levels become closer o he 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. We should 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, can converge 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 (). 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. 9

22 Proof: Snce he resul for he case 3, where b =, follows rvally from equaon (38), we concern ourselves wh he oher cases, where b < or b >. Mulplyng and dvdng he Euler equaon, (3), by / y and usng he feasbly condon, (3), we oban s s where s = c / y and h = ( r ) / y. We defne he funcon = β h, (48) r ( ) a h ( ) = = f (,) a a b b +. (49) Noce ha h'( ) < f b < and ha h'( ) > f b >. Noce ha, n he lm where b =, h ( ) = a and h'( ) =. We use he monooncy of he sequence n any soluon o he onesecor socal planner s problem () 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 (), hen s s srcly decreasng and, f h s srcly decreasng, hen s s srcly ncreasng. We begn wh he

23 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 (48) mples ha: Snce st st, hs mples ha s > s. (5) T+ T st st s T+ > st, (5) st st whch mples ha st+ > st. Ierang, we fnd ha, for all > T, he sequence s s srcly ncreasng. Usng equaon (5), for all > T, we oban: h s s s = > = β s β s β +. (5) In he lm, he sequences h and ( s ) / β boh converge o he same lm, ( sˆ ) / β. Equaon (5) mples ha h sˆ >, (53) β whch conradcs our assumpon ha h s srcly ncreasng. We prove ha, when h s srcly decreasng, s s srcly decreasng, usng he same argumen and jus reversng he nequales. The nex proposon provdes our man resuls 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 / β > da / b, hen here s convergence n relave ncome levels.

24 . If b < and / β da, hen here s dvergence n relave ncome levels f ˆ < and / b convergence n relave ncome levels f ˆ >. 3. If b =, relave ncome levels say consan. 4. If b > and / / b β > da, hen here s convergence n relave ncome levels f dvergence n relave ncome levels f ˆ >. 5. If b > and / β da / b, hen here s convergence n relave ncome levels. < ˆ and 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 he 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. We are lef wh he queson: When are here corner soluons n nvesmen for ndvdual counres, whch mae he negraed economy approach and he characerzaon of equlbra n proposons 3, 4, 5, and 6 nvald? The answer s found n he nex proposon. 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

25 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 =. (54) 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, (55) 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 =, (56) 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 (44) mples ha, z z = =. (57) z z Snce assumpon A. mples ha >, we now ha ( )/ >. If z s wealy ncreasng, or f z s srcly decreasng bu condon (55) holds, hen x = > for all and all. If, on he oher hand, f z s srcly decreasng and condon (55) 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

26 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 (). 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 (). 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, he equlbrum canno be solved usng he negraed approach n general. In wha follows, we characerze he cone of dversfcaon for some specfc versons of he model 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. Numercal expermens are needed o deermne he behavor of he counres dsrbuon of ncome. 5.. The C.E.S. model We frs consder he model n whch b b ( ) / y = φ (, ) = θ α + ( α ) (58) b b ( ) / y = φ (, ) = θ α + ( α ) (59) b b 4

27 b b ( ) / f( y, y ) = d a y + a y, (6) where b, 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. Seng hs elascy equal o ha of he producon funcon for nvesmen good f allows us o analycally solve for he funcon F. We refer o hs as he C.E.S. model. Here he parameer b deermnes he common elascy of subsuon σ = /( b), and he producon funcon F defned n () s also a C.E.S. producon funcon, wh he same elascy of subsuon and wh he share parameers ha are combnaons of he share parameers of he producon funcons φ, φ, and f : b b ( ) / b b F (, ) = D A + A (6) 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 = (6), A A b b b b b b b b b b b D= d ( aαθ ) + ( aαθ ) + ( a( α) θ ) + ( a( α) θ ). (63) 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. One procedure would be o solve (7) o deermne he secor-specfc capal-labor raos as funcons of relave prces, κ ( p / p) and κ ( p / p), hen use proposon 4 o deermne he prces n he negraed economy equlbrum, p ( ) f ( y (,), y (,)) = and = ( ), and, fnally, calculae κ ( p ( )/ p ( )) and κ ( p ( )/ p ( )) p ( ) f y (,), y (,) he C.E.S. model, he deermnaon of he cone of dversfcaon of he negraed economy s far smpler han hs. Solvng he maxmzaon problem ha defnes F, (), we fnd ha κ ( ( )/ ( )) j j. In p p = κ, j =,, (64) 5

28 where he consans κ, κ ha deermne he cone of dversfcaon have he form b b b b ( a( α) θ b ) + ( a( α) θ ) b b ( aαθ ) ( aαθ ) α κ j = α + b b, 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. The frs proposon gves suffcen condons for facor prces o equalze along he equlbrum pah, gven ha facor prce equalzaon occurs n a gven perod T. The ey parameer s, once agan, b. In parcular, when b >, facor prce equalzaon a T ensures facor prce equalzaon n any subsequen perod, a leas for he economcally neresng cases eher where ˆ and or where goes whou bound. The second proposon gves suffcen condons under whch facor prce equalzaon canno hold forever. I saes ha, when b <, and when / β DA and < ˆ, f facor prce equalzaon holds a a perod T and he capal-labor rao of one of he counres s close enough o he boundary of he cone of dversfcaon, facor prce equalzaon canno hold for all subsequen perods. The nuon s smple. If facor prce equalzaon were o occur forever, he analyss n he Venura model would apply, and he dsrbuon of capal-labor raos would become more dspersed over me. Snce he boundares of he cone of dversfcaon for he negraed economy are lnear funcons of he world capal-labor rao, however, f he capal-labor rao of one of he counres s close enough o he boundary of he cone, he dsrbuon of capal-labor raos canno become more dspersed f all capal-labor raos are o reman n he cone. 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. / b Proof: Assume ha all counres are n he cone of dversfcaon a perod T. Defne usng he formula, > T, 6

29 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 Snce we have assumed ha sequence s s wealy decreasng, we now from he proof of proposon 7 ha he sequence s wealy decreasng. 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. Proposon 9: 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

30 zˆ z mn κ (7) zˆ z max κ, (7) hen here exss an equlbrum wh facor prce equalzaon n every perod. If, however, eher of he condons (7) or (7) s volaed, here s no equlbrum wh facor prce equalzaon n every perod. When here exss an equlbrum wh facor prce equalzaon n every perod, s he unque such equlbrum. 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) or (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, can cause us o mss 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. We brefly explan how o compue equlbrum for he generalzed Venura model n secon 6 and presen examples of economes for whch facor prces are no equalzed along he equlbrum pah n secon 7. 8

31 5.. The Cobb-Douglas model In hs secon, we consder he lmng case of he C.E.S. model wh complee deprecaon and wh b =, ha s, wh producon funcons ha are Cobb-Douglas. = φ(, ) = y α α θ (73) = φ(, ) = y α α θ (74) a a f ( y, y ) dy y =. (75) In hs case, he funcon F s also Cobb-Douglas: F (, ) A A = D (76) A = aα + aα, A = A (77) α α a α α a ( ) ( ) d θ aα α θ a α α D =. (78) A A A A The consans ha deermne he cone of dversfcaon n he Cobb-Douglas case are α j A κ j = α j A, j =,. (79) Proposon : 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, (8) T where γ = / and + = β AD for T. T A The proof of hs proposon s a specal case of he proof of proposon 8. Noce ha, when all are n he cone of dversfcaon, we can use proposon o oban analyc soluons for all varables. Le γ = /, where κ γ κ. r = AD A, w = A D (8) A 9

32 a α ( κ ) θκ p = ad α ( κ) θκ a( α α ), a α ( κ) θκ p = ad α ( κ ) θκ a( α α ) (8) x = = γβad (83) A + a α D ( κ) θκ = α + d ( κ ) θκ ( γ γ β ) c A A A α a D ( κ ) θκ = α + d ( κ) θκ ( γ γ β ) c A A A α α,, a α D ( κ) θκ = α γβ d ( κ ) θκ x x A α a D ( κ ) θκ = α γβ d ( κ) θκ A α α (84) (85) = κ κ y γ κ θ κ ( ) α = κ κ y κ γ θ κ ( ) α γ κ κ, =, = κ κ κ κ κ γ κ, =, = κ κ κ κ γ κ (86) κ γ (87) where A A ( β ) ( ) /( ) A A = βad = AD. (88) 6. Compuaon of equlbrum In characerzng he cone of dversfcaon of he negraed economy as he se of capallabor raos ha sasfy κ κ, we have reled heavly, no jus on he assumpon of specfc funconal forms for φ, φ, and f, bu also on he assumpon ha all counres produce boh goods. Under hese assumpons, we derve opmal capal-labor raos n each ndusry as funcons of he world capal-labor rao. If capal-labor raos for all counres are nsde he cone of dversfcaon, we are jusfed n usng he negraed economy approach. If no, and a leas one of he counres specalzes, p / p does no, n general, equal o s value n he negraed equlbrum. Consequenly, we canno use κ ( ( )/ ( )) p p = κ o characerze he j j cone of dversfcaon and o deermne he paern of specalzaon. Insead, we mus calculae κ ( p/ p) and p p κ ( / ) by solvng (7). In he nex secon, we provde an example ha llusraes how κ j( p / p ) dffers from κ j when one of he counres specalzes. 3

33 In he C.E.S. model, he cone of dversfcaon s deermned by he capal-labor raos κ /( b) / b /( b) b/( b) /( b) b/( b) α ( α) ( θp / p) ( α) θ ( p / p) = /( b) b/( b) /( b) b/( b) α α θ α ( θp/ p) (89) /( b) α α p p = κ p p α α κ ( / ) ( / ). (9) In Cobb-Douglas model, he lmng case where b =, hese become κ ( p / p ) α α θp / p α α = θ α α /( α α ) (9) α α κ( p / p) = κ( p / p). (9) α α If he negraed economy approach s vald because here are no corner soluons n nvesmen and all counres have capal-labor raos ha reman n he cone of dversfcaon, hen we need only solve for he sequence of capal socs for he negraed economy by solvng he one-secor socal planner s problem () o solve for equlbrum. Gven hs soluon, we can hen use proposons 3 and 4 calculae he equlbrum prces and he formula z T T = zt T (93) o dsaggregae consumpon and nvesmen decsons. We dsaggregae producon decsons by requrng ha all counres use capal and labor n he opmal proporons, / = κ, and j j j sasfy he feasbly condons (9) and (). If he negraed economy approach s no vald, he suaon s far more complcaed. To eep our dscusson smple, we gnore he possbly of corner soluons n nvesmen and nsead focus on he case where complcaons arse because of specalzaon n producon. The approach ha we ae s o guess he sequence of capal socs for all counres and he sequence of prces for good. Gven he prce of good, we use he frs-order condons f ( x, x ) = f ( c, c ) = p (94) 3