Federal Reserve Ban of Mnneapols Research Deparmen Saff Repor 377 Sepember 6 Demographcs n Dynamc Hecscher-Ohln Models: Overlappng Generaons versus Infnely Lved Consumers* Clausre Bajona Unversy of Mam Tmohy J. Kehoe Unversy of Mnnesoa, Federal Reserve Ban of Mnneapols, and Naonal Bureau of Economc Research ABSTRACT Ths paper conrass he properes of dynamc Hecscher-Ohln models wh overlappng generaons wh hose of models wh nfnely lved consumers. In boh envronmens, f capal s moble across counres, facor prce equalzaon occurs afer he nal perod. In general, however, he properes of equlbra dffer drascally across envronmens: Wh nfnely lved consumers, we fnd ha facor prces equalze n any seady sae or cycle and ha, n general, here s posve rade n any seady sae or cycle. Wh overlappng generaons, n conras, we consruc examples wh seady saes and cycles n whch facor prces are no equalzed, and we fnd ha any equlbrum ha converges o a seady sae or cycle wh facor prce equalzaon has no rade afer a fne number of perods. * A prelmnary verson of hs paper wh he le On Dynamc Hecscher-Ohln Models I: General Framewor and Overlappng Generaons was crculaed n January 3. 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 Ths paper sudes he properes of dynamc Hecscher-Ohln models combnaons of sac Hecscher-Ohln rade models and wo-secor growh models wh wo dfferen demographc envronmens, an nfnely lved consumer envronmen and an overlappng generaons envronmen. In he model, a fne number of counres ha dffer only n populaon szes and nal endowmens of capal nerac wh each oher by exchangng wo raded goods, whch are produced usng capal and labor. The raded goods are used eher n consumpon or n he producon of a nonraded nvesmen good. Consumers supply labor nelascally and choose levels of consumpon and capal accumulaon o maxmze her lfeme uly. We fnd ha he equlbrum properes of he model depend crucally on he assumpons made on nernaonal capal mares and on he choce of demographc envronmen. If nernaonal borrowng and lendng are permed, facor prces equalze afer he frs perod, ndependenly of he envronmen. Furhermore, he levels of capal and of nernaonal borrowng are no deermned n equlbrum. A any gven pon n he equlbrum pah, here s a connuum of possble connuaon pahs ha have all he same prces and aggregae varables, bu dffer n he dsrbuon of capal and nernaonal borrowng across counres and n he paern of producon and rade. If nernaonal borrowng and lendng are no permed, hen he equlbrum properes vary dependng on he demographc envronmen. In he nfnely lved consumer envronmen, f a nonrval seady sae exss, here s a connuum of nonrval seady saes, ndexed by he dsrbuon of capal across counres. To whch seady sae he world economy converges depends on nal endowmens of capal. Facor prces equalze n all seady saes and rade s posve n all seady saes excep he one where capal-labor raos are equal across counres. In he overlappng generaons envronmen, any nonrval seady sae wh facor prce equalzaon s auarc and has no rade. Furhermore, we show usng examples ha facor prce equalzaon does no need o occur n seady sae. Boh envronmens also dffer n he behavor of equlbrum pahs ha converge o a seady sae. In he overlappng generaons envronmen, any equlbrum convergng o a seady sae where facor prces equalze becomes auarc n a fne number of perods. No correspondng resul exss for he nfnely lved consumer envronmen. As n wo-secor closed economy models, equlbrum pahs may exhb cycles and chaoc behavor. In he nfnely lved consumer envronmen, we show ha facor prces equalze n any
equlbrum cycle. In he overlappng generaons envronmen, we show usng examples ha here may exs equlbrum cycles n whch facor prces are no equalzed. The paper develops a mehodology for consrucng wo-counry, wo-secor overlappng generaons models from closed one- and wo-secor economes n such a way ha preserves her properes n erms of mulplcy of equlbra or cyclcal behavor. The leraure on dynamc Hecscher-Ohln models was poneered by On and Uzawa (965), Bardhan (965), Sglz (97), and Deardorff and Hanson (978). In her models, counres produce wo raded goods, a consumpon good and an nvesmen good, usng producon funcons ha dffer across secors bu no across counres. In addon o dfferences n endowmens, hese papers also assume ha counres have dfferen savngs raes or raes of me preference, so ha he seady sae s deermned ndependenly of nal condons. Ther assumpons preven facor prces from equalzng n he seady sae. Our model dffers from hers n wo crucal aspecs: we consder uly-maxmzng consumers and we do no mpose any modelng assumpons resrcng he equlbrum behavor of facor prces. Baxer (99) sudes he long-run behavor of a dynamc Hecscher-Ohln model where counres dffer n ax polcy and shows how changes n ax polcy may lead o reversals n comparave advanage. Cuña and Maffezzol (4a) calbrae a specfc dynamc rade model and sudy ssues of convergence n ncome levels across counres under he assumpon ha facor prces do no equalze over me. In conras o he prevous sudes, more recen papers n he leraure mpose, by assumpon or by he choce of producon funcons, facor prce equalzaon along he equlbrum pah. Cuña and Maffezzol (4b) nroduce echnology shocs and sudy he busness cycle properes of a dynamc Hecscher-Ohln model under he assumpon of facor prce equalzaon. Chen (99) sudes he equlbrum properes of a dynamc Hecscher-Ohln model wh elasc labor supply under he assumpon ha facor prces equalze along he equlbrum pah. Venura (997) adds addonal srucure o he model ha guaranees ha facor prces equalze n equlbrum, ndependenly of nal condons, and ha rules ou he possbly of cyclcal and chaoc behavor. He derves resuls regardng convergence of ncome dsrbuon across counres over me. Bajona and Kehoe (6) sudy he properes of a generalzed verson of Venura s model whou mposng facor prce equalzaon. They show ha he convergence resuls of Venura (997) depend crucally on he facor prce equalzaon assumpon.
A relaed leraure consders wo-secor growh models wh nfnely lved consumers under he small open economy assumpon. In he nfnely lved consumer envronmen, hese papers nclude Fndlay (97), Mussa (978), Smh (984), Aeson and Kehoe (), Chaerjee and Shuayev (4), and Obols-Homs (5). The leraure on dynamc Hecscher-Ohln models n an overlappng generaons envronmen s less abundan. Galor (99) characerzes he dynamcs of a wo-secor, woperod-lved overlappng generaons model of growh n a closed economy. Two-counry models of rade wh an overlappng generaons envronmen assume some dfference across counres besdes facor endowmens. For example, Bancon (995) assumes dfferences n ax raes across counres; Galor and Ln (997) and Mounford (998) assume ha counres dffer n her raes of me preference; Sayan (5) assumes dfferences n populaon growh raes. All hese papers sudy he long-run properes of he model under he facor prce equalzaon assumpon. Papers ha sudy he wo-secor overlappng generaons envronmen under he small open economy assumpon nclude Serra (99), Goceus and Tower (998), Kemp and Wong (995), and Fsher (99). A recen leraure consrucs dynamc Hecscher-Ohln models ha exhb endogenous growh. In he nfnely lved envronmen, Nshmura and Shmomura () and Bond, Tras and Wang (3) derve some resuls regardng ndeermnacy of equlbra. Gulló (999) and Mounford (999) nroduce producon exernales n he overlappng generaons envronmen.. The model There are n counres n he model, whch dffer n her populaon szes and her nal endowmens of capal. Each counry can produce hree goods: wo raded goods a capal nensve good and a labor nensve good and a nonraded nvesmen good. The echnologes avalable o produce hese goods are he same across counres. Each raded good j, j =,, s produced usng capal and labor accordng o he producon funcon y = φ (, ) () j j j j A.. The funcons φ j are ncreasng, concave, connuously dfferenable, and homogeneous of degree one. 3
We assume ha φ j s connuously dfferenable o smplfy he exposon, and we le addonal subscrps φ (, ), φ (, ) denoe paral dervaves. I s an open queson wheher jk j j jl j j any subsanve concluson depends on hs assumpon. In parcular, our analyss s easly exended o he fxed coeffcen producon funcons, y = mn[ / a, / a ]. j j jk j jl Producers mnmze coss ang prces as gven and earn zero profs: r p φ (, ), = f > () j jk j j j w p φ (, ), = f > (3) j jl j j for j =,. Here r s he renal rae, w s he wage, and p and p are he prces of he raded goods. A.. Good s relavely capal nensve and here s no capal nensy reversal. Tha s, j φl( /,) φl( /,) < φ ( /,) φ ( /,) K K for all / >. (4) Ths condon and he concavy of φ and φ mply ha for any wage-renal rao w/ r, he prof maxmzng capal-labor raos sasfy >. (5) Noce ha, f he producon funcons φ j, j =,, dsplay consan elasces of subsuon, assumpon A. mples ha boh producon funcons have he same elascy of subsuon. The nvesmen good s produced usng he wo raded goods: x = f( x, x ). (6) A.3. The funcon f s ncreasng, concave, connuously dfferenable, and homogeneous of degree one. Capal deprecaes a he rae δ, δ >. The frs-order condons for prof maxmzaon are p qf( x, x), = f x > (7) 4
p qf( x, x), = f x >, (8) where q s he prce of he nvesmen good. A.4. Labor and capal are no moble across counres, bu are moble across secors whn a counry... Infnely lved consumers In he envronmen wh nfnely lved consumer-worers, each counry, =,..., n, has a connuum of measure L of consumers, each of whom s endowed wh > uns of capal n perod and one un of labor a every perod, whch s suppled nelascally. Consumers have he same uly funcons, whn counres and across counres. In each perod, he represenave consumer n counry decdes how much o consume of each of he wo raded goods n he economy, c, c, how much capal o accumulae for he nex perod, +, and how much o lend, b +. Consumers derve her ncome from wages, w, reurns o capal, b r. The represenave consumer n counry solves he problem = max β uc (, c ) b + r, and reurns o lendng, s.. p c + p c + q x + b w + r + ( + r ) b (9) ( + δ ) x c, x, j b, b. B Here B s a posve number large enough so ha he consran bu does no oherwse bnd n equlbrum. b B rules ou Ponz schemes A.5. The perod uly funcon uc (, c ) s homohec, srcly ncreasng, srcly concave, and connuously dfferenable, wh lm c uj( c, c) =, and lm c uj( c, c) =. j j The frs-order condons of hs consumer s problem (9) mply ha 5
u ( c, c ) p = () u c c p (, ) u ( c, c ) p ( q+ ( δ ) + r+ ), = f x > () βu c c p q ( +, + ) + Furhermore, he reurns o capal and o nernaonal bonds, f permed, have o be he same: q ( δ ) + r + r, = f >. () b + + + q The feasbly condons for each raded good j, j =,, n perod, =,,..., s x. (3) n n L( c ) j + x j = Ly = = j Here y j and x j denoe, respecvely, he oupu and npu no nvesmen of raded good j n counry, boh expressed n per worer erms. Noce ha, because each consumer-worer s endowed wh one un of labor n every perod, hese quanes are also he same quanes per un of labor. I s easy o modfy he model, as does Venura (997), so ha he endowmen of labor per worer dffers across counres, as long as hese dfferences reman consan over me. The feasbly condons for facors and for he nvesmen good are + (4) (5) + ( + δ ) x. (6) The mare clearng condon for nernaonal bonds s n Lb = (7) = when nernaonal borrowng and lendng are permed. If no, hs condon becomes b =... Overlappng generaons In he envronmen wh overlappng generaons, a new generaon of consumer-worers s born n each perod n each counry. Consumers n generaon, =,,... are born n perod and 6
lve for m perods. Each of hese generaons n counry has a connuum of measure consumers. In perod of lfe h, h=,..., m, each consumer s endowed wh L of h uns of labor, whch are suppled nelascally. Consumers can save hrough accumulaon of capal and bonds. We assume ha hey are born whou any nal endowmen of capal or bonds. The represenave consumer born n counry n perod, =,,..., solves m β huhc h c h= + + h max (, ) s.. p c + p c + q x + b w + r + ( + r ) b (8) h b + h + h + h + h + h + h + h + h + h + h + h + h ( δ ) x + h + h + h c +, x + j h, b, x + ( δ ) +, b +, m where u h s he uly funcon n perod of lfe h and sasfes he analogue of assumpon A.5: h m m A.5'. For every h, h=,..., m, he uly funcon u ( h c, ) c s homohec, srcly ncreasng, srcly concave, and connuously dfferenable, wh lm c uhj( c, c) = and lm (, ) cj uhj c c =. j In addon, here are m generaons of nal old consumers alve n perod. Each generaon s, s = m+,...,, n counry has a connuum of measure L of consumers, each of whom lves for m+ s perods and s endowed wh h= m+ s. h s uns of labor n perod h,,..., s s Each nal old consumer s also endowed wh capal and bonds b. The represenave consumer of generaon, = m+,...,, n counry solves m β huhc h c h= + + h max (, ) s.. p c + p c + q x + b w + ( + r ) b + r (9) h b + h + h + h + h + h + h + h + h + h + h + h + h ( δ ) x + h + h + h c +, x + j h h h, b b, x + ( δ ) +, b +. m m m 7
Noce ha n each counry and each perod, he oal populaon s h oal supply of labor per perod o be, = he oal amoun of labor s m h= ml and normalzng he The feasbly condon for raded good j, j =,, n perod, =,,..., s Noce ha, n hs noaon, + ( ) n L m c h n j + x h j = Ly = = = j L.. () y j and x j are expressed n erms of per un of labor, no n erms of per capa. The oupu per capa of raded good j n counry n perod s y / m, for example. The feasbly condons for facor npus and for he nvesmen good are he same as n he nfnely lved consumer envronmen, and are gven by equaons (4), (5), and (6). Gven nergeneraonal heerogeney, we need o mpose addonal feasbly condons n he overlappng generaons envronmen: j x m h x + h= = () m = +. () h= h We also need o mpose a mare clearng condon on bonds. If nernaonal borrowng and lendng are permed, hen r b If he nal nomnal asses sasfy = r and hs condon s b ( ( ) s= s ). (3) n m h+ b n h L b h = + r L b = = = h= m hen he world has fa money. If, however,, (4) n h L b = h= m =, (5) n h L b = h= m hen he world does no have fa money. If nernaonal borrowng and lendng are no permed, hen he mare clearng condon on bonds whn each counry becomes ( ( ) s= s ), =,..., n. (6) m h + b b h h = + r b = h= m 8
If h b, (7) h= m hen counry has fa money, and, f h b =, (8) h= m hen counry does no have fa money. 3. Equlbrum We gve unfed defnons of equlbrum and of seady saes for Hecscher-Ohln models wh nfnely lved consumers and wh overlappng generaons. Defnon. There are n counres of dfferen szes, = n, and dfferen nal L,,..., endowmens of capal and bonds: and b, =,..., n, n he envronmen wh nfnely lved s s consumers and and b, s = m+,...,, =,..., n, n he envronmen wh overlappng generaons. An equlbrum s sequences of consumpons, nvesmens, capal socs, and bond holdngs, { c, c, x,, b } n he envronmen wh nfnely lved consumers and s s s s s { c, c, x,, b }, s= m+,...,, n he envronmen wh overlappng generaons, oupu and npus for each raded ndusry, { y,, l }, j =,, oupu and npus for he nvesmen secor {,, } j j j b x x x, and prces { p, p, q, w, r, r }, =,..., n, =,,..., such ha b. Gven prces { p, p, q, w, r, r }, he consumpon and accumulaon plan { c, c, x,, b } solves he consumers problem (9) n he envronmen wh nfnely lved consumers, and he s s s s s consumpon and accumulaon plan { c, c, x,, b } solves he consumers problems (8) and (9) n he envronmen wh overlappng generaons. b. Gven prces { p, p, q, w, r, r }, he producon plans { y,, l } and { x, x, x } sasfy j j j he cos mnmzaon and zero prof condons (), (3), (7), and (8). s s s s s 3. The consumpon, capal soc, { c, c, x,, b } or { c, c, x,, b }, and producon plans, { y,, l } and { x, x, x }, sasfy he feasbly condons (), (6) and (3) (7) n he j j j 9
nfnely lved consumer envronmen and () (8) n he overlappng generaons envronmen. Defnon. A seady sae s consumpon levels, an nvesmen level, a capal soc, and bond holdngs, ( ˆ ˆ ˆ ˆ ˆ c, c, x,, b ) n he envronmen wh nfnely lved consumers and ( cˆ, cˆ, xˆ, ˆ, b ˆ ), s=,..., m, n he envronmen wh overlappng generaons, oupu and npus s s s s s for each raded ndusry, ( ˆ, ˆ, ˆ y l ), j =,, oupu and npus for he nvesmen secor, ( ˆ, ˆ, ˆ ) x x x, and prces, j j j b ( pˆ, pˆ, qˆ, wˆ, rˆ, r ˆ ), =,..., n, ha sasfy he condons of a compeve equlbrum for approprae nal endowmens of capal and bonds, ˆ =, ˆ b = b n he envronmen wh nfnely lved consumers and s ˆs = and b s ˆs = b n he envronmen wh overlappng generaons. Here we se ν = ˆ ν for all, where ν represens a generc varable. Defnon 3. An equlbrum dsplays susanable growh f here exss a consan < γ < such ha lmnf + / = lmnf c+ / c = γ, =,..., n. We wll ofen assume ha he nal condons are such ha all counres produce a posve amoun of he nvesmen good n every perod, x >, and we normalze q = q = for all. We mae wo remars regardng hs assumpon: Frs, snce he nvesmen good s produced usng he wo raded goods, and hese prces are equalzed across counres by rade, even f a counry does no produce he nvesmen good n a gven perod, he prce of he nvesmen good s he same as n counres ha produce he nvesmen good. Second, n he examples presened n hs paper, we assume complee deprecaon, δ =, and he assumpon of posve nvesmen becomes an assumpon of posve capal, x = >. Posve capal n all counres n every perod can be ensured by assumng Inada condons on he producon funcons φ j. Bajona and Kehoe (6) consruc an example where here are corner soluons n nvesmen and n capal n a model where producon funcons do no sasfy Inada condons. The characerzaon and compuaon of equlbrum of he models descrbed above s dffcul n general because nvolves deermnng he paern of specalzaon n producon over an nfne horzon. In parcular, for any prces of he raded goods, p, p, here exss hreshold
values κ ( p / p ) and κ ( p / p ) such ha a counry produces posve amouns of boh raded goods f and only f s capal-labor rao sasfes κ ( p / p ) > > κ ( p / p ). (9) The se of capal-labor raos ha sasfy wea versons of hese nequales s called he cone of dversfcaon. Fgure, nown as he Lerner dagram, depcs he cone of dversfcaon graphcally. Any wo counres wh endowmens n he cone of dversfcaon use capal and labor n he same proporons and face he same facor prces, r and w. If he assumpon of no facor nensy reversals, A., s volaed, here can be more han one cone of dversfcaon and more han one par of facor prces compable wh producon of boh goods n equlbrum. If all counres have endowmens n he cone of dversfcaon a some prces of he raded goods, we say ha facor prce equalzaon occurs a hose prces. Ths resul s he facor prce equalzaon heorem of sac Hecscher-Ohln heory. Noce ha, gven he endowmens of capal and labor n each counry n each perod, he producon of raded goods s dencal o ha n a sac, wo-secor Hecscher-Ohln model. Consequenly, he Rybszyns heorem and he Solper-Samuelson heorem also hold n our model. To prove a verson of he Hecscher-Ohln heorem for hs model ha, n any perod, a counry expors he good ha s nensve n he facor n whch s abundan we would need o mae assumpons o ensure ha rade s balanced for each counry and ha expendures on he wo raded goods are proporonal across counres. In he nex secon, we assume ha here s no nernaonal borrowng and lendng, whch ensures ha rade s balanced for each counry. Assumpon A.3 ensures ha x / x s equal across counres. In he nfnely lved consumer envronmen, assumpon A.5 ensures ha c / c s equal across counres, bu our assumpons do no mpose any resrcons on ( c )/( + x c + x). If we assume ha uc (, c) = v( f( c, c)), where v s srcly concave and ncreasng, hen we now ha c / c = x / x, whch mples ha ( c + x )/( c + x ) s equal across counres and ha he Hecscher-Ohln heorem holds. Smlarly, n he overlappng generaons envronmen, we would need o srenghen assumpon A.5' by assumng ha uh( c, c) = vh( f( c, c)) for all h o be able prove he Hecscher-Ohln heorem. If we do no mae hs sor of assumpon on he relaon beween consumpon and
nvesmen, here s no reason o expec he Hecscher-Ohln heorem o hold. I s easy o consruc examples n whch a capal abundan counry expors he labor nensve good. If facor prce equalzaon occurs n every perod, he equlbrum prces and aggregae varables of he model can be deermned by solvng for he equlbrum of he negraed economy, a closed economy wh facor endowmens equal o he world endowmens of he facors of producon. (See Dx and Norman 98 for a descrpon of he mehodology.) The equlbrum of he n -counry model s hen compued by dsaggregang he consumpon, producon, and nvesmen allocaons of he negraed economy across counres n a way ha s conssen wh nal condons. The negraed economy approach grealy smplfes he characerzaon of equlbrum n sac models. As we wll see, s even more useful n he dynamc models consdered n hs paper. The queson arses of how general s a suaon where facor prces are equalzed along he equlbrum pah. The exsng leraure absracs away from hs queson eher by assumng facor prce equalzaon along he equlbrum pah as n Chen 99, Venura 997, and Cuña and Maffezzol 4a or by no allowng for facor prces o equalze as n Baxer 99 and Cuña and Maffezzol 4b. Ths paper derves general resuls regardng facor prce equalzaon n long-run equlbra and along equlbrum pahs. 4. General model wh nernaonal borrowng and lendng In hs secon, we oban wo resuls for models wh boh nfnely lved consumers and overlappng generaons when nernaonal borrowng and lendng are permed: Frs, facor prce equalzaon occurs afer he nal perod. Second, he equlbrum paerns of producon, rade, capal accumulaon, and nernaonal borrowng are no deermnae. All proofs are n appendx. Proposon : In boh a model wh nfnely lved consumers ha sasfes A. A.5 and a model wh overlappng generaons ha sasfy assumpons A. A.4 and A.5', assume ha nernaonal borrowng and lendng are permed. Also assume ha x > for all and all. Then facor prce equalzaon occurs for all =,,.... Proposon : In boh a model wh nfnely lved consumers ha sasfes A. A.5 and a model wh overlappng generaons ha sasfy assumpons A. A.4 and A.5', assume ha nernaonal
borrowng and lendng are permed. Also assume ha x > for all and all. Then counres producon plans and nernaonal rade paerns are no deermnae n any perod >. The nuon for hese proposons s he classc resul n sac Hecscher-Ohln heory ha rade n goods s a subsue for facor mobly. (See, for example, Mundell 957 and Marusen 983.) The assumpon ha x > for all and all s a far sronger assumpon han we need o prove proposon as we explan n he proof of he proposon n he appendx. In he res of he paper, we sudy versons of he general model where nernaonal borrowng and lendng are no permed. A.6. In he nfnely lved consumer envronmen, assume ha b = for all =,,..., =,..., n. A.6'. In he overlappng generaons envronmen, assume ha ( ( ) s= s ) for all =,,..., =,..., n. m h+ b h b h = + r b = h= m In addon, we wll somemes assume ha consumers aggregae he wo raded goods o oban uly n he same way ha frms aggregae hese goods o oban he nvesmen good. Defnon 4. A model wh nfnely lved consumers ha sasfes A. A.6 s one-secor aggregaable f uc (, c) = v( f( c, c)) for some v ha s connuously dfferenable, srcly concave, and srcly ncreasng. Smlarly, a model wh overlappng generaons ha sasfes A. A.4 and A.5' A.6' s one-secor aggregaable f uh( c, c) = vh( f( c, c)) for some v h, h=,..., m, ha sasfy hese properes. As we have seen, hs assumpon whch s very resrcve guaranees ha rade paerns obey he Hecscher-Ohln heorem. As we shall see, s also useful n resrcng he possble dynamc behavor of equlbra. 5. Resuls for economes wh nfnely lved consumers In hs secon, we sudy he behavor of equlbrum pahs for he model wh nfnely lved consumers. We defne a socal planner s problem for counry and prove ha n equlbrum he allocaon n each counry solves hs planner s problem. In fac, he planner s problem s ha 3
of a small open economy ha aes he sequence of prces of he wo raded goods as exogenously gven. We use properes of he value funcon for hs problem o derve resuls on he evoluon of relave facor abundances along equlbrum pahs. 5.. Counry socal planner s problem In hs secon, we argue ha equlbrum allocaons n counry solve a counry planner s problem n a one-secor growh model wh a me varyng producon funcon. We begn by aggregang consumpon of he wo raded goods. The homohecy assumpon A.5 mples ha ( ) uc (, c) = v gc (, c), where g s srcly ncreasng, concave, and homogeneous of degree one, and v s srcly ncreasng and srcly concave. Defne c= g( c, c) o be an aggregae consumpon good and p( p, p ) o be s un cos funcon p( p, p ) = mn pc + p c s.. g( c, c) (3) c. j We nex aggregae producon of he wo raded goods by defnng he revenue funcon π( p, p, ) = max pφ (, ) + p φ (, ) s.. + (3) +,. j Ths revenue funcon ndcaes, for any gven prces of he wo raded goods, he maxmum ncome ha a counry can oban by allocang capal and labor over he producon of he raded goods. (See, for example, Dx and Norman 98, who refer o hs funcon as he revenue funcon; many oher auhors refer o as he GDP funcon.) Fgure shows how he revenue funcon s consruced. As s seen n he fgure, hs funcon s srcly ncreasng, concave, bu no srcly concave, and connuously dfferenable, bu no wce connuously dfferenable. Usng he frs-order condons for he revenue maxmzaon problem (3), we oban a characerzaon of he relaon beween facor endowmens, facor prces, and producon paerns. j 4
We sae he followng resul, whch s sandard n Hecscher-Ohln heory, because s useful n he res of he paper. Lemma. The opmal capal-labor raos n neror soluons o he revenue maxmzaon problem depend only on relave prces: j j = κ ( p / p ) j, j =, (3) If κ p p κ p p ( / ) ( / ), hen facor prces only depend on goods prces, r( p, p ) = pφ ( κ ( p / p ),) = p φ ( κ ( p / p ),) (33) K K wp (, p) = pφ ( κ ( p/ p),) = pφ ( κ ( p/ p),). (34) L L > ( / ), hen counry produces only good, r ( p, p, ) = pφ (,) < r( p, p), and If κ p p w ( p, p, ) = pφ (,) > w( p, p ). If < κ( p / p), hen counry produces only good, L r ( p, p, ) = p φ (,) > r( p, p ), and w ( p, p, ) = pφ (,) < w( p, p). K L K Gven a sequence of prces p (( p, p ),( p, p ),...) = and an nal endowmen of + + capal, he counry socal planner aes he sequence of prces as gven and solves β = V( ; p ) = max v( c ) s.. p( p, p ) c + x π ( p, p, ) (35) ( + δ ) x c, x gven. Noce ha hs problem s le ha of a planner n a one-secor model, excep ha he analogue of he producon funcon, π ( p, p, ), changes every perod as prces change and consumpon s weghed by he prce ndex p( p, p ). 5
Lemma : In a model wh nfnely lved consumers ha sasfes A. A.6, le he sequence {,,,,, } c c x p p be he equlbrum consumpon, nvesmen, and capal soc n counry and he equlbrum prces for he raded goods. Then, for any equlbrum prces ((, ),(, ),...) =, he counry value funcon V( ; p ) s connuous, srcly p p p p + p+ ncreasng, and srcly concave n for all >. Furhermore, for each =,..., n, he sequence {,, } c x, where c = g( c, c ), solves he counry planner s problem (35) n whch he prces are he equlbrum prces and he nal capal soc s. Lemma says ha he counry value funcon V( ; p ) compleely summarzes he suaon of a counry. We have no mposed condons on φ, φ, f, and u o ensure ha an equlbrum of he world economy exss. I may be ha he economy s so producve ha he represenave consumer n some counry can aan nfne uly, or may be ha he economy s so unproducve ha consumpon n some counry converges so qucly o zero ha he consumer can aan no more han uly mnus nfny. Wha lemma says s ha, f an equlbrum exss, < V( ; p ) < +, no jus for he nal endowmens, =,..., n, bu for all >, and ha V( ; p ) has he characerscs of a dynamc programmng value funcon. The counry socal planner s problem and assocaed value funcon V( ; p ) are even closer o hose of a planner n a one-secor growh model when he model s one-secor aggregaable. In hs case, p( p, p ) = for any possble p and p. 5.. Relave facor abundance In hs secon, we show ha counres manan her relave facor abundance along equlbrum pahs. Noce ha he n counres socal planner s problems one for each counry have he same sequence of prces and dffer only n he nal endowmens of capal,. Therefore, comparng equlbrum allocaons of capal across counres s equvalen o dong comparave sacs wh respec o on he planner s problem (35). 6
Proposon 3. In a model wh nfnely lved consumers ha sasfes assumpons A.- A.6, le { }, { ' }, =,,..., be he equlbrum capal socs for wo counres and '. Assume ha ' >. Then for all. Furhermore, f x >, hen ' > mples ' >. ' + + The proof of proposon 3 apples a monooncy argumen smlar o hose of Mlgrom and Shannon (994) o he counry socal planner s problem. 5.3. Seady saes In hs secon, we analyze he properes of seady saes of he model wh nfnely lved consumers. Defnon 5. A nonrval seady sae s a seady sae n whch aggregae capal s posve,, ha s, ˆ n ˆ n ˆ / = L L > = = >, for some =,..., n. Bajona and Kehoe (6) consruc an example n whch φ (, ) = and n whch one counry has zero capal n he seady sae bu he oher counry has posve capal. Proposon 4: In a model wh nfnely lved consumers ha sasfes A. A.6, here s facor prce equalzaon n any nonrval seady sae. Proposon 5: In a model wh nfnely lved consumers ha sasfes A. A.6, f here exss a nonrval seady sae, here exss a connuum of hem. These seady saes have he same prces and he same aggregae capal-labor rao, ˆ. The seady saes are parameerzed by he dsrbuon of capal per worer across counres, ˆ,..., ˆn. Furhermore, nernaonal rade occurs n every seady sae n whch ˆ ˆ for some =,..., n. Proposon 5 s a sandard resul n dynamc Hecscher-Ohln models where counres only dffer n her nal facor endowmens. Chen (99), Baxer (99) and Bond, Tras and Wang (3) derve smlar resuls n envronmens only slghly dfferen from ours. The nex example llusraes he dependence of he seady sae dsrbuon of capal on he nal dsrbuon. 7
Example. Consder a dscree-me verson of he model suded by Venura (997). There are n counres. We assume ha he producon funcons for he raded goods each use one facor of producon: φ (, ) = (36) φ (, ) =. (37) Ths assumpon ensures ha facor prce equalzaon always holds: r = r = p (38) w = w = p (39) We furher assume ha he model s one-secor aggregaable wh a Cobb-Douglas nvesmenconsumpon funcon: f ( x, x ) = dx x. (4) a a Mang he addonal assumpons ha δ = and ha vc ( ) = log c, we can wre he counry socal planner s problem (35) as max = β log c s.. c + p + p (4) + c,. Snce facor prce equalzaon holds, as long as here are no corner soluons n nvesmen, we can use he negraed approach o solve for equlbrum. To fnd he equlbrum of he negraed economy, we solve he socal planner s problem max = β log c a + s.. c + d (4) c,. 8
The exboo soluon o hs problem, frs obaned by Broc and Mrman (97), s = ( ) /( ) a a a a β = + + + ( β ), ( ) ( ) ( ) ( ) /( ) a a a + a c βa d βa d βad ad ad = =. (43) Usng he frs-order condons, he feasbly condons, and he soluon (43), we oban c ( β ) = a, c = β a (44) x βa x =, = βa (45) p = ad a, p = ( a) d. (46) a To dsaggregae across counres, we sar by comparng he frs-order condons for he counry socal planner s problem (35) wh hose for he negraed economy equlbrum. (I s here ha he assumpon of no corner soluons n nvesmen s mporan.) c c = = =. (47) + βp βr + + + c c Usng (47) and he budge consran n (4), we can wre he demand of he consumer n counry n perod as c s ( ) p p = β s= τ =+ s + τ p τ. (48) Subracng he analogous condon for he negraed economy, we oban c c = ( β ) p ( ). (49) The budge consrans n (4) for counry and for he negraed economy mply ha c c + + + = ( + r δ )( ). (5) We can combne (47), (49), and (5) o oban Seng z = c / = ( βa)/( β a),,,... c = ( ). (5) + + c =, and ( ) z = c / β r = ( βa)/( βa), we oban 9
z z = = = + z z + + + +, (5) whch mples ha / s consan. Leng γ = /, we can solve for c / c o produce ( γ γ β ) c = a+ a a, c = a+ γ a γβa (53) x = γ βa, x = γ β a. (54) oban Comparng levels of ncome per capa, measured n curren prces, across counres, we y p + p γ ad + ( a) d = = = a+ γ a. (55) y p + p ad + ( a) d We now see he srong consequences of proposon 3. In a world of closed counres, we expec every counry o converge o he same seady sae capal-labor rao and level of ncome per capa, ˆ ( ) /( a = βad ) a, ˆ ( ) /( a y d βad ) =, (56) no maer wha s nal endowmen of capal. In a world of counres open o rade, however, dfferences n nal endowmens of capal lead o perssenly dfferen capal socs and ncome levels. In hs example, n fac, dfferences say proporonally fxed. As he world economy converges o s seady sae, each counry converges o a seady sae ha depends on s nal endowmen of capal relave o he world average, γ = /, ˆ ( ) /( a βad ) a = γ, ˆ ( ) ( ) /( a y a γ a d βad ) = +. (57) 5.4. Susaned growh pahs Equlbra n boh one- and wo-secor growh models can exhb susaned growh. (See, for example, Rebelo 99.) Snce our model generalzes hese closed economy models o a world wh rade, susaned growh s also possble here. The nex proposon exends he resuls n he prevous secon o he lmng behavor of equlbra ha exhb susaned growh.
Proposon 6: In a model wh nfnely lved consumers ha sasfes A. A.6, assume ha here exss an equlbrum n whch lm / lm / + = c+ c = γ for < γ < for all and γ > for some. Then γ = γ for all. In hs equlbrum wh susaned growh, facor prces are equalzed n he lm. Furhermore, f here exss a susaned growh pah, here s a connuum of hem, all of whch have he same prces and aggregae capal-labor rao,, bu dffer n he nal allocaon of capal per worer,, and he lmng dsrbuon of capal across counres, lm /. Inernaonal rade occurs n he lm of any equlbrum wh susaned growh n whch lm / for some =,..., n. I s worh nong a lmaon of proposon 6. Whle susaned growh s defned n erms of he nfmum lm of / + and / c + c, he proposon, whch characerzes he lmng behavor of equlbra, requres ha he lms of hese varables exs. On he oher hand, raher han assumng ha hese lms are equal across counres, he proposon proves ha hey are equal. I s also worh nong ha he proposon does no rule ou he possbly ha lm w =. If lm w = for some, however, he proposon proves ha does so for all. Furhermore, s easy o show ha, even f lm w =, w lm. (58) p y + py = Consequenly, even f he wage grows whou bound, does so slowly enough ha, n he lm, he economy behaves le an economy wh no labor. (See Bajona and Kehoe 6.) The nex example shows ha he lmng dsrbuon of capal n an equlbrum wh susaned growh depends on he nal dsrbuon, jus as example does n he case of an equlbrum ha converges o a seady sae. Example. Consder a world economy dencal o ha n example excep ha he producon funcon for consumpon and nvesmen s of he general consan-elascy-of-subsuon form b b ( ) / gc (, c) = f( c, c) = d ac + ( ac ). (59) b
Assume ha b > and β a / b he socal planner s problem d >. To fnd he equlbrum of he negraed economy, we solve max β log c = b b ( ) / s.. c + d ac + ( a) c (6) + c,. The soluon o pah for hs problem exhbs susaned growh. Bajona and Kehoe (6) show ha along hs pah z = c / decreases and ha b The analogue of equaon (5) holds: c β zˆ = lm =. (6) β + z =, (6) z alhough z / z s no equal o as s n example. The lmng dsrbuon of capal s deermned by he equaon ˆ ˆ zˆ ˆ z =. (63) 5.5. Cycles and chaos Equlbra n wo-secor growh models do no need o converge o a seady sae or o a susaned growh pah. Insead, he equlbrum may exhb cycles or complex dynamcs. General condons for he exsence of wo-perod cycles n wo-secor growh models are presened by Benhabb and Nshmura (985) and condons for chaos are presened by Denecere and Pelan (986), Boldrn and Monruccho (986), and Boldrn (989), among ohers. In wha follows, we presen a specfc example ha has complex dynamcs based on he wo-secor closed economy model developed by Boldrn and Denecere (99).
Example 3. Consder a world wh wo counres, each of whch has a measure one of consumers. Consumers have he perod uly funcon uc (, c) = cc, (64) α α where α =.3. The producon funcon of he nvesmen good uses he raded goods n fxed proporons: [ ] f( x, x ) = mn x, x / γ, (65) where γ =.9. The producon for he raded goods s such ha each of he raded goods uses only one facor of producon as n (36) and (37). Furhermore, δ =. Boldrn and Denecere (99) show ha such an economy exhbs sable wo-perod cycles for β [.93,.95] and chaos for β [.99,.]. If =, hen he equlbrum of he wo-counry economy concdes exacly wh he equlbrum of he closed economy. If one counry fnds opmal o ncrease s capal soc, so does he oher counry. Therefore, capal-labor raos n boh counres cycle n he same drecon, mmcng he oscllaons of he negraed equlbrum. Chen (99) maes a smlar argumen for a slghly dfferen model. Noce ha n hs example u s no srcly concave and f s no connuously dfferenable. Gven ha he propery of havng cycles or chaos s srucurally sable, however, would be easy o perurb he uly and nvesmen funcons o consruc examples ha sasfy assumpons A.3 and A.5 and ha have equlbra wh cycles or chaos. The mehodology used n hs example s general and allows he consrucon of a wocounry rade model sarng from any closed economy model wh a consumpon secor and an nvesmen secor, such as he model developed by Uzawa (964). Le g (, ) be he producon funcon for he consumpon good, f (, ) be he producon funcon for he nvesmen good, and ν () c be he uly funcon n he wo-secor closed economy model. We se φ (, ) =, φ (, ) =, and uc (, c) v( gc (, c) ) =. In he cycle n example 3, facor prces are equalzed n every perod. In fac, hs s he only sor of equlbrum cycle ha s possble, a leas f nvesmen s posve., f 3
Proposon 7: In a model wh nfnely lved consumers ha sasfes A. A.6, assume ha here exss an equlbrum s-perod cycle, < s < wh x > for all and all. Then facor prce equalzaon occurs n every perod of he cycle. To prove hs proposon n he appendx, we argue ha, f an equlbrum wh a fne cycle exss, counres have o change relave facor abundance a leas once over he cycle. Ths mples ha hey have o change relave facor abundance an nfne number of mes along he equlbrum pah, whch conradcs proposon 3. As we have seen n lemma and example, f consumers aggregae he wo raded goods o oban uly n he same way ha frms aggregae hese goods o oban nvesmen, hen he equlbrum allocaon of he negraed economy solves a one-secor socal planner s problem. Alhough he negraed economy approach o characerzng equlbra apples only when we can ensure facor prce equalzaon, he assumpon of one-secor aggregaably coupled wh proposon 7 pus srong resrcons on equlbrum dynamcs. Proposon 8: Assume ha a model wh nfnely lved consumers sasfes A. A.6 and s one- secor aggregaable. Also assume ha x > for all and all. Then here canno exs an equlbrum wh cycles. 6. Resuls for economes wh overlappng generaons In hs secon, we derve some general resuls for equlbrum pahs of he overlappng generaons model. In a closed economy seng, overlappng generaons models exhb a rcher varey of possble behavor han do nfnely lved consumer models. In parcular, mulple seady saes and cyclcal behavor can appear n one-secor, closed economy models wh homohec perod uly funcons and C.E.S. producon funcons. Ths rchness n behavor of he closed economy models carres over o Hecscher-Ohln models wh overlappng generaons, and mae such models sgnfcanly dfferen from nfnely lved consumer models. I s also worh nong ha, n closed economy models wh nfnely lved consumers, equlbra are genercally deermnae, whle n closed economy models wh overlappng generaons, here are robus examples wh ndeermnae equlbra. (See Kehoe and Levne 985 and he relaed leraure.) Alhough hese sors of resuls can be expeced o carry over o dynamc Hecscher-Ohln models, we do no pursue hese maers n hs paper. 4
6.. Seady sae analyss In conras wh nfnely lved consumer models, an overlappng generaons model can have a seady sae n whch facor prces do no equalze. If prces do equalze n a seady sae, however, hen all counres behave n exacly he same way, and here s no rade. Proposon 9: In a model wh overlappng generaons ha sasfes assumpons A. A.4 and A.5' A.6', assume ha here s a nonrval seady sae n whch facor prces equalze. Then ˆ = ˆ s n he neror of he cone of dversfcaon and here s no nernaonal rade n hs seady sae. Proposon : In a model wh overlappng generaons ha sasfes assumpons A. A.4 and A.5' A.6', any equlbrum ha converges o a seady sae n whch here s facor prce equalzaon reaches facor prce equalzaon and no rade whn a fne number of perods. In parcular, he equlbrum becomes auarc once all generaons alve have been born under facor prce equalzaon. As n he nfnely lved consumer envronmen, hese resuls can be exended o economes wh equlbra ha converge o susaned growh pahs. Susaned growh, however, can occur only for economes ha are no one-secor aggregaable and under srong condons. See Jones and Manuell 99 and Fsher 99 for deals. Proposon : Suppose ha a model wh overlappng generaons sasfes assumpons A. A.4 and A.5' A.6' and s one-secor aggregaable. Then an equlbrum canno dsplay susaned growh. Ths proposon follows drecly from heorem n Fsher (99). The nuon s ha, because he producon funcons are concave, he rao of wage ncome relave o he capal soc converges o zero as capal goes o nfny. Therefore, wage ncome s no able o purchase an ever growng capal soc. Jones and Manuell (99) presen an example of a wo-secor overlappng generaons economy wh susaned growh and Fsher (99) derves necessary condons for equlbra wh susaned growh o occur. These condons are srong and nvolve boh producon echnologes 5
and uly funcons. I s easy o urn he Jones-Manuell (99) example no a wo-counry Hecscher-Ohln example as n example 3. 6.. General srucure of examples As we have menoned, overlappng generaons economes can have seady saes where facor prces do no equalze across counres. In wha follows, we descrbe a general mehodology for he consrucon of model economes wh such properes sarng from one-secor closed economy models. We hen use our mehodology o derve four dfferen examples ha have seady saes whou facor prce equalzaon. The general srucure of our examples s ha of a model wh wo counres, =, wh C.E.S. producon funcons for he raded goods: for ρ <, ρ, and ρ ρ ( ) / φ (, ) = θ α + ( α ), (66) j j j j j j ρ j = θ j j j j j φ (, ) α α (67) for ρ =. We assume ha α = α, α = α + ε, θ = θ, and θ θ λε = + for > α >, ε >, θ >, and λ. Ths relaonshp beween he parameers allows us o express he capal-labor raos ha deermne he cone of dversfcaon as funcons of α, ε, θ, and λ. Noce ha ε > guaranees ha good s he capal nensve good. The producon funcon for he nvesmen good s Cobb-Douglas: f ( x, x ) = dx x (68) a a where > a >. Capal deprecaes compleely, δ =. There s a measure one of consumers of each generaon n each counry. The represenave consumer n each generaon n counry lves for wo perods, has labor endowmens (, ), and has he uly funcon β ( c ) ( c ) β log ( c + ) ( c + ) log a a a a +. (69) Noce ha, snce he parameer a n he uly funcon s he same as ha n he producon funcon for he nvesmen good, he model s one-secor aggregaable. 6
To fnd he cone of dversfcaon when ρ, we calculae he opmal capal-labor raos n neror soluons o he revenue maxmzaon problem (3). When ρ, he soluon s κ /( ρ ) / ρ /( ρ) ρ/( ρ) /( ρ) ρ/( ρ) α ( α) ( θp / p) ( α) θ ( p / p) = /( ρ) ρ/( ρ) /( ρ) ρ/( ρ) α α θ α ( θp / p) (7) and When ρ =, hese funcons become /( ρ ) α α p p = κ p p α α κ ( / ) ( / ). (7) κ ( p / p ) α α θp / p α α = θ α α /( α α ) (7) and α α κ( p / p) = κ( p / p). (73) α α Togeher wh he endowmens of capal and labor, he parameers ε, λ, d, and a deermne he paern of specalzaon and rade. As α = α + ε and α = α become more smlar asε approaches zero, he cone of dversfcaon narrows, collapsng no a sragh lne whenε =. We se p / p =, α as ε ends o o oban = α, α = α + ε, θ = θ, and θ = θ + λε and use l Hôpal s rule o ae lms θ ( α) ρλ κ() = κ() = θ + αρλ / ρ (74) f ρ and κ = = (75) / () κ() e λ θ f ρ =. By choosng λ appropraely, we can mae hs degenerae cone of dversfcaon pass hrough any pon. 7
We se he parameers d and a so ha p = p =. The frs-order condons for uly maxmzaon and for prof maxmzaon n he nvesmen secor mply ha ac ax p ( ac ) ( ax ) p = =. (76) Feasbly and L = L = mply ha c x c + x y + y c x c x y y = = = + + (77) whch mples ha a = y + y y y y y + + +. (78) If p = p = q=, we can use he frs-order condon for prof maxmzaon no he nvesmen secor wh respec o x o oban a y + y = ad y + y (79) whch mples ha a + + + + + + + + y y y y y y y y d = = a y y y y y y y+ y y+ y + y+ y. (8) We derve each of our examples sarng wh a one-secor, closed-economy, overlappng generaons model ha has mulple seady saes. Le φ (, ) be he producon funcon of hs model. Le θ = θ and α = α be he parameers of hs producon funcon. We hen consruc anoher one-secor, closed economy model ha preserves he seady sae behavor of he orgnal model by slghly perurbng hs producon funcon. Le φ (, ) wh parameers α = α + ε and θ = θ + ελ be he producon funcon of hs perurbed model. The ey s o fnd values of ε and λ so ha here exss a seady sae of he orgnal model wh capal-labor rao and facor prces 8
ˆ rˆ w ˆ and a seady sae of he perurbed verson of he model wh capal-labor rao and (,, ) facor prces and ˆ rˆ w ˆ ha sasfy (,, ) ˆ > ˆ, rˆ < rˆ, and wˆ > wˆ (8) eher ˆ κ () > or ˆ κ () <, or boh. (8) Condon (8) mples ha, n a wo-secor, wo-counry economy wh producon funcons φ (, ) and φ (, ), a leas one counry specalzes n producon f prces are p = p = and counres capal-labor raos are ˆ, =,. Condon (8) mples ha rˆ and w ˆ are conssen wh lemma. We can choose λ and ε so ha condon (8) holds. Specfcally, we choose λ so ha he cone of dversfcaon passes beween he seady sae ˆ of he orgnal one-secor model and he seady sae ˆ of he perurbed one-secor model when ε = and hen ncrease ε. Wheher or no condon (8) holds n he consruced wo-secor model depends on he properes of he orgnal one-secor model, as we wll see n examples n he nex secon. By choosng d and a so ha p = p =, we ensure ha rade s balanced. In any seady sae of a one-secor model, we can add up budge consrans o oban We consruc he wo secors so ha. (83) c ˆ x ˆ c ˆ x ˆ w ˆ r ˆ ˆ w ˆ r ˆ ˆ h h h + = ( + ) = + = + h= h= wˆ + rˆ ˆ = pˆ yˆ + pˆ yˆ. (84) Snce p = p =, we now ha cˆ = d( cˆ ) ( cˆ ) = cˆ + cˆ and analogously for x ˆ. Consequenly, a a cˆ + cˆ + xˆ + xˆ = cˆ + xˆ = yˆ + yˆ. (85) ( yˆ cˆ xˆ ) + ( yˆ cˆ xˆ ) =. (86) 6.3. Seady sae examples Usng he mehodology developed n he prevous secon, we consruc four examples of wo-counry economes ha have seady saes where facor prces are no equal across counres. 9
In he frs wo examples, here s posve fa money n one of he counres bu no he oher. In he oher wo examples, here s no fa money. Example 4. Consder a one-secor, closed economy model where he represenave consumer n generaon has he uly funcon log c + log c + (87) and he labor endowmen s (, ) = (.8,.). The producon funcon s y = φ (, ) = 4.5.75. (88) Ths economy has wo seady saes: a seady sae wh no fa money where b ˆ =, ˆ =.4675, r ˆ =.75, and w ˆ = 3.39 and a seady sae wh posve fa money where b ˆ =.4, ˆ =, r ˆ =, and w ˆ = 3. Noce ha he seady sae whou fa money has a hgher capal-labor rao, a lower renal rae, and a hgher wage han does he seady sae wh fa money, allowng us o consruc an example ha sasfes lemma. We perurb he producon funcon φ o y = φ (, ) = 3.96.3.7. (89) Tha s, we se ε =.5 and λ =.8. The perurbed economy also has wo seady saes. The seady sae whou fa money has b ˆ =, ˆ =.49, r ˆ =.986, and w ˆ = 3.87. Consder now a wo-counry, wo-secor economy where good has producon funcon φ and good has he producon funcon φ. The cone of dversfcaon for he wo-counry economy for pˆ ˆ = p = s deermned by ˆ κ () =.757 and ˆ κ () =.383. Se he seady sae capal-labor rao n counry o ˆ =.49 and n counry o ˆ =. We now have a seady sae of he wo-counry, wo-good economy where counry specalzes n he producon of good, producng y ˆ = 4.4 and y ˆ =, and counry specalzes n he producon of good, producng y ˆ = and y ˆ = 4. Fgure 3 depcs he cone of dversfcaon for hs world economy. To ensure ha p = p =, se d =.9977 and a =.539. Facor prces do no equalze n hs seady sae: r ˆ =.986 and ˆ 3.87 w =, bu r ˆ = and w ˆ = 3. 3
The calculaon of he oher varables s sraghforward. Snce c ˆ =.47 and c ˆ =.9364 n he seady sae of he perurbed one-secor economy, for example, and ( yˆ + yˆ ) /( yˆ + yˆ + yˆ + yˆ ) = 4.4/8.4 =.539, we se c ˆ = (.539).47 =.546, c ˆ = (.539).47 =.4965, c ˆ =.44, and c ˆ =.9. Smlarly, snce xˆ.49 = ˆ =, we se x ˆ =.7449 and x ˆ =.677. Noce ha rade s balanced: Counry expors yˆ cˆ cˆ xˆ = 4.4.546.44.7449 =.955 of good and mpors cˆ + cˆ + xˆ yˆ =.4965 +.9 +.677 =.955 of good. Example 5. Consder a modfcaon of example 4 where θ = 3.9. Tha s, we se λ =.6. The cone of dversfcaon s now deermned by ˆ κ () =.379 and ˆ κ () =.6944, and ˆ =.457 s n s neror, as depced n fgure 4. In hs seady sae, counry dversfes n producon, producng y ˆ =.6 and y ˆ =.7736, and counry specalzes n he producon of good, producng y ˆ = and y ˆ = 4. To ensure ha pˆ ˆ = p =, we se d =.633 and a =.93. Facor prces do no equalze: and w ˆ = 3. r ˆ =.83 and ˆ 3.43 w =, bu, once agan, r ˆ = Example 6. Consder now a model wh he same uly funcon and labor endowmens as n examples 4 and 5, bu where he producon funcon s 3 3 ( ) /3 y = φ (, ) = 4.5 +.75 (9) The one-secor closed economy model has hree seady saes. In one seady sae, b ˆ =.4 and ˆ =. In he oher wo, b ˆ = and eher ˆ =.5675 or ˆ =.3355. When ˆ =.5675, r ˆ =.75 and w ˆ = 3.39, and, when ˆ =.3355, r ˆ =.75 and w ˆ = 3.39, allowng us o consruc an example ha sasfes lemma. The producon funcon for he perurbed economy s 3 3 ( ) /3 y = φ (, ) = 4.3 +.7. (9) Tha s, we se ε =.5 and λ =. The perurbed economy has also hree seady saes, one seady sae wh posve fa money and wo seady saes wh no fa money. 3
The cone of dversfcaon s now deermned by ˆ κ () =.9687 and ˆ κ () =.35. Se he capal-labor rao n counry o ˆ.398 =, whch s he perurbaon of he seady sae of he one-secor model where ˆ =.3355, and se he capal-labor rao n counry o ˆ =.5675. Counry specalzes n good, producng y ˆ = 4.884 and y ˆ =, and counry specalzes n he producon of good, producng y ˆ = and ˆ 3.47 y =. To ensure ha pˆ = pˆ =, we se d =.9749 and a =.5793. Facor prces do no equalze: r ˆ =.437 and ˆ 3.699 w =, bu r ˆ = 3.545 and w ˆ =.9. The essenal sep n he consrucon of hs example s o sar wh a one-secor closed economy model ha has mulple seady saes whou fa money. I s mpossble o do hs wh a model wh logarhmc uly and Cobb-Douglas producon. Consder such a model wh producon funcon (, ) α φ = θ α, uly funcon βlog c + β log c+, and labor endowmens (, ). The unque seady sae whou fa money s he soluon o a a β a ( αθ ) = w( ) c ( w( ), r( )) = ( αθ ) ( αθ ) + a β β αθ. (9) + I s ˆ α β( α) αθ = β( α) + ( β+ β) α. (93) Example 7. Ths example uses a dfferen perurbaon of he one-secor closed economy model of example 6. In parcular, hs example explos he fac ha a one of he seady saes whou fa money he one where ˆ =.5675 he seady sae capal soc and he wage ncrease and he renal rae decreases as we ncrease α o α + ε, hus allowng us o sasfy lemma. I s worh nong ha we canno do hs a he oher seady sae whou fa money, where ˆ =.3355. Nor can we do hs n a model wh logarhmc uly and Cobb-Douglas producon where ( ) β ( ) ( ) + + +. (94) ˆ β α β β α β β α rˆ = αθ α = = + β( α) β β( α) Increasng α o α + ε, he renal rae ncreases, rˆ > rˆ, whch mples ha lemma s volaed. 3